Analysis on ∂-Manifolds of Bounded Geometry, Hodge-De Rham Isomorphism and L^{2}-Index Theorem

Dissertation zur Erlangung des Grades

”Doktor der Naturwissenschaften”

am Fachbereich Mathematik der Johannes Gutenberg Universität Mainz

Thomas Schick geb. 22. Mai 1969 in Alzey

Mainz, Juli 1996

Berichterstatter: Prof. Dr. Wolfgang Lück Prof. Dr. Matthias Kreck Dr. Elmar Schrohe

Termin der mündliche Prüfung: 27.9.1996

Contents

1 Introduction 1 Organisation of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Notation 10

I Bounded geometry and Hodge-De Rham theory 11

3 Bounded geometry 11 Bounded geometry: Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . 11 Technical properties of manifolds of bounded geometry . . . . . . . . . . . . . . 17 Sobolev theory for manifolds of bounded geometry . . . . . . . . . . . . . . . . 20 Normalisation to a product metric . . . . . . . . . . . . . . . . . . . . . . . . . 23 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Ellipticity and bounded geometry 30

5 The Hodge decomposition 42 Hodge decomposition for complete ∂-manifolds . . . . . . . . . . . . . . . . . . . 42 Hodge decomposition for ∂-manifolds of bounded geometry . . . . . . . . . . . . 45

6 Hilbert-de Rham complexes and L^{2}-cohomology 49 L^{2}-de Rham complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Sobolev-de Rham complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 L^{2}-de Rham cohomology and coverings . . . . . . . . . . . . . . . . . . . . . . . 57

7 L^{2}-cohomology and curvature 57 Curvature, coverings and symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 61

8 L^{2}-Hodge-de Rham theorem 68 L^{2}-de Rham theorem if ∂M = ∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 L^{2}-de Rham theorem for product metrics . . . . . . . . . . . . . . . . . . . . . . 70 De Rham isomorphism: general case . . . . . . . . . . . . . . . . . . . . . . . . 72

II L^{2}-index theorem for ∂-manifolds 75

9 Traces for A-module morphisms 75

10 L^{2}-Rellich lemma 84

11 Boutet de Monvel calculus 86

12 Proof of the L^{2}-index theorem 90 13 Explicit expressions for the analytical index 92

Index and adjoint boundary value problems . . . . . . . . . . . . . . . . . . . . 92 A formula for the Γ-dimension of ker P̃ . . . . . . . . . . . . . . . . . . . . . . . 94 Representations of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Bounded geometry: Proofs 96 Metric tensor in normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 97 Changes of normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

1 Introduction

The broad theme of this thesis is geometry, topology and analysis on non-compact manifolds with boundary. All the results will hold also if the boundary is empty. Of course, we have to focus on special aspects, and also, we have to restrict the class of objects. I started to work with coverings of compact manifolds, and ended with manifolds of bounded geometry (definition 3.1). These are (non-compact) Riemannian manifolds with bounds on curvature and second fundamental form and their derivatives. Also, they have positive injectivity radius and a collar of positive width.

These manifolds have nice analytical properties. For example, Sobolev spaces and Banach spaces of k-times differentiable bounded functions can be defined in a natural way. We develop this (uniform) Sobolev theory. Most of the results from the Sobolev theory of Euclidian space hold here, too. It is important to note that we have Sobolev embedding theorems, i.e. sections in H^{k+s} are k-times continuously differentiable if s > dim M/2. Differential operators (of order μ) extend to bounded operators H^{s} → H^{s−μ} in the usual fashion, and a bounded restriction map H^{s}(M) → H^{s−1/2}(∂M) (s > 1/2) exists. Also, there is an appropriate version of ellipticity for differential boundary value problems. One of the main results is the corresponding regularity theorem 4.15 which roughly says:

1.1. Theorem. If P = (A, p⃗) is a uniformly elliptic boundary value problem of order μ and u ∈ H^{t}(E) with P(u) ∈ H^{s−μ} then already u ∈ H^{s}. Moreover, we find C_{t,P} > 0 independent of u so that the following a priori estimate holds:

|u|_{H^{s}}^{2} ≤ C_{t,P} · (|Pu|_{H^{s−μ}}^{2} + |u|_{H^{t}}^{2})

Of course, t must be sufficiently large so that the boundary values p⃗(u) are defined.

This means: if the right hand side of the equation Pu = f admits derivatives up to order s − μ, then u admits μ orders of differentiation more. In particular, the kernel of P consists of smooth forms (employ the Sobolev embedding theorem 3.25).

Extending ideas of Lions and Magenes [40], we develop the concept of formal adjointness for boundary value problems. Here we get an “adjoint regularity theorem” and, as another key result, the following important theorem about adjointness on the Hilbert space L^{2}:

1.2. Theorem. (compare theorem 4.25) Let (A, p⃗) be uniformly elliptic of order μ with uniformly elliptic formal adjoint (B, q⃗). Let D_{A} := {f ∈ H^{μ}(E); p⃗f = 0}. Consider A as an unbounded operator on L^{2}(E) with domain D_{A}. Then: the L^{2}-adjoint of A is

A^{t} = B with domain D_{B} = {g ∈ H^{μ}(F); q⃗g = 0}.

For us, the most important example is the Dirichlet/Neumann problem for the Laplacian on differential forms (definition 5.2).

1.3. Theorem. (compare proposition 5.14) On an oriented manifold of bounded geometry, the Dirichlet/Neumann problem for the Laplace operator Δ is a uniformly elliptic boundary value problem.

Moreover, this boundary value problem is formally self adjoint and hence by the above theorem yields a self adjoint operator on L^{2}. Self adjointness is a very important functional analytical property, because the spectral theorem and the corresponding calculus apply only to self adjoint operators (formal self adjointness is not sufficient here). In general, self adjointness is not easy to obtain, so the result is an important achievement. Using self adjointness, we can extend the Hodge decomposition of L^{2}, which holds for arbitrary complete manifolds (5.10), to a corresponding decomposition of the higher Sobolev spaces on manifolds of bounded geometry. This is important, because it is impossible to define the de Rham integration map on L^{2}. But for sufficiently high Sobolev spaces, restriction of forms to simplexes of positive codimension is no problem in view of the restriction map or the Sobolev embedding theorem 3.25.

1.4. Theorem. (compare 5.19) Let M be a manifold of bounded geometry, ∂M = M_{1} ⨿ M_{2}. If H^{p}(M, M_{1}) is the space of square integrable harmonic forms which fulfill relative and absolute boundary conditions on M_{1} and M_{2} respectively (definition 5.7), then

H^{2k}(Λ^{p}(T^{∗}M), bd(2k)) = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{∞}^{p−1}} ⊕ \overline{δΩ_{∞}^{p+1}}

The space which we decompose consists of forms which fulfill Dirichlet/Neumann boundary conditions up to order 2k. The forms in the domains of d and δ fulfill infinitely many boundary conditions. The closures are taken in the topology of the Sobolev space H^{2k}.

The L^{2}-splitting for complete manifolds is probably well known, but I did not find a suitable reference in the literature, so I included the result with proof here.

In the following part of the paper we apply these results to geometry and topology. We introduce the L^{2}-de Rham complex and L^{2}-de Rham cohomology for complete Riemannian manifolds with boundary. This cohomology fulfills Poincare duality and is a vb-homotopy invariant, where vb-maps are Lipschitz maps which are ”volume bounded” (explicitly: vol(f^{−1}(B)) ≤ C vol(B)). Using the Hodge decomposition, we show that this cohomology is naturally isomorphic to the L^{2}-harmonic forms. This makes it possible to give Bochner type vanishing results for L^{2}-de Rham cohomology (explained below).

A particularly interesting result is the L^{2}-Hodge-de Rham theorem for oriented ∂- manifolds. It says that the square integrable harmonic forms are isomorphic to simplicially defined L^{2}-cohomology. For this end, let K be a smooth triangulation of M. This triangulation has to be sufficiently regular (compare Dodziuk [26, 2.3]). We have to guarantee that the triangulation is neither too sparse nor to fine in different regions of our non-compact manifold (else the sets of L^{2}-cochains will not be comparable). Additionally, we put some weak curvature condition on the triangulating maps (this corresponds to the curvature assumptions for the metric):

1.5. Definition. A smooth triangulation K is called g-bounded (g the metric of M), if

(1) a lower bound v exists for the volume of the m-simplexes of K

(2) an upper bound D exists for the diameter of the m-simplexes of K

(3) there is c > 0 so that |dφ_{p}| ≤ c ∀p, if φ_{p} is the barycentric coordinate function for the vertex p of K (differentiable outside a subset of codimension 1)

Attie [9, 1.14] constructs similar triangulations for every manifold of bounded geometry without boundary. His methods also work on manifolds with boundary.

1.6. Theorem. Suppose (M, g) is an oriented manifold of bounded geometry with boundary ∂M = M_{1} ⨿ M_{2}. Either M_{1} or M_{2} or both may be empty. Let (K, L) (L = (K_{1}, K_{2})) be a g-bounded triangulation of (M, ∂M).

Let H^{∗}(M, M_{1}) be the space of smooth harmonic square integrable forms on M which fulfill absolute boundary conditions on M_{2} and relative boundary conditions on M_{1} (defined in 5.7). Then, the de Rham map (integration of forms over simplexes)

A : H^{∗}(M, M_{1}) → H_{(2)}^{∗}(K, K_{1})

yields an isomorphism between this space and the relative L^{2}-cohomology H_{(2)}^{∗}(K, K_{1}) of the triangulation K. This is true with real as well as complex coefficients.

1.7. Corollary. Let N be a compact Riemannian ∂-manifold with smooth triangulation. Then theorem 1.6 applies to any orientable normal covering M↓N with lifted metric and lifted triangulation K. Let Γ = π_{1}(N)/π_{1}(M) be the covering group. In this case, the simplicial L^{2}-cohomology H_{(2)}^{∗}(K, K_{1}) equals the relative L^{2}-cohomology H_{(2)}^{∗}(N, N_{1}; l^{2}(Γ)) of the finite CW-complex N with values in the N (Γ)-module l^{2}(Γ). Because, obviously, A is Γ-equivariant we conclude:

A : H^{∗}(M, M_{1}) → H_{(2)}^{∗}(N, N_{1}, l^{2}(Γ))

is an isomorphism of Hilbert-N (Γ)-modules.

Lott/Lück [41, section 5] prove that in the covering situation, L^{2}-de Rham cohomology and simplicially defined cohomology are Γ-isomorphic. They use different methods and do not show that the de Rham map (integration of forms over simplices) induces the isomorphism. To have a concrete isomorphism is important for instance for the definition of analytic torsion.

1.8. Remark. N (Γ) is the group von Neumann algebra of Γ. For more details about L^{2}- cohomology, which is a homotopy invariant of finite CW-complexes, see f.i. Lück [43]. One important feature is the existence of a real valued normalised dimension (due to the group operation), which allows us to define real valued L^{2}-Betti numbers b_{(2)}^{i}(M, M_{1}) := dim_{Γ} H^{i}(M, M_{1}). In the second part of this thesis, we will study Hilbert modules and such dimension functions in great detail. Now, we will use only the property b_{(2)}^{i} = 0 ⟺ H^{i} = {0}.

In the following example we compute some of the L^{2}-Betti numbers (this was originally performed by Donnelly/Xavier [28] for manifolds without boundary):

1.9. Theorem. (compare 6.14) Let M^{m} be a compact manifold with sectional curvature pinched between −1 and −1 + ϵ (0 ≤ ϵ < 1) and with convex boundary (i.e. the second fundamental form of ∂M is ≥ 0). Suppose π_{1}(M) is infinite. Then there are no absolute L^{2}-harmonic p-forms on the universal covering —i.e. b_{(2)}^{p}(M̃) = 0— if p < \sqrt{1 − ϵ}(m − 1)/2 or if p = m.

We show by an example (6.15) that the curvature condition for the boundary is necessary.

Another aspect of our work is the relation between topology and geometry. Note that, on a non-compact manifold, very different metrics may exist and the topology does not restrict the quality of the metrics very much. Therefore, we will concentrate on topological obstructions for Riemannian metrics in a given bilipschitz class, especially since our manifolds are by definition equipped with a (comparison) metric. This is proposed by Roe in [57]. We get the following Bochner-type vanishing theorem 7.7 for L^{2}-de Rham cohomology.

1.10. Theorem. If −R_{p}^{W} ≥ 0 and −S_{p} ≥ 0, then the p-th relative L^{2}-de Rham cohomology is zero. If −S_{m−p} ≥ 0, the same holds for the p-th absolute L^{2}-cohomology. Here R^{W} is the Weitzenböck operator, a pointwise linear operator which is defined in terms of the curvature operator. −R_{p}^{W} ≥ 0 is a certain positivity condition on the metric, for example −R_{1}^{W} = Ric and −R_{p}^{W} ≥ 0 if the curvature operator fulfills R ≥ 0. S is a pointwise linear operator on the boundary which is determined by the second fundamental form.

−S_{p} ≥ 0 is equivalent to certain convexity conditions of the boundary, f.i. −S_{1} ≥ 0 iff the boundary is convex in the mean. If the second fundamental form operator l ≥ 0 then −S_{p} ≥ 0 ∀p.

The theorem can answer the question of (non)-existence of a metric with positive Ricci curvature so that the boundary is convex in the mean.

With a slight (but necessary) additional condition, it is possible to weaken the positivity/convexity condition considerably, and in a surprising way, compared with the classical result.

1.11. Theorem. Suppose a metric g is bilipschitz to another Riemannian metrics which has an infinite discrete subgroup of the isometry group (for example to any infinite normal covering of a Riemannian manifold). Then the above Bochner type vanishing results hold even if the curvature assumptions (for the manifold and the boundary) are fulfilled only outside a compact set.

The surprising feature is the fact that the curvature can be arbitrary on the exceptional compact set. Example 7.15 shows that the theorem is not true if the infinite isometry group is missing.

A corollary is the following generalization of results of Roe [57]:

1.12. Theorem. (compare theorem 7.13) Suppose M is a compact manifold with boundary. Let M̄ be a normal covering of M. If the first relative L^{2}-Betti number b_{(2)}^{1}(M, ∂M) ≠ 0, then no metric g on M̄ exists so that: (M̄, g) has bounded geometry and lies in the bilipschitz class of the lifted metric, g has non-negative Ricci curvature outside a compact set and the trace of the second fundamental form is non-negative outside a compact subset.

1.13. Corollary. If M has dimension 2, or M has dimension 4 and is closed, and if χ(M) < 0, then b_{(2)}^{1}(M) > 0 and the statement of the theorem holds.

Especially, these results give an affirmative answer to a question [57, 6.3] of Roe (he could prove the theorem only for amenable normal coverings and manifolds without boundary). However, we prove that a manifold without boundary with Ric ≥ 0 outside a compact set has polynomial growth. This property is a quasi-isometry invariant, so that the covering group has polynomial growth, too. Gromov shows that the group must be not only amenable but even virtually nilpotent. Especially, Roe’s theory applies. Polynomial growth does not follow if ∂M ≠ ∅.

In the second part of the thesis, we concentrate on normal coverings of compact ∂- manifolds. We study elliptic differential boundary value problems on these spaces. Let M be a compact Riemannian manifold with boundary ∂M. Suppose E, F↓M and X, Y ↓∂M are Riemannian vector bundles. Let A : C^{∞}(E) → C^{∞}(F) be a differential operator and T : C^{∞}(E) → C^{∞}(Y) a differential boundary operator so that the pair P := (A, T) is elliptic. Define

ker P := {f ∈ L^{2}(E); f ∈ C^{∞}, Af = 0 = T f}  and
coker P := {(F, f) ∈ L^{2}(F) ⊕ L^{2}(Y); (F, Aφ)_{L^{2}(F)} + (f, T φ)_{L^{2}(Y)} = 0 ∀φ ∈ C_{0}^{∞}(E)}.

The classical theory of elliptic boundary value problems states that the dimensions of kernel and cokernel are finite and that ind(P) := dim ker P − dim coker P is an interesting invariant, which does not change under small perturbations of the operator. Knowledge of the index is important for instance if we explicitely have to solve boundary value problems: if the index is zero, the Fredholm alternative holds. Moreover, the index theorem (stated below) provides deep connections between topological, geometrical and analytical properties of the manifold.

Suppose M̃ ↓M is a normal covering of M with covering group Γ := π_{1}(M)/π_{1}(M̃), acting on M̃ by deck transformations. Pull the bundles back to M̃ and lift the operators and metrics. (Note that this is possible, since we deal with local differential operators). As a rule, corresponding objects on M̃ have the same notation decorated with an additional tilde. Note that Γ operates on the bundles, their sections and that P̃ = (Ã, T̃) is Γequivariant. Define the kernel and cokernel of P̃ literally in the same way as for P. They are in general infinite dimensional. But ker(P̃) and coker(P̃) are Hilbert modules over the group von Neumann algebra N (Γ). For details concerning von Neumann algebras and Hilbert modules compare [23, 41, 43, 44]. For these Hilbert modules, a normalised dimension dim_{Γ} is defined. It has the following properties:

(1) dim_{Γ} ∈ [0, ∞]

(2) dim_{Γ}(X) = 0 ⟺ X = {0}

(3) dim_{Γ}(X ⊕ Y) = dim_{Γ} X + dim_{Γ} Y

(4) dim_{Γ}(l^{2}(Γ)) = 1

(5) |Γ| < ∞ ⟹ dim_{Γ} = \frac{1}{|Γ|} dim_{ℂ}

We give an explicit formula for dim_{Γ}(ker P̃) in proposition 13.7.

The following is the main result of the second part:

1.14. Theorem. In the situation above,

dim_{Γ} ker(P̃) < ∞, dim_{Γ} coker(P̃) < ∞

ind_{Γ}(P̃) := dim_{Γ} ker(P̃) − dim_{Γ} coker(P̃) = ind(P)

Especially ind_{Γ}(P̃), which is the difference of two reals, is integral.

The theorem is particularly interesting, because for ind(P) on M a well known purely topological expression exists. This is a version of the classical index theorem of Atiyah, compare Atiyah/Bott [2, theorem 2] or Atiyah’s [48, Appendix I]. The index theorem has the usual form: every elliptic boundary value problem (A, p) defines a K-theoretic symbol class [σ(A, p)] ∈ K(B(M), S(M) ∪ B(M)|_{∂M}) (here B(M) and S(M) are the disc bundle and sphere bundle of T M, respectively.) Unfortunately, the construction of this symbol class is very complicated. In the usual way we can assign to this symbol class the topological index. For differential boundary value problems this can be computed cohomologically as (compare [2])

ind_{t}(A, p) = ∫_{S(M)} ch(A)π^{∗}T (M) + ∫_{B(M)|_{∂M}} ch(A, p)π^{∗}T (M)

Here π : T M → M is the projection, T (M) is a Todd class of M and ch denotes a Chern character.

Rempel/Schulze [53, section 3.2] give an extension of these results to arbitrary pseudodifferential boundary value problems. However, the main problem with these formulae is that we find few explicit examples. Note that also for closed manifolds the value of the index theorem lies not in the formula for the index of arbitrary operators but much more in the specific examples such as (twisted) Dirac operators, the Laplacian and the signature operator.

1.15. Corollary. The index of elliptic differential boundary value problems is multiplicative under finite coverings.

Proof. This is immediate from the multiplicativity of dim_{Γ} [41, 3.9] and from the fact that every finite covering is finitely covered by a normal covering: let M̃ be an d-sheeted normal covering of M, then

ind(Ã) = dim_{ℂ}(ker Ã) − dim_{ℂ}(coker Ã)

= |Γ|(dim_{Γ}(ker Ã) − dim_{Γ}(coker Ã)) = |Γ| ind_{Γ} Ã = |Γ| ind A (1.16)

In theorem 1.14, we can replace coker(P̃) by the kernel of an adjoint boundary value problem Q̃ in virtue of the following

1.17. Proposition. (compare theorem 13.1) Let Q be an adjoint boundary value problem to P. Then coker(P̃) is isomorphic (as Hilbert N (Γ)-module) to ker(Q̃). Especially, dim_{Γ}(coker P̃) = dim_{Γ}(ker Q̃).

This is important because often it is much easier to deal with the kernel of an adjoint than with some cokernel, f.i. if we want to compute the Euler characteristic of M: it is well known that

χ(M) = ind((d + δ, b_{abs}) : Ω^{ev}(M) → Ω^{odd}(M) ⊕ Ω^{∗}(∂M))

Here d is the differential of forms, δ its adjoint, and b_{abs}(ω) = i^{∗}(∗ω) with i : ∂M ↪ M the inclusion. Therefore, with our index theorem 1.14, χ(M) can be computed on the universal covering of M (with induced metric).

χ(M) = χ_{Γ}(M̃) := ind_{Γ}(d̃+ δ̃, b̃_{abs})

For details, compare theorem 13.5. The advantage is that a single computation can yield results for many compact manifolds. Dodziuk [25] and Donnelly/Xavier [28] have computed the sign of the Euler characteristic of closed negatively curved manifolds in this way, using Atiyah’s L^{2}-index theorem for manifolds without boundary.

Another possible application is the following: If we know that ind_{Γ}(P̃) > 0 then ker(P̃) is neccesarily non-trivial. In this way the purely topological computation of ind(P) on the compact manifold M can yield interesting infinite dimensional Γ-representations. Even better, in section 13 we show how one can construct representations of semisimple Lie groups in this way, using ideas of Atiyah andAtiyah/Schmid [1, 7].

Our index theorem is the generalisation of Atiyah’s L^{2}-index theorem [1] to manifolds with boundary and differential boundary value problems. The proof is along the lines of Atiyah’s proof. In order to deal with boundary value problems, we must replace the calculus of pseudodifferential operators by the Boutet de Monvel calculus. We also try to clarify the exposition. For this reason, section 9 studies a special aspect of the general theory of von Neumann algebras A: we introduce traces for endomorphisms of arbitrary Hilbert A-modules and study their general properties. We also use the theory of Sobolev spaces to simplify the work with regularizing operators and especially with their traces. The main result in this context is the following:

1.18. Theorem. (compare theorem 10.5) If r > dim M/2, the inclusion of Sobolev spaces

H^{s+r}(M̃) ↪ H^{s}(M̃)

is a Γ-trace class operator (i.e. a trace class operator in the sense of the group von Neumann algebra N (Γ) with its standard trace).

This theorem should be valuable also in other contexts. For example, the expositions of Roe [55, chapter 13] and Ramachandran [50] could be considerably simpler.

The general idea for the proof of the index theorem is the following: to P construct an inverse Q (modulo smoothing operators) in the BdM calculus which has the special property that it can be lifted to M̃, i.e. PQ = 1 − S_{1}, QP = 1 − S_{0} and P̃Q̃ = 1 − S̃_{1}, Q̃P̃ = 1 − S̃_{0}. Then the following two results prove the theorem:

• ind_{Γ}(P̃) = Sp_{Γ} S̃_{0}−Sp_{Γ} S̃_{1} (and the corresponding formula on the base with Γ = {1})

• For lifts of smoothing operators, we have Sp_{Γ} S̃ = Sp S.

Note that our index theorem does not generalize the Atiyah-Patodi-Singer index theorem [4]. Atiyah, Patodi, Singer deal with a specific non-local boundary condition for Dirac type operators (which should have a product structure near the boundary). The index of these boundary value problems is, in general, not multiplicative under finite coverings. There is also an L^{2}-version of this type of index theorem, proved by Ramachandran [50]. This theorem is disjoint with ours: Ramachandran deals only with Dirac type operators with a product structure near the boundary and a specific fixed boundary condition. But this boundary condition is —contrary to our theorem— non-local. Our result is valid for arbitrary elliptic differential boundary value problems, and without constraints near the boundary. However, we only deal with local boundary conditions. Also, the method of proof is completely different.

The signature is the index of a non-local boundary value problem. It can not be the index of a differential boundary value problem, because by corollary 1.15 it would then be multiplicative under coverings — and this is false (compare [4, 5] for an explicit example).

Organisation of the paper

We start in section 2 with some notational conventions which will be used throughout the text.

In section 3 we introduce the concept of a ∂-manifold of bounded geometry and derive the main (analytical) properties. At the heart of the theory lies the existence of nice charts (so called normal charts) and subordinate partitions of unity. With the aid of these, one can study analytical properties locally. In this way, Sobolev spaces are defined and their properties are derived. Partly, technical proofs for that are contained in appendix A. Then, we construct to any metric a metric which is a product near the boundary and which is analytically very close to the given metric. This will be used in the proof of the L^{2}-de Rham theorem. Next, we introduce and study differential boundary value problems in a unified way — particularly well suited for manifolds and bundles.

Section 4 deals with elliptic boundary value problems. We define ”uniform ellipticity” as the right condition for manifolds of bounded geometry. Then we prove the elliptic regularity results and a priori estimates for uniformly elliptic boundary value problems and give the self adjointness condition.

In section 5 we prove the Hodge decomposition results. For the decomposition of higher Sobolev spaces (with higher boundary conditions) on manifolds of bounded geometry, we use the ellipticity theory developed in section 4. To apply the latter, we derive the (technical) fact, that the Laplacian with Dirichlet/Neumann boundary conditions is a uniformly elliptic boundary value problem.

In section 6 we define L^{2}-de Rham cohomology and derive its principal properties. F.i. we show that Poincare duality holds and that L^{2}-de Rham cohomology is an vb-homotopy invariant. We compute L^{2}-de Rham cohomology for certain negatively curved spaces with boundary, using methods of Donnelly/Xavier.

Section 7 examines the relations between L^{2}-de Rham cohomology and curvature. This is based on the Hodge isomorphism and the Weizenböck formula Δ = ∇^{∗}∇ − R^{W} with its integrated consequence (for ∂-manifolds) |∇ω|_{L^{2}}^{2} = |dω|+|δω|+(R^{W} ω, ω)+∫_{∂M}(Sω, ω)_{x} dx.

We derive algebraic properties of the operators S and R^{W} in these formulae, especially sufficient conditions for negativity. With this at hand, it is an easy task to get the vanishing results for square integrable harmonic forms. Next, we concentrate on coverings of compact manifolds. We prove the vanishing results for metrics which fulfill the curvature assumptions only outside a compact set. We show that this is not true without the regularity assumption stated above. Additionally, we prove polynomial growth if Ric ≥ 0 outside a compact set and ∂M = ∅. This implies that Roe’s methods are sufficient to answer Roe’s question in this case.

In section 8 we prove the L^{2}-de Rham theorem. The strategy is the following: first prove the theorem if ∂M = ∅ (this was done by Dodziuk [26]). Then use this result and a standard doubling-trick to prove the result if the metric is a product near the boundary. To achieve this, we repeat some facts from representation theory of finite groups. The last step consists in the comparison of an arbitrary metric with a suitable product metric near the boundary (constructed in section 3). Important is, that the isomorphism between the harmonic forms with respect to the two different metrics (which is not to hard to construct) commutes with the de Rham integration map. This requires the Sobolev theory we introduced.

The next sections are preparations for the proof of the L^{2}-index theorem: In section 9 we construct a regularized trace for A-module endomorphisms on an arbitrary Hilbert Amodule V. Here A is a finite von Neumann algebra with fixed trace φ, and the constructed regularised trace Sp_{φ} is based on φ, but on nothing more. If V = l^{2}(A) then Sp_{φ} = φ. If A = ℂ then Sp_{φ} is the ordinary trace on the Hilbert space V. Sp_{φ} has all the expected properties of a trace, esp. the Sp_{φ}-trace class operators form an ideal and Sp_{φ}(fg) = Sp_{φ}(gf).

To gain greater flexibility, we define the property to be of Sp_{φ}-trace class even for A-module morphisms f : V → W. They form an (extended) ideal and prove very useful in this way, although it is impossible to define Sp_{φ}(f) as a number if V ≠ W. In section 10 we prove the Rellich type theorem 1.18. In section 11 we repeat the basics of the Boutet de Monvel calculus. In view of our application we compute the trace (and Γ-trace) of smoothing operators and show one of the key results: tr_{Γ} of a lift equals the trace on the base.

In section 12 we complete the proof of the L^{2}-index theorem 1.14, using the comparison theorem 9.16, originally proved by Atiyah [1]. The last section contains some augmentations: we show how to replace the cokernel in the definition of the index by the kernel of an adjoint boundary value problem and apply this to computations of Euler characteristics. Moreover, we give the explicit formula for the Γ-dimension of the kernel of elliptic boundary value problems. At last, we apply the L^{2}-index theorem to the construction of geometric representations of Lie groups.

Acknowledgements

It is a pleasure for me to thank my advisor Prof. Dr. Wolfgang Lück for his support during the work on my thesis, for his confidence in my mathematical abilities and the freedom he let me to follow my own mathematical paths. Further, I thank Eckehard Hess for proof reading parts of the manuscript and for hundrets of fruitful discussions about any mathematical and non-mathematical subject. Last but not least I thank Gudrun Ewertz and Ainsley McNab for checking the English of this text. All the remaining mistakes are my fault, due to changes after they had done their job.

2 Notation

2.1. Definition. We will very often use inequalities to show equivalences of norms. To abbreviate: if a, b, c ∈ ℝ with c > 0 set

a ≤^{c} b : ⟺ a ≤ c · b.

If c is not explicitly specified, a suitable positive constant has to be chosen. In a longer chain of inequalities, the same symbol (i.g. c) may be used for different constants. Note that the relation is transitive in the following sense:

a ≤^{c} b and b ≤^{c} d ⟹ a ≤^{c} d.

2.2. Lemma. We will very often use the elementary inequality

(a + b)^{2} ≤ 2a^{2} + 2b^{2} a, b ∈ ℝ

which is true because it is equivalent to 0 ≤ (a − b)^{2}. For the same reason the following holds

\sqrt{a} + \sqrt{b} ≤ \sqrt{2}\sqrt{a + b} a, b ≥ 0

2.3. Definition. We will come up with equations where the sign of some of the terms does not matter (in most of the cases because the value is zero). Since the formally correct sign is sometimes quite complicated (e.g. depending on degrees of forms) we simply use “±” in these instances.

2.4. Definition. M will always be a smooth manifold of dimension m with boundary ∂M and E, F ↓M, X, Y ↓∂M are vector bundles. Their pullbacks to a covering M̃ of M are denoted Ẽ, …. If we talk of the Hodge-∗-operator we require that the corresponding manifold is oriented. We allow that the boundary of a ∂-manifold, i.e. a manifold with boundary, is empty.

2.5. Notation. If M is a manifold, we write M^{∘} = M − ∂M for the interior of M.

2.6. Definition. Throughout the paper we use the notation of multi-indices (normally denoted by small Greek letters α, β, …). A multi-index α = (α_{1}, …, α_{m}) is a tupel of non-negative integers, |α| := max{α_{1}, …, α_{m}} and we use the differentiation operator D^{α} := \frac{∂^{α_{1}}}{∂x_{1}^{α_{1}}} … \frac{∂^{αm}}{∂x_{m}^{αm}}.

2.7. Definition. Let M be a metric space. B(p, r) := {x ∈ M; d(x, p) ≤ r} ⊂ M. If we write B(x′, r) ⊂ ∂M, we consider ∂M as a metric space with induced Riemannian metric and construct the ball in this metric.

2.8. Notation. Let M be a manifold of bounded geometry, F ↓M a vector bundle of bounded geometry over M. A constant C is called M-universal and F -universal, respectively, if it does depend on M and F only, but on nothing more, especially not on specific charts.

2.9. Notation. A covering p : M̃ ↓M of CW-complexes is called normal, if p_{∗}(π_{1}(M̃)) is a normal subgroup of π_{1}(M). The quotient group Γ is called the covering group. It is isomorphic to the deck transformation group and acts transitively on the fiber.

2.10. Notation. Given a Riemannian manifold (M, g) and a chart x, we write g_{ij} = g(∂_{x_{i}}, ∂_{x_{j}}) for the covariant metric tensor, and g^{ij} for the contravariant metric tensor. These two matrices are inverse to each other.

2.11. Notation. Every Riemannian manifold is a measure space in the usual way (compare [29]). If f is a function, we simply write ∫_{M} f(x) dx for the corresponding integral. If E is a Riemannian bundle over M, we define the L^{2}-inner product for section f, g of E by

(f, g)_{L^{2}(E)} := ∫_{M}(f(x), g(x))_{E_{x}} dx

where (·, ·)_{E_{x}} denotes the inner product of the fiber E_{x}.

2.12. Notation. On a Riemannian manifold, we always use the Levi-Civita connection ∇ and the corresponding covariant derivative, without explicitly mentioning it.

2.13. Situation. Throughout the paper, A denotes a von Neumann algebra, H is a Hilbert space, B(H_{1}, H_{2}) denotes the bounded operators from H_{1} to H_{2}, B(H) = B(H, H).

Part I

Bounded geometry and Hodge-De Rham theory

3 Bounded geometry

Bounded geometry: Basic definitions

In this section we introduce the concept of a ∂-manifold of bounded geometry and derive the main (analytical) properties. At the heart of the theory lies the existence of nice charts (so called normal charts) and subordinate partitions of unity. With the aid of these, one can study analytical properties locally. In this way, Sobolev spaces are defined. For them, the Sobolev embedding theorem holds, differential operators extend to bounded operators H^{s} → H^{s−k}, a bounded restriction map H^{s}(M) → H^{s−1/2}(∂M) (s > 1/2) exists, ….

Manifolds of bounded geometry, but with empty boundary, have been examined and used by many authors, for instance the following: Aubin, [10], Shubin [64] and Kordyukov [38] study (pseudo)differential operators, Roe [55] derives an index theorem; Dodziuk [26] and Buttig [17] study the Laplacian (analytic and combinatorial); Triebel [67] introduces various function spaces; Attie [9] develops a surgery theory; Cheeger/Gromov/Taylor [21] investigate integral kernels of functions of the Laplacian. There is also related work of Eichhorn (compare the references in [17]).

Then, we construct to any metric a metric which is a product near the boundary and is analytically very close to the given metric. This will be used in the proof of the L^{2}-de Rham theorem in section 8. Next, we introduce and study differential boundary value problems in a unified way — particularly well suited for manifolds and bundles.

We start with the definition of manifolds of bounded geometry.

3.1. Definition. Suppose M is a manifold with boundary ∂M. It is of bounded geometry if the following holds:

(N) Normal collar: there exists r_{C} > 0 so that the geodesic collar

N = [0, r_{C}) × ∂M → M : (t, x) ↦ exp_{x}(tν_{x})

is a diffeomorphism onto its image, where ν_{x} is the unit inward normal vector at x ∈ ∂M. Equip N with the induced Riemannian metric. In the sequel we will identify N and its image. Denote im[0, r_{C}/3)×∂M by N_{1/3} and define N_{2/3} similarly.

(IC) Injectivity radius of ∂M: The injectivity radius of ∂M is positive: r_{inj}(∂M) > 0.

(I) Injectivity radius of M: There is r_{i} > 0 so that for x ∈ M − N_{1/3} the exponential map is a diffeomorphism on B(0, r_{i}) ⊂ T_{x}M. I.e. if we identify T_{x}M with ℝ^{m} via an orthonormal frame we have Gaussian coordinates ℝ^{m} ⊃ B(0, r_{i}) →^{κ_{x}} M around every point in M − N_{1/3}.

(B) Curvature bounds: For every K ∈ ℕ there is C_{K} > 0 so that |∇^{i}R| ≤ C_{K} and | ¯∇^{i}l| ≤ C_{K} ∀0 ≤ i ≤ K. Here R is the curvature and l second fundamental form tensor (with respect to ν), ∇ is the Levi-Civita connection of M and ¯∇ the one of ∂M

This is a generalisation of corresponding definitions for manifolds without boundary. The idea is, to obtain a minimal degree of homogeneity. Therefore, we exclude extreme curvatures and extreme changes (higher derivatives) of the curvature. Also, we require that the local describtion given by the exponential map is accurate to a definite degree (r_{i} > 0). A physicist who models a non-compact quantity but has only knowledge about a compact subset will naturally make assumptions like that, because if the physics elsewhere were radically different from the known part of the world, his rules should no longer apply. Now we introduce a boundary. Its behavior is described by the second fundamental form; again bounds on it will guarantee some homogeneity. Because the injectivity radius does not make sense near the boundary, we have to replace it by something different. The geodesic collar is the natural substitute.

Now we recall what Gaussian coordinates are and we explain their alternate near the boundary.

3.2. Definition. Choose 0 < r_{i}^{C} < r_{inj}(∂M). Choose x′ ∈ ∂M and an ONB of T_{x′}∂M. Define normal collar coordinates

κ_{x′} : \underbrace{B(0, r_{i}^{C})}_{⊂ℝ^{m−1}}×[0, r_{C}) → M : (v, t) ↦ exp_{exp_{x′}^{∂M} (v)}^{M}(tν).

(I.e. we iterate an exponential map of ∂M and an exponential of M.) Here, we have identified ℝ^{m−1} with T_{x′}∂M via the orthonormal frame; and ν is the inward unit normal vector field. For x ∈ M − N_{1/3} the exponential map yields Gaussian coordinates

κ_{x} : B(0, r_{i}) → M : v ↦ exp_{x}^{M}(v)

(identify T_{x}M with ℝ^{m} via an orthonormal base). We denote normal collar coordinates and Gaussian coordinates with the common name normal coordinates and the range κ_{x}(B(0, ϵ)) (if x ∈ M − N_{1/3}) as well as κ_{x}(B(0, ϵ) × [0, r_{C})) (if x ∈ ∂M) with NR(x, ϵ).

3.3. Convention. For the following we assume that r_{C}, r_{i} and r_{i}^{C} are sufficiently small (depending on the constant C). Especially we require: if x ∉ N_{2/3} then B(x, r_{i}) ∩ N_{1/3} = ∅.

There are other definitions of bounded geometry. We need the following

3.4. Definition. A Riemannian manifold M has (chart)bounded geometry iff (N), (IC), (I) of 3.1 hold and (instead of (B))

(B1) For any K ∈ ℕ an M-universal C_{K} > 0 exists, so that in arbitrary normal coordinates (if r_{C}, r_{i} and r_{i}^{C} are sufficiently small) the following holds for the metric tensor g_{ij} and its inverse g^{ij}:

|D^{α}g_{ij}| ≤ C_{K} and |D^{α}g^{ij}| ≤ C_{K} ∀|α| ≤ K

3.5. Remark. The proof of the following proposition shows that it suffices to check the condition for an atlas.

3.6. Remark. In the definition of bounded geometry and (chart)bounded geometry we could distinguish different degrees of boundedness in requiring bounds on the partial derivatives only up to some order K ∈ ℕ. This would extend to all of the statements concerning bounded geometry with the following pattern: a certain degree of boundedness in the assumptions yields a certain degree of boundedness in the consequences. We only had to check which assumptions we really need during the proof. It seemed not worth to work this out, because the notation necessarily would become more complicated. Moreover, the following result of Shi [62, 1.1 and 1.2] shows that boundedness and boundedness of degree zero are very close to each other: Suppose (M, g) is a complete Riemannian metric without boundary, and |R| ≤ C. Then, for any ϵ > 0 we find a metric g_{ϵ} on M with |∇^{K}R′| ≤ C_{K} ∀K and |g − g_{ϵ}| < ϵ. In particular, the two metrics are bilipschitz. Here g_{ϵ} is the solution at time t of the differential equation ∂_{t}g(t) = − Ric with g(0) = g.

This shows that the same method also works for boundary manifolds, if the metric g is a product near the boundary, and that g_{ϵ} has this product property, too. Combined with metric normalisation 3.26, which produces product metrics, we can get results similar to the one of Shi for ∂-manifolds. However, I do not see, and it is not clear, that the injectivity radius remains positive.

Cheeger and Gromov [20, 2.5] prove the following regularisation theorem for manifolds without boundary: if (M, g) has positive injectivity radius and |R| bounded, then we find another metric g_{ϵ} on M with |g − g_{ϵ}| < ϵ, so that the curvature and all the covariant derivatives of g_{ϵ} are bounded, the new injectivity radius is positive, too, and the difference between the two Levi-Civita connections is bounded.

3.7. Proposition. Let M be a Riemannian ∂-manifold. M has (chart)bounded geometry if and only if it has bounded geometry.

Proof. One implication follows directly form the formulae for R, l and their covariant derivatives in terms of g_{ij}, g^{ij} and their partial derivatives. The converse is proved in appendix A, theorem A.1.

3.8. Proposition. Let M be a ∂-manifold of bounded geometry. There is an M-universal C_{K} > 0 ∀K ∈ ℕ so that if κ_{x} and κ_{y} are normal coordinates, then the diffeomorphism κ_{x}^{−1}κ_{y} of two open subsets of ℝ^{m} (or of ℝ_{≥0}^{m}, if both κ_{x} and κ_{y} are normal boundary charts) has the following property: partial derivatives up to order K of κ_{x}^{−1}κ_{y} are bounded by C_{K}.

Proof. The proof is given in appendix A, theorem A.22.

On our manifolds of bounded geometry, we will study vector bundles, sections of these bundles and differential operators between them. All this has to be sufficiently regular. We start with the conditions on the bundles:

3.9. Definition. A bundle E↓M over a manifold of bounded geometry is called a bundle of bounded geometry if it is provided with the following additional structure: a trivialisation (called admissible) over each normal chart neighbourhood so that the transition (matrix) functions for the trivialisations pulled back via normal charts are bounded in the following sense: ∀K ∈ ℕ an E-universal C_{K} > 0 exists so that all partial derivatives of the transition functions up to order K are bounded by C_{K}.

3.10. Notation. We will frequently talk of functions f : M → X in ”normal coordinates” for manifolds M of bounded geometry. Then we always mean the composition of f with the corresponding chart. To keep notation simple we omit writing down this chart. In case of a section f : M → E of a bundle E of bounded geometry “in admissible trivialisations”, f stands for the composition t∘f ∘κ : ℝ^{m} → ℝ^{m} ×𝕂^{n} with normal chart κ and admissible bundle trivialisation t.

Similar remarks apply to other objects (like differential operators).

3.11. Definition. Let E↓M be a bundle of bounded geometry. Suppose it is equipped with a Hermitian metric. We denote this metric bounded and the bundle a bounded Hermitian bundle if the conjugate linear isomorphism E → E′ : x ↦ (·, x) is bounded with bounded inverse (in the sense of 3.16).

If we use admissible trivialisations, this means the following: We obtain functions b_{ij}(x) = (e_{i}, e_{j})_{x} on each domain of normal coordinates where e_{i} is the unit basis vector in ℂ^{n} (identified with the fiber over x via the admissible trivialisation) and (·, ·)_{x} is the Hermitian inner product in this fiber. Let b^{ij}(x) be the inverse matrix. Then we demand that E-universal constants C_{K} > 0 exist so that |∂_{x}^{α}b_{ij}(x)| ≤ C_{K} and |∂_{x}^{α}b^{ij}(x)| ≤ C_{K} ∀|α| ≤ K.

Similar definitions apply to Riemannian bundles.

3.12. Remark. The uniformity conditions for the bundles are just what we need technically. Here we follow Shubin [64]. If E has a natural connection (f.i. for twisted spinor bundles) we can reformulate the condition in terms of curvature bounds (compare [55]).

3.13. Example. (compare [64] and [55])

(1) Every compact Riemannian manifold M is of bounded geometry, and each bundle over M can be considered as a bundle of bounded geometry.

(2) If M̃ is the covering of a compact manifold M and the Riemannian metric is lifted then M̃ is of bounded geometry. Every pullback bundle Ẽ of a bundle E over M is a bundle of bounded geometry over M̃ if we lift the trivialisations.

(3) Any Lie group or symmetric space with (left)-invariant metric is of bounded geometry.

(4) Compact perturbations do not change the bounded geometry property. F.i. the connected sum of two manifolds of bounded geometry has bounded geometry.

(5) The leaves of a foliation of a compact Riemannian manifold with the induced metric are of bounded geometry.

(6) Given a manifold M of bounded geometry, the trivial bundle over M as well as the tangent bundle (with trivialisations induced by normal coordinates) are bundles of bounded geometry.

(7) The boundary of a manifold of bounded geometry is also of bounded geometry, as well as restrictions of bundles of bounded geometry to the boundary.

(8) If E, F are bundles of bounded geometry then so are E ⊕F, E ⊗F, Λ^{∗}E, Hom(E, F), E′ = Hom(E, 𝕂), …

Proof.

(1) To find admissible trivialisations for a bundle E↓K start with a finite covering {U_{i}} of K and accordingly finitely many trivialisations for E|_{U_{i}}. Then choose r_{C}, r_{i}, r_{i}^{C} sufficiently small so that for any x we have NR(x, r) ⊂ U_{i} for at least one of the U_{i}. Now we can pull back the trivialisation to E|_{NR(x,r)}. Since we started with finitely many trivialisations, this yields an admissible set of trivialisations.

(5) The functions M ∋ x ↦ ∇^{i}R_{(leaf through x)} and ∂M ∋ x′ ↦ ¯∇^{i}l_{(leaf through x′)} are continuous functions on the compact manifold M. The same is true for the injectivity radii. This yields the required bounds.

(6) The statement for the tangent bundle follows from the transition relation

\frac{∂}{∂x_{i}} |_{p} = ∑_{j} \frac{∂(y_{j} ∘ x^{−1})}{∂x_{j}}|_{x(p)} · \frac{∂}{∂y_{j}} |_{p}

for charts x, y : U → ℝ^{m} and p ∈ U.

(8) This is the case because the entries of the new transition matrices are polynomials in the old ones, and the polynomials are determined by the functor in question.

3.14. Notation. C^{∞}(E) denotes the smooth sections; C_{0}^{∞}(E) the smooth sections with compact support. Similarly defined are C^{k} and C_{0}^{k}.

3.15. Definition. Let E be a bundle of bounded geometry over M. Then

C_{b}^{k}(E) := {f ∈ C^{k}(E); |∂^{α}(t_{p} ∘ f ∘ κ_{p})| ≤ C_{k}(f) ∀|α| ≤ k}.

Here, κ_{p} runs through the normal charts, t_{p} the corresponding admissible trivialisations. C_{b}^{k}(E) is a Banach space. The norm of f is the smallest possible constant C_{k}(f) in the definition.

On non-compact manifolds, analyis is only possible if the operator classes and the function spaces match. F.i., analysists have introduced various classes of weighted function spaces on ℝ^{n}. Here, we will consider uniform function spaces and uniform operators. The prototype is the lift of an operator from a compact manifold to a covering.

3.16. Definition. Suppose E, F are bundles of bounded geometry over M. Let A : C_{0}^{∞}(E) → C^{∞}(F) be a differential operator of order k. In normal coordinates with admissible trivialisations A becomes a matrix (∑_{|α|≤k} a_{α}^{ij}(x)D^{α})_{ij} (this is the definition of a differential operator). We say A is bounded if ∀K ∈ ℕ

|D^{β}a_{α}^{i,j}(x)| ≤ C_{K} ∀|β| ≤ K

with (E, F)-universal constants C_{K}. Similar notions apply to boundary differential operators p : C_{0}^{∞}(E) → C^{∞}(X), where X is a bundle of bounded geometry over ∂M. A boundary differential operator is by definition locally a differential operator composed with restriction to the boundary.

3.17. Example.

• On a compact manifold, every differential operator between arbitrary bundles is bounded.

• On a covering manifold M̃ every lifted differential operator is bounded.

• On an arbitrary manifold of bounded geometry the exterior differentiation d, its formal adjoint δ, the Laplacian Δ and the Hodge-operator ∗ are bounded (all acting on differential forms).

Proof. Only the last statement requires a proof. Here, boundedness is clear for d. In view of the following lemma, since δ = ± ∗ d∗ and Δ = dδ + δd, we only have to check it for ∗. But the entries in the matrix for ∗, using trivialisations which come from charts (as we have to do), are polynomials in the metric tensors g_{ij} and g^{ij}.

3.18. Lemma. Let A and B be two bounded differential operators. Suppose their composition A ∘ B is defined. Then the composition is bounded.

Proof. The entries of the coefficient matrix for A∘B in normal coordinates are polynomials in the entries for A and in derivatives up to the order of A of the entries for B.

Technical properties of manifolds of bounded geometry

Now we collect some technical properties of manifolds of bounded geometry.

3.19. Lemma. Let M be a ∂-manifold of bounded geometry. We find r > 0 so that for every x ∈ M the metric ball B(x, r) is contained in some normal chart NR(y, ϵ). Moreover, 0 < D_{1}(r) < D_{2}(r) exist for any such r so that

D_{1}(r) ≤ vol(B(x, r)) ≤ D_{2}(r) ∀x ∈ M,
D_{1}(r) ≤ vol(NR(x′, r)) ≤ D_{2}(r) ∀x′ ∈ ∂M and
D_{1}(r) ≤ vol(B(x′, r) ⊂ ∂M) ≤ D_{2}(r) ∀x′ ∈ ∂M

Proof. Note that boundedness implies ∃C_{1}, C_{2} > 0 so that

∥(g^{ij})_{ij}∥ < C_{1}, ∥(g_{ij})_{ij}∥ < C_{1}, and C_{2} ≤ \sqrt{|det g_{ij}|} ≤ C_{1}

(in normal coordinates). The first assertion is trivial for x ∈ M − N_{1/3}, if we choose r ≤ r_{i} (r_{i} of the definition of bounded geometry). For x ∈ N_{1/3}, project x along the geodesic collar to x′ ∈ ∂M and consider NR(x′, r_{inj}). Because of the boundedness of the norm of the matrices (g^{ij})_{ij} and (g_{ij})_{ij}, every path which starts from x and does leave NR(x′, r_{inj}) has length at least r, for some M-universal constant r > 0. This proves the first statement. The assertion about the volumes follows from the upper and lower bounds of \sqrt{|det g_{ij}|}. (We have to integrate these functions over a fixed cylinder and ball, resp., in Euclidian space.) It remains to explain why we can find volume bounds for the metric balls if they do not lie in the range of a Gaussian chart. In this case, we can suppose that B(x, r) ⊂ NR(x′, ϵ) for some cylinder NR(x′, ϵ). We can not describe the image B = B(x, r) in terms of these coordinates. However, from the bounds on the metric tensor we know that M-universal r_{1}(r) and r_{2}(r) exist so that (if we equip the cylinder NR(x′, ϵ) with the Euclidian metric)

B_{Eucl.}(x, r_{1}) ⊂ B ⊂ B_{Eucl.}(x, r_{2})

The volume of B_{Eucl.}(x, r_{1}) in (M, g) is M-universally bounded by the same reason as above. This concludes the proof.

3.20. Lemma. Let M be a connected non-compact ∂-manifold of bounded geometry. We can find a function V_{0} : ℝ_{≥0} → ℝ_{≥0} with the following properties:

• V_{0} is monotonous

• V_{0}(r) → 0 if r → 0

• V_{0}(r) → ∞ if r → ∞

• V_{0}(r) ≤ vol(B(x, r)) ∀x ∈ M

Proof. Lemma 3.19 implies that we can construct V_{0}(r) for r sufficiently small. To extend it to all of ℝ_{≥0}, we only have to show that

w(r) := inf_{x∈M}{vol(B(x, r))} → ∞ if r → ∞

Actually, w(r) grows at least linearly: Choose any d > 0. Fix r > 0 and 0 < r_{ϵ} < d/3 so small that B(x, r_{ϵ}) is contained in some normal chart ∀x ∈ M. Then, we find M-universal C > 0 so that vol(B(x, r_{ϵ})) > C ∀x ∈ M. To given x ∈ M choose y ∈ M with d(x, y) ≥ 2r. (this is possible because M is complete but not compact). Choose a path from x to y with length d(x, y) (because ∂M ≠ 0, this is not necessarily a geodesic). Mark successive points x_{0} = x, x_{1}, …, x_{[r/d]} on the path with d(x_{i}, x_{i+1}) = d (distance on the path!).

Claim: B(x_{i}, r_{ϵ}) ∩ B(x_{j}, r_{ϵ}) = ∅ if i ≠ j. Else d(x_{i}, x_{j}) < 2r_{ϵ} < d, and the given path would not be length minimizing (in contradiction to its choice). Therefore

vol(B(x, r)) ≥ ∑_{i=0}^{[r/d]−1} vol(B(x_{i}, r_{ϵ})) ≥ (r/d − 1) · C.

3.21. Definition. The multiplicity of a covering of a set is the supremum of the number of members of the covering with non-empty intersection. Other authors use the term ”covering dimension” instead of multiplicity. Our terminology follows Shubin [64].

The following is a very important technical lemma for our analysis on manifolds of bounded geometry.

3.22. Lemma. (Partition of unity) Let M be a manifold of bounded geometry. There is r_{m} > 0 so that, if 0 < r < r_{m}, then a countable covering of M exists by sets {NR(x_{i}, r)}_{i∈ℤ} which has the following properties:

• x_{i} ∈ ∂M for i ≥ 0, x_{i} ∈ M − N_{2/3} for i < 0

• {NR(x_{i}, r)}_{i∈ℤ} has finite multiplicity, even more: there is M_{f} so that ∀s < r_{m} and ∀x the intersection NR(x, s) ∩ NR(x_{i}, r) ≠ ∅ for at most M_{f} of the x_{i}

• {NR(x_{i}, r/2)}_{i∈ℤ} is a covering of M, too.

(For technical reasons we assume that r_{C} is one half of the original r_{C}.) Denote with κ_{i} : B(0, r) → NR(x_{i}, r) (i < 0) and κ_{i} : B(0, r) × [0, r_{C}) → NR(x_{i}, r) for i ≥ 0 the corresponding charts.

To this covering, a subordinate smooth partition of unity {φ_{i}} exists which is uniformly bounded, i.e. ∀K ∈ ℕ we find C_{K} > 0 so that

|D^{α}φ_{i}| ≤ C_{K}  ∀i ∀|α| ≤ K  (in normal coordinates).

We get the following data:

• a covering NR(x_{i}, r)

• with charts κ_{i} : ℝ^{m} → NR(x_{i}, r)

• and a nice subordinate partition of unity φ_{i} (i ∈ ℤ).

Proof. If ∂M = ∅: compare Shubin [64, A1.2 and A1.3] for the first statement. We extend his argument to cover our more general situation. The idea of the proof is the following.

Choose a maximal set of disjoint balls of fixed radius r/2. Maximality implies that the corresponding balls with radius r cover M. If a fixed point z lies in N of the corresponding balls of radius 2r then, on the other hand, N disjoint balls of radius r/2 lie in B(z, 3r). The volume of balls is M-universally bound from above and from below, this yields an M-universal upper bound for N, i.e. the covering has finite multiplicity.

For the partition of unity choose a fixed cutoff function φ on ℝ^{m} which is 1 if |x| ≤ r and 0 if |x| ≥ 2r. Transport this function via the exponential map to of the covering balls. Since the change of coordinate functions have M-universally bounded derivatives, D^{α}φ is bounded in arbitrary normal coordinates. Dividing by the (locally finite) sum over all the transported cutoff functions yields the desired partition of unity.

Now we give the rigorous proof. In particular we have to take care of the boundary: Set r_{m} = min{r_{i}/3, r_{i}^{C}/3}. Let 0 < r < r_{m}. Now choose a maximal set of points {x_{i} ∈ ∂M}_{i∈ℕ} so that (in ∂M) all {B(x_{i}, r/4)} are disjoint (exists by Zorn’s lemma and is necessarily countable — note that Zorn’s lemma is not necessary here). Similarly, choose a maximal set of points {x_{i} ∈ M − N_{2/3}}_{i<0} so that all B(x_{i}, r/4)_{i<0} are disjoint. Then

⋃_{i<0} NR(x_{i}, r/2) = ⋃_{i<0} B(x_{i}, r/2)} covers M − N_{2/3} :

Else we find z ∈ M − N_{2/3} which has distance ≥ r/2 to all of the x_{i}. Then B(z, r/4) ∩ B(x_{i}, r/4) = ∅ ∀i < 0, violating the maximality of the set {x_{i}}_{i<0}. Similarly, {B(x_{i}, r/2)}_{i≥0} covers ∂M ⟹ {NR(x_{i}, r/2)}_{i≥0} covers the collar N. Now we have to show that the covering {NR(x_{i}, r/2)}_{i∈ℤ} has finite multiplicity. For this, choose 0 < ϵ < 3r_{m}. Lemma 3.19 shows that we find 0 < D_{1}(ϵ) < D_{2}(ϵ) so that

D_{1}(ϵ) ≤ vol(NR(x, ϵ)) ≤ D_{2}(ϵ) ∀x; D_{1}(ϵ) ≤ vol(B(x′, ϵ) ⊂ ∂M) ≤ D_{2}(ϵ) ∀x′ ∈ ∂M

Now, choose z ∈ M. If z ∈ N_{1/3}, z ∉ NR(x_{i}, r) ∀i < 0 (by the assumption 3.3 on r_{i}). Project z along the geodesic collar to z′ ∈ ∂M. If z′ ∈ B(x_{i}, r) ⊂ ∂M then B(x, r/4) ⊂ B(z′, 2r) ⊂ ∂M (2r < 3r_{m} ≤ r_{i}). For the number N_{1} of these x_{i} we have, because all B(x_{i}, r/4) ⊂ ∂M (i ∈ ℕ) are disjoint:

N_{1} ≤ \frac{sup_{z′∈∂M} vol(B(z′, 2r) ⊂ ∂M)}{inf_{x∈∂M} vol(B(x, r/4) ⊂ ∂M)} ≤ \frac{D_{2}(2r)}{D_{1}(r/4)}

Note z ∈ NR(x_{i}, r) ⟺ z′ ∈ B(x_{i}, r) ⊂ ∂M (i ≥ 0). If z ∈ M − N_{1/3}, it is (by the same argument) contained in at most N_{1} zylinders of type NR(x_{i}, r)_{i≥0}. Now, it is also possible that z ∈ NR(x_{i}, r) = B(x_{i}, r) (i < 0). Then B(x_{i}, r/4) ⊂ B(z, 2r). For the number N_{2} of these x_{i}, we have again N_{2} ≤ D_{2}(2r)D_{1}(r/4)^{−1}. Therefore, {NR(x_{i}, r)}_{i∈ℤ} is a covering of finite multiplicity.

Similarly to the proof of lemma 3.19 we show that an M-universal r_{m} > 0 exists so that

for x ∈ im(\underbrace{B(y′, r_{i}/2)}_{⊂∂M} ×[0, r_{C}/2)) holds: B(x, r_{m}) ⊂ im(\underbrace{B(y′, r_{i})}_{⊂∂M} ×[0, r_{C})) (image under

the geodesic collar map). Replacing now r_{C} by r_{C}/2, we can prove the existence of M_{f} just as we proved finite multiplicity.

It remains to construct the subordinate partition of unity. Choose a smooth cutoff function φ : ℝ^{m} → [0, 1] with φ(x) = 1 if |x| ≤ r/2 and φ(x) = 0 if |x| ≥ r. Denote the restriction to ℝ^{m−1} also with φ. Choose smooth ψ : ℝ → [0, 1] with ψ(x) = 0 if x ≥ r_{C} and ψ(x) = 1 if x ≤ 2r_{C}/3. Via the normal coordinates this yields cutoff functions f_{i} on NR(x_{i}, r) with f_{i}κ_{i}(y′, t) = φ(y′)ψ(t) if i ≥ 0 and f_{i}κ_{i} = φ if i < 0. Therefore, if κ is any normal chart, f_{i} ∘ κ = φ ∘ (κ_{i}^{−1}κ) (i < 0) and f_{i} ∘ κ = (φ · ψ) ∘ (κ_{i}^{−1}κ) (i ≥ 0). The chain rule shows that the M-universal bounds on derivatives up to order K of the coordinate changes (proposition 3.8) yields M-universal bounds on the partial derivatives up to order K of f_{i} in normal coordinates. To construct the partition of unity, set F = ∑_{i∈ℤ} f_{i} (note that finite multiplicity implies that pointwise this is a finite sum with at most N_{1} + N_{2} non-trivial summands). Since

M − N_{2/3} ⊂ ⋃_{i<0} NR(x_{i}, r/2) and N_{2/3} ⊂ ⋃_{i≥0} NR(x_{i}, r/2),

for each z ∈ M at least one of f_{i}(z) = 1 ⟹ F ≥ 1. Define

φ_{i} := f_{i}/F.

Obviously, {φ_{i}}_{i∈ℤ} is a smooth partition of unity subordinate to our covering. Choose arbitrary φ_{i} and normal chart κ. For partial derivatives up to order K in normal coordinates observe

\begin{aligned}|D^{α}(φ_{i} ∘ κ)| & = |D^{α} \frac{f_{i} ∘ κ}{F ∘ κ}| = \frac{|P_{α}(D^{β}(f ∘ κ), D^{γ}(F ∘ κ); |β|, |γ| ≤ |α|)|}{|F ∘ κ|^{2^{|α|}}} \\ & ≤^{|F |≥1} |P_{α}(D^{β}(f ∘ κ), D^{γ}(F ∘ κ))| \end{aligned}

P_{α} is a polynomial entirely determined by α. At every point v, D^{γ}(F ∘ κ)|_{v} is the sum of at most N_{1} + N_{2} summands of the type D^{γ}(f_{s} ∘ κ)|_{v}. Therefore, for all the entries of P_{α}, we have M-universal bounds. This yields an M-universal bound C for |D^{α}(φ_{i} ∘ κ)| if |α| ≤ K.

Sobolev theory for manifolds of bounded geometry

Now, we introduce the class of Sobolev spaces which corresponds to bounded operators and bounded functions. We do this via localisation, although in many cases a global expression will do, too (compare proposition 3.25). Of course, the uniformity condition for bounded geometry will be needed here.

3.23. Definition. Let M be a bounded manifold with boundary. Choose data as in lemma 3.22. For s ∈ ℝ define the Sobolev norm

|f|_{H^{s}(M)}^{2} := ∑_{i} |φ_{i}f ∘ κ_{i}|_{H^{s}(κ_{i}^{−1}(NR(x_{i},r)))}^{2} f ∈ C_{0}^{∞}(M).

Here | · |_{H^{s}(κ_{i}^{−1}(NR(x_{i},r)))} is the Sobolev norm on Euclidian space.

In the following, we will often abbreviate by writing H^{s}(ℝ^{m}) or H^{s} instead of H^{s}(κ_{i}^{−1}(·)) and even omit the κ_{i}.

Let H^{s}(M) be the completion of C_{0}^{∞}(M) with respect to the Sobolev norm | · |_{H^{s}}. In a similar way (using admissible trivialisations) we define H^{s}(E) if E is a bundle of bounded geometry over M.

3.24. Lemma. The choice of other data in definition 3.23 yields equivalent Sobolev norms.

Proof. If {NR(x_{i}, r)}_{i∈ℤ} is a covering with subordinate partition {φ_{i}}, {NR(y_{j}, r′)}_{j∈ℤ} another one with partition {ψ_{j}}, consider the covering U_{ij} = NR(x_{i}, r) ∩ NR(y_{j}, r′) with subordinate partition φ_{i}ψ_{j}. Set J_{i} := {j ∈ ℤ; NR(y_{j}, r′)∩NR(x_{i}, r) ≠ ∅}. Then |J_{i}| ≤ M_{f} by lemma 3.22. Now, if we use the chart κ_{i}:

∑_{ij}|φ_{i}ψ_{j}f ∘ κ_{i}|_{H^{s}}^{2} = ∑_{i} ∑_{j∈J_{i}}|ψ_{j}φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} ≤^{M_{f} ·sup_{j}∥ψ_{j}·∥_{B(H^{s})}} ∑_{i}|φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} = |f|_{H^{s}(M)}

Note that the operators ”multiplication with ψ_{j}” on H^{s}(ℝ^{m}) (−K ≤ s ≤ K) have norms bounded by the suprema on partial derivatives of ψ_{j} up to order K (this is clear for s ∈ ℕ, the general case follows from interpolation and duality). Therefore, sup_{j}∥ψ_{j}·∥_{B(H^{s})} < ∞. On the other hand

|f|_{H^{s}(M)} = ∑_{i}|φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} = ∑_{i}|∑_{j} ψ_{j}φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} = ∑_{i}|∑_{j∈J_{i}} ψ_{j}φ_{i}fκ_{i}|_{H^{s}}^{2}

≤^{2^{M_{f}}} ∑_{i,j∈J_{i}}|ψ_{j}φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} = ∑_{ij}|φ_{i}ψ_{j}f ∘ κ_{i}|_{H^{s}}^{2}

It remains to compare the norm induced from one partition (again denoted {φ_{i}}) and one covering, but with two different systems of normal charts {κ_{i}} and {x_{i}}:

∑_{i}|φ_{i}f ∘ κ_{i}|_{H^{s}}^{2} = ∑_{i}|φ_{i}f ∘ x_{i} ∘ x_{i}^{−1} ∘ κ_{i}|_{H^{s}}^{2} ≤^{sup_{i}∥·∘x_{i}^{−1}∘κ_{i}∥_{BH^{s}}} ∑_{i}|φ_{i}f ∘ x_{i}|_{H^{s}}^{2}

Here, we used the fact that composition with a diffeomorphism is a bounded isomorphism of H^{s}(ℝ^{m}) with norm bounded by the suprema of partial derivatives of the diffeomorphism up to order max{|s|, 1}: again this is clear for s ∈ ℕ, and the general statement follows from interpolation and duality. Theorem A.22 of the appendix provides us with bounds on the suprema.

In the following proposition, the elementary properties of our Sobolev spaces are listed.

3.25. Proposition. Let E be a bundle of bounded geometry over M. Suppose X is a bounded vector bundle over ∂M. Then the following holds for the Sobolev spaces H^{s}(E) (s, t ∈ ℝ):

(1) H^{s}(E) is a Hilbert space (the inner product depends on the choices)

(2) The usual Sobolev embedding theorems hold: we have a bounded embedding

H^{s}(E) ↪ C_{b}^{k}(E) whenever s > m/2 + k

(3) For differential forms, the following is true: If k ∈ ℕ_{0} and ∇ is the exterior differential of the Riemannian manifold M then

|ω|_{k}^{2} := ∑_{i=0}^{k} ∫_{M} |∇^{i}ω(x)|_{x}^{2} dx

is equivalent to the Sobolev norm on H^{k}(Λ^{p}T^{∗}M).

(4) For s < t we have a bounded embedding with dense image

H^{t} ⊂ H^{s}.

This map is compact if and only if M is compact. We define

H^{∞}(E) := ⋂ H^{s}(E) H^{−∞}(E) := ⋃ H^{s}(E).

(5) Let A : C^{∞}(E) → C^{∞}(F) be a k-bounded differential operator of order k. Then A extends to a bounded operator

A : H^{s}(E) → H^{s−k}(F) ∀s

(6) Let p : C^{∞}(E) → C^{∞}(X) be a k-bounded boundary differential operator of order k. Then p extends to an bounded operator

p : H^{s}(E) → H^{s−k−1/2}(X) if s > k + \frac{1}{2}.

In particular, we have the bounded restriction map

res : H^{s}(E) → H^{s−1/2}(E|_{∂M}) if s > \frac{1}{2}.

(7) H^{s}(E) and H^{−s}(E′) are dual to each other by extension of the pairing

(f, g) = ∫_{M} g(f(x)) dx; f ∈ C_{0}^{∞}(E), g ∈ C_{0}^{∞}(E′)

where E′ denotes the dual bundle of E.

(8) If E is a bounded Hermitian or Riemannian bundle, then the norm on L^{2}(E) we have defined is equivalent to the usual L^{2}-norm

|f|^{2} := ∫_{M}(f, f)_{E_{x}} dx f ∈ C_{0}^{∞}(E).

Moreover, H^{s}(E) and H^{−s}(E) are dual to each other by extension of

(f, g) = ∫_{M}(f, g)_{x}d x f, g ∈ C_{0}^{∞}(E).

Here (·, ·)_{x} denotes the inner product in the fiber E_{x} over x ∈ M.

Remember that we work with a covering with finite multiplicity N and with a subordinate partition of unity {φ_{i}} with certain boundedness properties. Of course, in every case we have to reduce the statement to the well known equivalent in Euclidian space. Lemma 3.22 will be an essential ingredient of the proofs.

Since the actual proofs have to be done in quite an obvious way we will omit them here. In flavour, they are similar to the proof of lemma 3.24.

Normalisation to a product metric

Now, we will construct to every metric with bounded curvature a nearby metric, which is a product near the boundary and has bounded geometry, too. This will be used to show that the it suffices to prove the de Rham theorem for product metrics.

3.26. Theorem. Let (M, g) be a manifold of bounded geometry with boundary ∂M. Suppose K : ∂M ×[0, r_{C}) → M is a metric collar. Let g′ = K_{∗}(g|_{∂M} ×g_{ℝ}) be the corresponding product metric on the collar. Set ḡ := φg′ + (1 − φ)g, where φ : [0, r_{C}) → [0, 1] is a cutoff function, i.e. 1 in a neighbourhood of 0 and 0 outside another (sufficiently small) neighbourhood of 0. Then (M, ḡ) is a manifold of bounded geometry, every bundle E↓M which is a bundle of bounded geometry with respect to g has bounded geometry also with respect to ḡ, and the Sobolev spaces do not depend on the metric: H^{s}(E, g) = H^{s}(E, ḡ). For every ϵ > 0, by a suitable choice of φ, we can obtain ḡ so that |ḡ − g| < ϵ. (The latter pointwise estimate holds with respect to both of the metrics).

Proof. First, observe g|_{∂M} = ḡ|_{∂M}. Moreover, the inward unit normal vector fields ν and ν̄ are equal. The geodesics in (M, g) with starting velocity ν and length < r_{C} are by the very construction of g′ also geodesics of (im(K), g′). Let α(t) be such a geodesic with v_{t} = α′(t). Then α is a geodesic of (M, ḡ) too, since for arbitrary vector fields Z (compare Gallot/Hulin/Lafontaine [29, 2.51]):

2ḡ(∇ḡ ; v_{t}v_{t}, Z) =2v_{t}.ḡ(v_{t}, Z) − Z.ḡ(v_{t}, v_{t}) − 2ḡ([v_{t}, Z], v_{t})
=^{ḡ=φg′+(1−φ)g}φ2g′(\underbrace{∇_{v_{t}}^{g′}v_{t}}_{=0}, Z) + (1 − φ)2g(\underbrace{∇_{v_{t}}^{g}v_{t}}_{=0}, Z) + 2v_{t}(φ)(g′(v_{t}, Z) − g(v_{t}, Z))
+ Z(φ)(g′(v_{t}, v_{t}) − g(v_{t}, v_{t})) = 0

The latter is true since the Gauss lemma A.4 says g(v_{t}, Z) = 0 ⟺ g′(v_{t}, Z) = 0 and moreover g(v_{t}, v_{t}) = 1 = g′(v_{t}, v_{t}). Therefore ∇ḡ ; v_{t}v_{t} = 0 and α is a geodesic. It follows that normal boundary coordinates for (M, g) and (M, ḡ) coincide. If we take supp φ and the injectivity radius r_{i} to be sufficiently small, the same holds for Gaussian coordinates centered at a point in M − N_{1/3} (because g and ḡ are equal there). If we choose supp φ small enough, ḡ will be arbitrarily close to g. This follows from the fact that near the boundary g(t) → g(0) for t → 0 (t the boundary defining normal coordinate), and this convergence is uniform on ∂M because of the M-universal bounds for first derivatives of the metric tensor in normal coordinates.

To show that (M, ḡ) is a manifold of bounded geometry, it suffices to compute the metric tensors ḡ_{ij} and ḡ^{ij} in (the old!) normal boundary coordinates and find M-universal bounds for them and their partial derivatives. For ḡ_{ij}(x′, t) = φ(t)g_{ij}(x′, 0) + (1 − φ(t))g_{ij}(x′, t) this is immediate since φ is M-universal and we have by assumption the bounds for g_{ij} and its derivatives. Now, lemma A.11 of the appendix shows that it suffices to find bounds for |ḡ^{ij}| only. Equivalently, produce a bound for the norm of the matrix (ḡ^{ij})_{ij}, i.e. a lower bound for the smallest eigenvalue of the positive self adjoint (inverse) matrix (ḡ_{ij})_{ij}. By assumption, we find corresponding M-universal bounds for (g_{ij}(x′, t))_{ij} and (then trivially) for (g_{ij}(x′, 0))_{ij}. The smallest eigenvalue of a positive self adjoint matrix G is equal to inf_{|x|=1}(Gx, x). For convex combinations, if (G_{1}x, x) ≥ c and (G_{2}x, x) ≥ c then also ((φG_{1} + (1 − φ)G_{2})x, x) = φ(G_{1}x, x) + (1 − φ)(G_{2}x, x) ≥ c. This implies that we get an M-universal bound for |(ḡ^{ij})|; and (M, ḡ) has bounded geometry. In the definition of the Sobolev spaces H^{s}(E), we have seen right now that we can choose identical data for g and for ḡ.

Boundary value problems

The contents of this subsection will deal not only with bounded geometry but in the first part will apply to arbitrary smooth manifolds with corresponding bundles.

3.27. Definition. Let E, F be vector bundles over a manifold M and X_{0}, …, X_{r} bundles over ∂M. A system

(A, p_{0}, …, p_{r}) : C_{0}^{∞}(E) → C^{∞}(F) ⊕ C^{∞}(X_{0}) ⋯ ⊕ C^{∞}(X_{r})

is called a (differential) boundary value problem (of order μ), if A is an ordinary differential operator (of order μ) and p_{i} are boundary differential operators. If not stated otherwise, we demand that ord p_{i} = i —this may always be achieved by using direct sums, adding the zero bundle and rearranging terms. We set p⃗ = (p_{0}, …, p_{r}) and P = (A, p⃗).

3.28. Definition. (Composition of boundary value problems) Let E, F, G be smooth vector bundles over a manifold M and X_{i}, Y_{j} bundles over its boundary ∂M. Let (A, p⃗) : E → F ⊕ X_{0} ⊕ ⋯ ⊕ X_{r} and (B, q⃗) : F → G ⊕ ⨁_{j} Y_{j} be differential boundary value problems of order μ and μ′ respectively. Their composition is defined as

(B ∘ A, q⃗ ∘ A, p⃗) : C^{∞}(E) → C^{∞}(G) ⊕ ⨁_{j} C^{∞}(Y_{j}) ⊕ ⨁_{i} C^{∞}(X_{i}).

Adjoint boundary value problems

We will need an adjoint regularity theorem in our applications. Therefore, we set up the theory concerning this. In the following we generalize to arbitrary manifolds and bundles the presentation of Lions and Magenes [40] on adjointness for differential boundary value problems on ℝ^{m}, and we extend it slightly. From now on,

P = (A, p_{0}, …, p_{r}) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁_{i} C^{∞}(X_{i})

is a boundary value problem of order (μ, 0, 1, …, r) with dim X_{i} = n_{i}, dim E = n = dim F.

3.29. Definition. We say (p_{0}, …, p_{r}) is normal if n_{i} ≤ n and in all charts around ∂M of the bundle the matrix (a_{i,j}(x, ξ))\begin{matrix}i=1,...,n_{α} \\ j=1,...,n \end{matrix} of the principal part of p_{α} has rank n_{α} for fixed x

and for ξ = (0, …, 0, ξ_{m}); ξ_{m} ≠ 0. (Obviously, for fixed x it suffices to check this for only one chart.)

If (p_{0}, …, p_{r}) is normal and dim X_{i} = n ∀i, it is called a Dirichlet system.

Normality says that the different boundary operators p_{i} are independent of each other, i.e. we can always find a section f with prescribed boundary values p_{i}(f) (compare 3.32). Dirichlet systems are maximal normal systems and therefore in some sense complete. They are unique up to transformations of lower order, i.e. we can restrict attention to the prototype Dirichlet system

(∂_{ν}, ∂_{ν}^{2}, ∂_{ν}^{3}, …) · 1.

The main property of a Dirichlet system is that it has a unique adjoint.

3.30. Lemma. Suppose (p_{0}, …, p_{r}) is normal. Then a (normal) set of boundary differential operators (t_{0}, …, t_{r}) : E → Y_{0} ⊕ ⋯ ⊕ Y_{r} exists so that p⃗ ⊕ t⃗ is a Dirichlet system.

Proof. Choose a Riemannian metric on M and on E. Then p_{i} = A_{i}∂_{ν}^{i} + B_{i} + C_{i} where ∂_{ν} denotes differentiation in inward normal direction, B_{i} is a differential operator of order i of ∂M (a tangential differential operator), C_{i} is a boundary differential operator of order < i and A_{i} : E → X_{i} is a linear bundle morphism which is surjective by normality. Let Y_{i} := ker A ⊂ E and t_{i} := pr_{Y_{i}} · ∂_{ν}^{i} (where pr_{Y_{i}} is the orthogonal projection E → Y_{i}). The assertion follows.

3.31. Lemma. If p⃗ = (p_{0}, …, p_{r}) and q⃗ = (q_{0}, …, q_{r}) are two Dirichlet systems then Φ = (Φ_{i,j})_{0≤i,j≤r} and Ψ = (Ψ_{i,j})_{0≤i,j≤r} exist, where

Φ_{i,j} : X_{i} → Y_{j} and Ψ_{i,j} : Y_{i} → X_{j}

are tangential differential operators (i.e. d.o. of ∂M) of order j − i (especially Φ_{i,j} = 0 = Ψ_{i,j} for i > j), so that

Φ_{i,i} : X_{i} → Y_{i} and Ψ_{i,i} : Y_{i} → X_{i}

are linear bundle isomorphisms and

p⃗ = Ψq⃗; q⃗ = Φp⃗

(this implies Ψ ∘ Φ = 1 = Φ ∘ Ψ).

Proof. First observe that it suffices to construct Φ and Ψ locally (we may glue them together using a partition of unity). In a next step we may replace q⃗ by D⃗ : U × 𝕂^{n} → U × (𝕂^{n})^{r+1} with D_{i} = 1 · ∂_{m}^{i} (the general case is then established by comparing p⃗ with D⃗ and then D⃗ with q⃗ and composing the matrices we get). It follows directly from the required properties of p⃗ that p⃗ = ΦD⃗ with appropriate Φ. We construct Ψ_{i,j} inductively in j: For j = 0 set Ψ_{0,0} = p_{0}^{−1} (the zero-order differential operator p_{0} is invertible by definition of Dirichlet systems). For i > 0 set Ψ_{i,0} = 0. Suppose Ψ_{i,j} is already constructed with the required properties for j < j_{0}. We have

p_{j_{0}} = A_{0} ∘ ∂_{ν}^{j_{0}} + ∑_{i=1}^{j_{0}} A_{i}∂_{ν}^{j_{0}−i}

with A_{i} tangential differential operators of order ≤ i and A_{0} invertible (by the Dirichlet property). Hence

\begin{aligned}∂_{ν}^{j_{0}} & = A_{0}^{−1}p_{j_{0}} − ∑_{i=1}^{j_{0}} A_{0}^{−1} ∘ A_{i} ∘ ∂_{ν}^{j_{0}−i} \\ & =^{ind.} A_{0}^{−1}p_{j_{0}} − ∑_{i=1}^{j_{0}} A_{0}^{−1} ∘ A_{i} ∘ ∑_{k=0}^{j_{0}−i} Ψ_{k,j_{0}−i}p_{k}. \end{aligned}

Rearranging terms yields Ψ_{k,j_{0}} with the required properties.

3.32. Lemma. Suppose p⃗ is a Dirichlet system. For arbitrary sections φ_{i} ∈ C_{0}^{∞}(X_{i}) we find f ∈ C_{0}^{∞}(E) with p_{i}f = φ_{i}. Moreover, the support of f may be confined in any open set containing the support of each of the φ_{i}.

Proof. It suffices to consider the local case. Let p⃗ = ΦD⃗ and D⃗ = Ψp⃗ as in lemma 3.31. Let φ⃗ = (φ_{0}, …, φ_{μ−1}) and supp φ⃗ = ⋃ supp φ_{i}. Set α⃗ = Ψ(φ⃗). By a well known result (Whitney [69]), we find smooth f with compact support arbitrarily close to supp α⃗ so that D⃗f = α. Then

p⃗f = ΦD⃗f = Φ(α⃗) = ΦΨ(φ⃗) = φ⃗.

3.33. Remark. We can also define infinite Dirichlet systems (p_{0}, p_{1}, …). 3.31 and 3.32 remain true without change of the proofs. Note that [69] also covers this more general case.

3.34. Definition. Suppose (A, p⃗) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁_{i} C^{∞}(X_{i}) is a boundary value problem with normal p⃗. Let t⃗ : C^{∞}(E) → ⨁_{i} C^{∞}(Y_{i}) be a supplement so that p⃗ ⊕ t⃗ is Dirichlet (according to 3.30). We denote (A, p⃗, t⃗) = (A, p⃗ ⊕ t⃗). A similar system (but with possibly permuted orders) (B, s⃗, q⃗) : C^{∞}(F) ⊕ ⨁ C^{∞}(X_{i}) ⨁ C^{∞}(Y_{i}) is called the (L^{2})-formal adjoint system (with respect to given inner products on E, F, X_{i}, Y_{i} and a given Riemannian metric on M) if

(Af, g) − (f, Bg) = (p⃗f, s⃗g) − (t⃗f, q⃗g) ∀f ∈ C_{0}^{∞}(E), g ∈ C_{0}^{∞}(F)  (3.35)

Here (p⃗f, s⃗g) := ∑_{i} ∫_{∂M}(p_{i}f(x), s_{i}g(x))_{X_{i}|_{x}}dx and similarly for the other expressions. It follows especially that B is the formal adjoint of A, i.e. (Af, g) = (f, Bg) for f, g ∈ C_{0}^{∞}(M^{∘}). Also, we call (B, q⃗) the formal adjoint system to (A, p⃗) (with respect to the so called Greenian formula (3.35)).

3.36. Theorem. For every Dirichlet system (A, p⃗, t⃗) a unique adjoint system (B, s⃗, q⃗) exists.

Proof. Uniqueness: Only considering f with supp f ∩ ∂M = ∅ (which are dense in L^{2}(E)), we see that B must be the (unique!) formal adjoint of A. Let (B, s⃗, q⃗) and (B, s′, p′) be two adjoint systems. By lemma 3.32 and the Greenian formula:

\begin{matrix}(φ, s⃗g) − (ψ, q⃗g) = (φ, s′g) − (ψ, q′g) ∀φ, ψ ∈ C_{0}^{∞} \\ ⟹ s⃗ = s′ q⃗ = q′ \end{matrix}

Existence: Using charts and a partition of unity, it suffices to construct the adjoint locally. Here, ordinary integration by parts yields

(Af, g)_{L^{2}(F)} − (f, Bg)_{L^{2}(E)} = (D⃗f, N⃗g)_{∂M}

for g with support in a chart neighbourhood. (D⃗ is defined in the proof of 3.31). The explicit calculations can be found in Lions/Magenes [40, p. 218]. Now, by 3.31 D⃗ = Φ(p⃗⊕t⃗) with a tangential d.o. Φ. Let Φ^{∗} denote the formal adjoint of Φ (as an operator on ∂M). Then

(Af, g) − (f, Bg) = (D⃗f, N⃗g) = (p⃗ ⊕ t⃗f, Φ^{∗}N⃗g) = (p⃗f, (Φ^{∗}N⃗)_{1}g) + (t⃗f, (Φ^{∗}N⃗)_{2}g)

As in the proof of Lions/Magenes one checks that (s⃗, q⃗) := ((Φ^{∗}N⃗)_{1}, (Φ^{∗}N⃗)_{2}) has the required properties. By uniqueness, these locally defined operators patch together to the global operators we need.

3.37. Remark. Note that the adjoint (B, p⃗) to (A, p⃗) is not unique (there are many ways to complete p⃗ to a Dirichlet system).

3.38. Lemma. Suppose the manifold and all the bundles in question have bounded geometry. Let (A, p⃗, t⃗) and (B, s⃗, p⃗) be formally adjoined to each other in the sense of 3.34. Then

(Af, g) − (f, Bg) = (p⃗f, s⃗g) − (t⃗f, q⃗g) ∀f ∈ H^{μ}(E), g ∈ H^{μ}(F).

Proof. This follows from the fact that C_{0}^{∞} is dense in H^{μ} and from the boundedness A : H^{μ}(E) → L^{2}(F), p⃗ : H^{μ}(E) → L^{2}(⨁_{i} X_{i}), …

3.39. Lemma. Let (A, p⃗, s⃗) and (B, q⃗, t⃗) be Dirichlet boundary value problems with formal adjoint (A^{t}, s^{t}, p^{t}) and (B^{t}, t⃗^{t}, q^{t}), respectively. Moreover, suppose A : C^{∞}(E) → C^{∞}(F) and B : C^{∞}(F) → C^{∞}(G) are elliptic. Then the composition

(BA, q⃗A, t⃗A, p⃗, s⃗)  (3.40)

is a Dirichlet boundary value problem with formal adjoint (A^{t}B^{t}, s^{t}B^{t}, p^{t}B^{t}, t⃗^{t}, q^{t}). The formal adjoint of (BA, q⃗A, p⃗) with resp. to the Greenian formula induced from (3.40) is (A^{t}B^{t}, p^{t}B^{t}, q^{t}).

Proof. The compositions are Dirichlet boundary value problems by the ellipticity of A and B^{t} which especially says that the principal symbol evaluated at a normal non-zero cotangent vector at the boundary is invertible. Straightforward calculations show that the two systems are adjoint to each other.

Approximation with prescribed boundary values

From now on, we return to manifolds and bundles of bounded geometry. We will construct functions with prescribed boundary values (of order > k) which are small in the Sobolevnorm H^{k}. In some arguments, then we can replace arbitrary functions by functions which have a nice behaviour near the boundary. First, we consider the Euclidian case, then we use the local results to deal with manifolds of bounded geometry. The local result is probably well known, but I could not find a reference in the literature. 3.41. Lemma. Let k ∈ ℕ be given. For any ϵ > 0 and collection f_{j} ∈ C_{0}^{∞}(ℝ^{m−1}) with supp f_{j} ⊂ [−R, R]^{m−1}, j = k, k + 1, …; f_{j} = 0 for j = 0, …, k − 1 we can find f_{ϵ} ∈ C_{0}^{∞}(ℝ_{≥0}^{m}) with

\frac{∂^{j}f_{ϵ}}{(∂x_{m})^{j}} |_{x_{m}=0} = f_{j} for j ≥ 0  and |f_{ϵ}|_{H^{k}} < ϵ.

Proof. By [69] there exists f ∈ C_{0}^{∞}(ℝ^{m}) with ∂_{x_{m}}^{j}f = 0 for j < k and ∂_{x_{m}}^{j}f = f_{j} for j ≥ k. We may assume that supp f ⊂ [−2R, 2R]^{m}. Now take a cutoff function λ ∈ C_{0}^{∞}(ℝ) with λ(t) ≡ 1 for t ≤ 0 and λ(t) ≡ 0 for t ≥ 1. Set λ_{n}(t) := λ(nt).

Set M := max{|λ^{(r)}(t)|; t ∈ [0, 1], r = 0, …, k} and observe λ_{n}^{(r)}(t) = n^{r}λ^{(r)}(nt) ⟹ |λ_{n}^{(r)}(t)| ≤ M · n^{r} ∀t ∈ ℝ.

Set f_{ϵ}(x) := f(x) · λ_{N}(x_{m}) for sufficiently large N (we will specify N in a moment). Then by the choice of λ we have ∂_{x_{m}}^{j}f_{ϵ}|_{x_{m}=0} = ∂_{x_{m}}^{j}f|_{x_{m}=0}, thus the boundary conditions are fulfilled.

|f_{ϵ}|_{H^{k}}^{2} = ∑_{|α|≤k} |D^{α}f_{ϵ}|_{L^{2}}^{2} ≤^{f_{ϵ}=fλ_{N}} 2^{k} ( ∑_{|α|≤k} ∑_{r=0}^{α_{m}} |(D^{α−re_{m}}f) · λ_{N}^{(r)}|_{L^{2}}^{2})

Now we make use of the fact that D_{x_{m}}^{j}f = 0 for j < k. Then, by Taylor expansion

f(x) = x_{m}^{k}ψ(x) with ψ ∈ C_{0}^{∞}(ℝ^{m}).

Let W = max{|D^{α}ψ(x)|; x ∈ ℝ_{≥0}^{m}; |α| ≤ k}. Then

|D^{α}f(x)| ≤ |x_{m}|^{k−α_{m}} · \underbrace{k!2^{k}W}_{=:W′} ∀x ∈ ℝ_{≥0}^{m} for |α| ≤ k.

Therefore,

|f_{ϵ}|_{H^{k}}^{2} ≤ 2^{k} ∑_{|α|≤k} ∑_{r=0}^{α_{m}} |x_{m}^{k−(α_{m}−r)} · χ_{[0,1/N]×[−2R,2R]^{m−1}} |_{L^{2}}^{2} · N^{r}W′M (as λ_{N}(t) = 0 if t > \frac{1}{N})

= 2^{k}W′M(4R)^{m−1} ∑_{|α|≤k} ∑_{r=0}^{α_{m}} ∫_{0}^{1/N} x_{m}^{2(k−(α_{m}−r))} · N^{r} dx_{m}

= 2^{k}W′M(4R)^{m−1} ∑_{|α|≤k} ∑_{r=0}^{α_{m}} \frac{N^{r−2k+2α_{m}−2r−1}}{2(k − α_{m} + r) + 1}

≤^{α_{m}≤k} 2^{k}W′M(4R)^{m−1} ∑_{|α|≤k} ∑_{r=0}^{α_{m}} N^{−1} ≤ ϵ^{2} for N chosen appropriately.

3.42. Theorem. Let E↓M and X_{i}↓∂M be bundles of bounded geometry. Suppose

p⃗ = (p_{0}, p_{1}, …) : C^{∞}(E) → C^{∞}(X_{0}) ⊕ C^{∞}(X_{1}) ⊕ ⋯

is an infinite Dirichlet system and k ∈ ℕ. Then, to ϵ > 0 and

u⃗ = (0, …, 0, u_{k}, u_{k+1}, …) with u_{i} ∈ C_{0}^{∞}(X_{i}) and supp u_{i} ⊂ K

(K a fixed compact set) we find u ∈ C_{0}^{∞}(E) with

p_{i}u = u_{i} ∀i < l so that |u|_{H^{k}(E)} < ϵ.

3.43. Remark. The theorem is true for k = 0 if M is an arbitrary complete Riemannian manifold.

Proof. Let {φ_{i}}_{i∈ℕ} be an admissible partition of unity subordinate to an admissible covering of M, used to define the Sobolev norm on E (following lemma 3.22). Let κ_{i} : ℝ^{m} → M be the charts and t_{i} : ℝ^{m} × 𝕂^{n} → E the bundle trivialisations. We will produce u_{i} with support in the chart neighbourhood of φ_{i} which solves the problem for φ_{i}u⃗ and with |u_{i}|_{H^{k}(E)} < ϵ·2^{−i}. Then u = ∑_{i} u_{i} has the required properties by the linearity of p⃗ and the triangle inequality. To find u_{i} we can entirely work in admissible coordinates. We have to be careful about Sobolev norms because |u_{i}|_{H^{k}(E)} and |u_{i}|_{H^{k}(ℝ^{m})} do not coincide. But:

3.44. Lemma. In the situation of the theorem, there is an E-universal C_{k} > 0 so that if supp u is contained in the range of a normal chart κ_{i} then

|u|_{H^{k}(E)}^{2} ≤^{C_{k}} |t_{i}u ∘ κ_{i}|_{H^{k}(ℝ^{m})}^{2} ≤^{C_{k}} |u|_{H^{k}(E)}^{2}.

Proof of the lemma. Suppose N is the multiplicity of the covering.

|u|_{H^{k}(E)}^{2} = ∑_{j}|t_{j}^{−1}(φ_{j}u) ∘ κ_{j}|_{H^{k}(ℝ^{m})}^{2} = ∑_{j}|(t_{j}^{−1}u ∘ κ_{j}) · (φ_{j} ∘ κ_{j})|_{H^{k}(ℝ^{m})}^{2}

= ∑_{j ≤ N + 1 summands}|(t_{j}^{−1}t_{i}t_{i}^{−1}u ∘ κ_{i}κ_{i}^{−1}κ_{j}) · (φ_{j} ∘ κ_{j})|_{H^{k}(ℝ^{m})}^{2}

≤^{C(N+1)} |t_{i}^{−1}uκ_{i}|_{H^{k}(ℝ^{m})}^{2} = |u|_{H^{k}(ℝ^{m}).}^{2}

Note that composition with the diffeomorphism κ_{i}^{−1}κ_{j} is H^{l}-bounded with norm bounded by polynomials in derivatives up to order k of κ_{i}^{−1}κ_{j} and its inverse. These bounds are provided by lemma A.22. Composition with the bundle isomorphism t_{j}^{−1}t_{i} is H^{k}-bounded with norm bounded by polynomials in derivatives of t_{j}^{−1}t_{i} and its inverse up to order k (these norms are given by definition). Multiplication with a smooth function is bounded with norm bounded by polynomials in derivatives up to order l of the function (bounds given by lemma 3.22). This implies that C can be chosen independently of u.

Now, we continue the proof of the theorem. In view of the lemma we may actually work in Euclidian space. Here we find Φ and Ψ with p⃗ = ΦD⃗ and D⃗ = Ψp⃗ so that Φ and Ψ are upper triangular (compare lemma 3.31). Set v⃗ = Ψu⃗. Since Ψ is triangular v_{0} = 0 = ⋯ = v_{k−1}. Now D_{j} = 1_{n}∂_{x_{m}}^{j} is diagonal. Therefore, the result for functions (lemma 3.41) yields the existence of a section u with D⃗u = v⃗ and |u|_{H^{k}(ℝ^{m})} ≤ ϵ. But p⃗u = ΦD⃗u = Φv⃗ = ΦΨu⃗ = u⃗.

4 Ellipticity and bounded geometry

This section deals with elliptic boundary value problems. We define ”uniform ellipticity” as the right condition for manifolds of bounded geometry. We obtain elliptic regularity results and an a priori estimate for solutions of uniformly elliptic boundary value problems, and we give a sufficient condition for essential self adjointness of elliptic unbounded operators (defined on the kernel of the boundary operator).

In this section, every manifold and bundle shall be bounded as well as every operator, if not stated otherwise. Some of the computations to check uniformity are quite technical. However, if M is a covering of a compact manifold and the operators are lifted, then all these results are fullfilled trivially (f.i. 4.3, 4.10, 4.11, 4.14, 4.16).

4.1. Notation. If M is a manifold of bounded geometry, x′ ∈ ∂M, then

Z(x′, ϵ) := K(\underbrace{B(x′, ϵ)}_{⊂∂M} ×[0, ϵ))

is the cylinder with height and radius ϵ (K is the geodesic collar). Note that Z(x, ϵ) ⊂ ℝ^{m} is mapped to Z(x, ϵ) ⊂ M under a normal collar chart.

4.2. Definition. A bounded differential operator A : C^{∞}(E) → C^{∞}(F) of order μ ≥ 1 between k-dimensional bundles of bounded geometry on a manifold M is called uniformly elliptic if for its principal symbol a_{μ}(x, ξ) in admissible coordinates we have

(1) the matrix a_{μ}(x, ξ) is invertible for all (x, ξ) with ξ ≠ 0 (ellipticity)

(2) There is an M-universal constant C so that ∥a_{μ}^{−1}(x, ξ)∥ · |ξ|^{μ} ≤ C ∀(x, ξ)

We use the norms induced by admissible trivialisations.

As an example we consider the formal adjoint of a differential operator:

4.3. Lemma. Suppose A : C^{∞}(E) → C^{∞}(F) is bounded, where E and F are bounded Riemannian bundles. Then the formal adjoint A^{t} : C^{∞}(F) → C^{∞}(E) is also bounded. If A is uniformly elliptic then A^{t} is uniformly elliptic, too.

Proof. The coefficients of A^{t} in admissible coordinates are polynomials in derivatives up to order μ of the Riemannian metric tensors and the coefficients of A. This implies directly that A^{t} is bounded. For uniform ellipticity, note that it is well known that if A is elliptic then so is A^{t}. The principal symbol of A^{t} is just the adjoint of the principal symbol of A composed with the Riemannian tensors of E and F and their inverse, respectively. Since the norm of a matrix and its adjoint are equal we get the desired uniformity.

Now we will define uniformly elliptic boundary value problems. The idea is, as for elliptic differential operators, to copy the local definition (for elliptic boundary value problems) and to require uniformity so that local constructions patch together to global ones on manifolds of bounded geometry. The definition of a (local) elliptic boundary value problem is the following (Hörmander [36, 10.6.2])

4.4. Definition. A boundary value problem (A, p) : C^{∞}(E) → C^{∞}(F) ⊕ C^{∞}(X) is elliptic, if the following conditions are fulfilled:

• A is an elliptic operator of order μ

• p is a normal system of boundary conditions

• the homogeneous local frozen boundary value problem (transported to ℝ^{m} via a chart)

∑ A_{ij}^{0}(x_{0}, D)u_{j} = 0 in ℝ_{≥0}^{m} ∑ p_{ij}^{0}(x_{0}, D)u_{j} = 0 in ℝ^{m−1}

has no solution of the form

u_{j}(x) = e^{i(x,ξ′)}w_{j}(x_{n}) with ξ′ ≠ 0

which is bounded in ℝ_{≥0}^{m}. (A^{0} denotes the principal part).

• μ dim E = 2 dim X and the order of p_{ij} with respect to D_{n} is smaller than μ.

4.5. Remark. Normality of p actually follows from the rest.

4.6. Remark. There are other (equivalent) definitions for elliptic boundary value problems, compare f.i. Atiyah [48, Appendix I].

Unfortuantly, it is not easy to see how one could require uniformity in this definition. Instead, we use the existence of fundamental solutions for elliptic boundary value problems and pose uniformity conditions on these. This is technically convenient because we will work exactly with these fundamental solutions. On the other hand, the conditions are hard to check. In proposition 5.13 we will give an ellipticity condition which can be checked more directly on the coefficients of the operators in local coordinates.

4.7. Definition. Let (A, p_{0}, …, p_{r}) : C^{∞}(E) → C^{∞}(F) ⊕ C^{∞}(X_{0}) ⊕ ⋯ ⊕ C^{∞}(X_{r}) be a differential boundary value problem of order μ ≥ 1, where E, F are bundles of bounded geometry over M and X_{i} are bundles of bounded geometry over ∂M. Then (A, p⃗) is called uniformly elliptic if it is an elliptic boundary value problem in the classical sense 4.4, if A is a uniformly elliptic d.o. and if the following holds: Since (A, p⃗) is elliptic there exists ∀x ∈ ∂M a 0 < ϵ_{x} < r_{C} and a bounded fundamental solution (compare Hörmander [36, Theorem 10.4.1])

R_{x} : L^{2}(F |_{Z(x,ϵ_{x})}) ⊕ ⨁_{i=0}^{r} H^{μ−i−1/2}(X_{i}|_{∂M∩Z(x,ϵ_{x})}) → H^{μ}(E|_{Z(x,ϵ_{x})}).

Set

M^{k} := H^{k}(F |_{Z(x,ϵ_{x})}) ⊕ ⨁_{i=0}^{r} H^{k+μ−i−1/2}(X_{i}|_{∂M∩Z(x,ϵ_{x})}).

The fundamental solution fulfills

(1) (A, p⃗) ∘ R_{x}(F⃗) ≡ F⃗ on Z(x, ϵ_{x})

(2) R_{x} ∘ (A, p⃗)u = u if supp u ⊂ Z(x, ϵ_{x}) is compact and u ∈ H^{μ}(E|_{Z(x,ϵ_{x})}).

For uniform ellipticity we require that it is possible to choose ϵ independently of x and R_{x} in a way that the norm of R_{x} is bounded uniformly in x.

4.8. Remark. It suffices to find a corresponding set of fundamental solutions on a covering by charts as in lemma 3.22.

4.9. Remark. If (A, p_{0}, …, p_{r}) is an elliptic boundary value problem of order μ, ellipticity implies that r ≤ μ − 1, i.e., without loss of generality, we may assume r = μ − 1.

The next lemma implies that for a uniformly elliptic boundary value problem, not only the induced map H^{μ} → L^{2} has uniform fundamental solutions, but every H^{μ+k} → H^{k}. We will need this property if we compose such boundary value problems.

4.10. Lemma. Let (A, p⃗) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁ C^{∞}(X_{i}) be an elliptic boundary value problem of order μ with local fundamental solution R : M^{0} → H^{μ}(E|_{Z(x_{0},ϵ)}) near a point x_{0} ∈ ∂M and suppose k ∈ ℕ. Let ϵ′ < ϵ′′ < ϵ and ψ be a smooth cutoff function with ψ ≡ 1 on Z(x_{0}, ϵ′) and ψ ≡ 0 outside Z(x_{0}, ϵ′′). Set R_{1} = ψR which is also a fundamental solution of the given boundary value problem. Then: If f⃗ ∈ H^{k}(F) ⊕ ⨁_{i} H^{k+μ−i−1/2}(X_{i}) and supp f⃗ ⊂ Z(x_{0}, ϵ′) then R_{1}(f) ∈ H^{k+μ}(E) and

|R_{1}f⃗|_{H^{k+μ}} ≤ C · |f⃗|_{M^{k}}.

The constant C depends only on the operator norm ∥R∥_{B(M^{0},H^{μ})}, on the derivatives of ψ and the coefficients of A and p⃗ up to order k and on the ellipticity constant (bounding the inverse of the principal symbol of A).

4.11. Corollary. In view of the preceding lemma we may suppose, without loss of generality, that the local fundamental solutions in the definition of uniformly elliptic boundary value problems are M-uniformly bounded maps from M^{k} → H^{μ+k}(E|_{Z(x,ϵ)}) ∀k, that their range consists of functions with compact support, and that they fulfill properties of an inverse (compare definition 4.7) for section with support in Z(x, ϵ/2).

Proof of the lemma. (Induction on k) k = 0 is the definition of uniform ellipticity. Suppose that the proposition is true for 0 ≤ i < k. Choose a cutoff function ψ_{2} with ψ_{2} ≡ 1 on Z(x_{0}, ϵ′′), i.e. especially on supp ψ and which is zero outside another cylinder in Z(x_{0}, ϵ). Then the induction hypothesis applies to the parametrix R_{2} = ψ_{2}R as well. By [36, p. 270] if g ∈ H^{μ}(E), supp g ∈ Z(x, ϵ) and Pg ∈ M^{k} then g ∈ H^{k+μ}(E). In our case take f⃗ ∈ M^{k} with supp f⃗ ∈ Z(x_{0}, ϵ′). Obviously R_{1}f⃗ ∈ H^{μ}(E) and we can apply Hörmander’s lemma. Now

PR_{1}f⃗ = f⃗ + P′Rf⃗  (4.12)

where P′ = Pψ − ψP is a differential operator of order ≤ μ − 1 with support in Z(x, ϵ′′). Especially, P′R = P′R_{2}. By induction R_{2}f⃗ ∈ H^{k+μ−1}(E) ⟹ P′Rf⃗ ∈ M^{k}. Since f⃗ lies in the same space by assumption, Hörmanders lemma reveals R_{1}f⃗ ∈ H^{k+μ}(E). Now for the norm (remember that we work in admissible coordinates) (set D_{i} = ∂/∂_{x_{i}}):

|R_{1}f⃗|_{H^{μ+k}}^{2} = ∑_{|α|≤1}|D^{α}R_{1}f⃗|_{H^{μ+k−1}}^{2} = |D_{m}R_{1}f⃗|_{H^{μ+k−1}}^{2} + ∑_{i≠m}|D_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

≤ ∑_{α≤μ−1}|D^{α}D_{m}R_{1}f⃗|_{H^{k}}^{2} + ∑_{i≠m}|D_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

≤^{μ} |D_{m}^{μ}R_{1}f⃗|_{H^{k}}^{2} + ∑_{\begin{matrix}i≠m \\ |α|≤μ−1 \end{matrix}}|D^{α}D_{i}R_{1}f⃗|_{H^{k}}^{2} + ∑_{i≠m}|D_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

≤^{2+K} |b(x)AR_{1}f⃗|_{H^{k}}^{2} + |(b(x)A − D_{m}^{μ})R_{1}f⃗|_{H^{k}}^{2} + ∑_{i≠m}|D_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

(Here b(x) = a_{μ}(x, (0, …, 0, 1))^{−1}, where a_{μ} is the principal symbol of A, so that the coefficient of D_{m}^{μ} in b(x)A is 1. Uniform ellipticity and C^{∞}-boundedness of A imply that b(x) and its derivatives up to a given order are uniformly bounded by the coefficients of A and the ellipticity constant, independently of the basepoint x_{0}. K ≤ m^{μ−1} is due to the second summand.)

≤^{C(b(x))} |AR_{1}f⃗|_{H^{k}}^{2} + ∑_{\begin{matrix}i≠m \\ |α|≤μ−1 \end{matrix}}|ba_{α+e_{i}}D^{α}D_{i}R_{1}f⃗|_{H^{k}}^{2} + ∑_{|α|≤μ−1}|ba_{α}D^{α}R_{1}f⃗|_{H^{k}}^{2}

+ ∑_{i≠m}|D_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

(multiplication by b(x) is an operator of order zero)

≤^{C(b,a_{α})+K} |PR_{1}f⃗|_{M^{k}}^{2} + ∑_{i≠m}|R_{2}PD_{i}R_{1}f⃗|_{H^{μ+k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

(|Pg|_{M^{k}}^{2} = |Ag|_{H^{k}}^{2} + ⋯, and if supp g ⊂ Z(x_{0}, ϵ′′) then R_{2}Pg = g and here supp R_{1}f⃗ ⊂ Z(x_{0}, ϵ′′))

≤^{∥R_{2}∥_{B(M^{k−1},H^{μ+k−1})}^{2}} |PR_{1}f⃗|_{M^{k}}^{2} + ∑_{i≠m}|PD_{i}R_{1}f⃗|_{M^{k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

(induction hypothesis applied to R_{2})

≤^{2} |PR_{1}f⃗|_{M^{k}}^{2} + ∑_{i≠m}|D_{i}PR_{1}f⃗|_{M^{k−1}}^{2} + ∑_{i≠m}|P_{i}R_{1}f⃗|_{M^{k−1}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

(P_{i} is defined by PD_{i} = D_{i}P + P_{i} and is an operator of order μ. Here we need i < m because P is an abbreviation for many operators, some of which are boundary differential operators. Therefore, D_{i}P is only defined for a tangential d.o. D_{i}.)

1+m+∑∥P_{i}∥_{B(H^{μ+k−1},M^{k−1})} ; ≤ |PR_{1}f⃗|_{M^{k}}^{2} + |R_{1}f⃗|_{H^{μ+k−1}}^{2}

≤^{2+∥R_{1}∥_{B(M^{k−1},H^{μ+k−1})}^{2}} |f⃗|_{M^{k}}^{2} + |P′R_{2}f⃗|_{M^{k}}^{2} + |f⃗|_{M^{k−1}}^{2}

(PR_{1}f⃗ = f⃗ + P′R_{2}f⃗ by (4.12) and induction hypothesis applied to R_{1})

≤^{2+∥P′∥_{B(H^{k+μ−1},M^{k})}^{2}} |f⃗|_{M^{k}}^{2} + |R_{2}f⃗|_{H^{k+μ−1}}^{2} (The operator P′ has order μ − 1)

≤^{2+∥R_{2}∥_{B(M^{k−1},H^{μ+k−1})}^{2}} |f⃗|_{M^{k}}^{2} (induction hypothesis applied to R_{2})

It remains to check the constants. Here (by induction) only the norms of P′ and P_{i} are critical. And these are clearly bounded by derivatives of coefficients of P.

4.13. Example.

• If we deal with a covering manifold M̃↓M with pullback bundles then the lift of an elliptic differential operator and boundary value problem resp. is clearly uniformly elliptic.

• The same is true for any G-equivariant elliptic d.o. on a homogeneous space G/H, e.g. for the Laplacian.

Small perturbations of uniformly elliptic boundary value problems remain uniformly elliptic, as shows the next theorem:

4.14. Theorem. Suppose P = (A, p⃗) (notation following 3.27 and 4.7) is uniformly elliptic. Suppose Q = (B, q⃗) is another boundary value problem which yields a bounded map Q : H^{μ}(E) → M^{0} such that the norm of this map is small enough. Suppose the C_{b}^{μ}-norm of the coefficients of Q is sufficiently small (this implies the previous condition). Then P + Q remains uniformly elliptic.

Proof. It is a classical result that the new operator is elliptic. Uniform ellipticity of A + B will not be shown (follows form the condition on the coefficients). We have to construct local fundamental solutions. This will be done using purely functional analytical methods. We give only the key ideas (with lax notation): Let R be a fundamental solution of P. Suppose the (local) norm of Q is smaller than half the one of R. Then the von Neumann sum ∑_{k=0}^{∞}(−RQ)^{k} converges in the Banach space of bounded maps on H^{μ}(E|). Define the fundamental solution

S = (∑_{k=0}^{∞}(−RQ)^{k}) R,

whose norm is twice the norm of R at the most. We have to check only that this is a fundamental solution for P + Q. If supp u is sufficiently small then

S(P + Q)u =^{RPu=u} ∑_{k}(−RQ)^{k}(RQu + u) =^{von Neumann series} u.

On the other hand, for arbitrary F⃗,

\begin{aligned}(P + Q)SF⃗ & = (P(RQ + 1) + (1 − PR)Q) ∑_{k}(−RQ)^{k}RF⃗ \\ & = \underbrace{PRF⃗}_{= F⃗ on …} + \underbrace{(1 − PR)}_{= 0 on …}(Q ∑_{k}(−RQ)^{k}RF⃗) ≡ F⃗ on … \end{aligned}

We will now derive elliptic regularity, the main result on uniformly elliptic boundary value problems. The corresponding result for compact manifolds is well known, so it is not surprising that it holds under suitable uniformity conditions also for non-compact manifolds.

4.15. Theorem. (elliptic regularity) Let P = (A, p⃗) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁_{i} C^{∞}(X_{i}) be uniformly elliptic of order μ (notations follow definition 3.27). If u ∈ H^{t}(E) and P(u) ∈ M^{s−μ} (here M^{s} = H^{s}(F) ⊕ ⨁_{i} H^{s+μ−i−1/2}(X_{i}) denotes global sections) then

u ∈ H^{s}(E) and |u|_{H^{s}(E)}^{2} ≤ C_{t,P} · (|Pu|_{M^{s−μ}}^{2} + |u|_{H^{t}(E)}^{2})

with C_{t,P} independent of u. The inequality is called the a priori estimate for uniformly elliptic boundary value problems. Of course, t must be sufficiently large so that P(u) is defined.

Proof. First note that the hypotheses implies ([36, p. 270]) u ∈ H_{loc}^{s}(E), i.e. φu ∈ H^{s} for every smooth function φ with compact support. It remains only to check the norm inequality. Take a normal covering (with centers {x_{i}}) with subordinate admissible partition of unity {φ_{i}} as in lemma 3.22 and so small, that we have local fundamental solutions R_{i}. Let N be the multiplicity of the partition. In the following we abbreviate |·|_{H^{s}} and |·|_{M^{s}} by |·|_{s}.

|u|_{s}^{2} = ∑_{i}|φ_{i}u|_{s}^{2} = ∑_{i}|R_{i}Pφ_{i}u|_{s}^{2}

≤ C ∑_{i}|Pφ_{i}u|_{s−μ}^{2} (uniform ellipticity: condition on fundamental solutions)

≤ 2C(∑_{i}|φ_{i}Pu|_{s−μ}^{2} + ∑_{i}|P_{i} \underbrace{∑_{j} φ_{j}}_{=1} u|_{s−μ}^{2}) (P_{i} are defined by Pφ_{i} = φ_{i}P + P_{i})

≤ 2^{N+2}C(|Pu|_{s−μ}^{2} + ∑_{i} ∑_{j}|P_{i}φ_{j}u|_{s−μ}^{2})

(P_{i} are boundary differential operators of order strictly lower than P with support in supp φ_{i}. Therefore, each sum P_{i} ∑_{j} φ_{j} has only N + 1 non-trivial summands, and this, together with the triangle inequality |a + b|^{2} ≤ 2(|a|^{2} + |b|^{2}), yields the additional factor. As the derivatives of the coefficients of P_{i} are bounded independently of i we get:)

≤ \underbrace{C2^{N+2}(N + 1)(1 + sup_{i}∥P_{i}∥_{B(H^{s−1},M^{s−μ})}^{2})}_{=:C_{0}}(|Pu|_{s−μ}^{2} + ∑_{j}|φ_{j}u|_{s−1}^{2})

≤ C_{0}(|Pu|_{s−μ}^{2} + \frac{1}{2C_{0}} \underbrace{∑_{j}|φ_{j}u|_{s}^{2}}_{=|u|_{s}^{2}} +C_{t} \underbrace{∑_{j}|φ_{j}u|_{t}^{2}}_{=|u|_{t}^{2}})

Here we used the fact that there is an E-universal constant C_{t} (depending on s, t and on C_{0}) so that

|f|_{s−1}^{2} ≤ \frac{1}{2C_{0}} |f|_{s}^{2} + C_{t}|f|_{t}^{2} ∀f ∈ H^{s}(E)

The a priori estimate follows.

4.16. Proposition. The composition of elliptic boundary value problems, as in 3.28, is elliptic again (this is valid for arbitrary bundles); the composition of uniformly elliptic ones remains uniformly elliptic.

Proof. First we prove ellipticity: The operator BA is elliptic because

det((BA)^{0}(ξ)) = det(B^{0}(ξ)A^{0}(ξ)) = det(B^{0}(ξ)) det(A^{0}(ξ)) ≠ 0

for ξ ≠ 0, where A^{0}(ξ) denotes the principal symbol of the differential operator A.

Following definition 10.6.2 of Hörmander [36], it remains to check that no bounded exponential solution u of the homogenous boundary value problem

(B ∘ A)^{0}u = 0; p^{0}u = 0, (qA)^{0}u = 0  (4.17)

exists. (The superscript 0 always will refer to the principal part.) By Hörmander [36, 10.1.1], this is the case if, for some N ∈ ℕ, every solution u ∈ C^{N} is actually smooth:

Take N = deg(B ∘ A). Suppose u ∈ C^{N} is a solution of (4.17). Then A^{0}u ∈ L_{loc}^{2} is a solution of the elliptic boundary value problem (B^{0}, q^{0}). Elliptic regularity implies A^{0}u ∈ C^{∞}. Since also p^{0}u = 0 ∈ C^{∞} and (A^{0}, p^{0}) is elliptic the regularity theorem implies u ∈ C^{∞} as desired. Now we check for uniformity. Clearly B∘A is a uniformly elliptic differential operator. The conditions on the parametrix are fulfilled by corollary 4.11 and the elementary observation that

R ∘ (S ⊕ id) : L^{2}(G) ⊕ ⨁_{j} H^{μ′−j−1/2}(Y_{j}) ⊕ ⨁_{i} H^{μ′+μ−i−1/2}(X_{i}) → H^{μ+μ′}(E)

is a local parametrix, where R is a parametrix for (A, p⃗) and S one for (B, q⃗).

Now we will derive some regularity theorems for adjoint boundary value problems (compare 3.34). We start with a theorem of Hörmander [36, theorem 10.4.3 and p. 270], describing the local situation:

4.18. Theorem. Let M be a Riemannian manifold and E, F↓M, X_{i}↓∂M be bundles with Riemannian metrics. Suppose (A, p⃗) is an elliptic boundary value problem as in definition 4.7. (We do not require any form of boundedness). Suppose f ∈ H_{loc}^{t}(F), g ∈ H_{loc}^{s}(E) and T_{i} ∈ H_{loc}^{−∞}(X_{i}). If

(f, Aφ) + ∑_{i}(T_{i}, p_{i}φ) = (g, φ) ∀φ ∈ C_{0}^{∞}(E)

then f ∈ H_{loc}^{s+μ}, A^{∗}f = g and T_{i} ∈ H_{loc}^{s+i+1/2}. Here A^{∗} is the formal adjoint of A and the pairings are extensions of the L^{2}-inner products for smooth sections with compact support (compare the corresponding property of Sobolev spaces).

We could now also derive global results (f in the Hilbert space H^{s+μ}), but we will not do so because the a priori estimate, which we have already derived, will be sufficient for our purposes.

We will need the following lemma to apply Hörmander’s theorem 4.18 in concrete situations.

4.19. Lemma. Again we do not require that the boundary value problem is bounded. Let (A, p⃗) be an elliptic boundary value problem as in 4.7. Suppose f ∈ L^{2}(F), g ∈ L^{2}(E) so that

(Au, f) = (u, g) ∀u ∈ C_{0}^{∞}(E) with p⃗u = 0  (4.20)

To arbitrary u ∈ C_{0}^{∞}(E) choose u_{i} ∈ C_{0}^{∞}(E) with p_{i}u_{i} = p_{i}u and p_{j}u_{i} = 0 if j ≠ i (this is possible by lemma 3.32 since p⃗ is normal). Set

T_{i}(p_{i}u) = (u_{i}, g) − (Au_{i}, f).

Then T_{i} ∈ H_{loc}^{−∞}(X_{i}). Especially

(Au, f) + ∑_{i=0}^{r} \underbrace{(T_{i}, p_{i}u)}_{=T_{i}(p_{i}u)} = (u, g)  (4.21)

and, by Hörmander’s theorem 4.18, f ∈ H_{loc}^{μ} and A^{∗}f = g.

Proof. It follows directly from (4.20) that T_{i} is well defined. Obviously T_{i} is linear. It remains to show that for an arbitrary real-valued C_{0}^{∞}-function φ (without loss of generality with support in some chart neighbourhood with bundle trivialisation) the functional φ · T_{i} is H^{N}-bounded for some N ∈ ℕ (then φT_{i} ∈ H^{−N} by the corresponding duality theorem).

Extend p⃗ to a Dirichlet system p⃗ ⊕ s⃗. By lemma 3.31 we find Ψ and Φ so that D⃗ = Ψ(p⃗ ⊕ s⃗) and p⃗ ⊕ s⃗ = ΦD⃗. Set α^{i} := Ψ(0, …, 0, φp_{i}u, 0, …, 0). By our condition on φ we can work in Euclidian space via a chart. We consider α^{i} as a tupel of functions on ℝ^{m−1} with values in 𝕂^{n}.

Fix v_{k} : [0, r_{C}) → ℝ with ∂_{t}^{j}v_{k}(t)|_{t=0} = δ_{kj}. Set

f_{α^{i}} := ∑_{k} α_{k}^{i}(x′) · v_{k}(x_{m}); x′ ∈ ℝ^{m−1}

⟹ D_{k}f_{α^{i}}|_{x_{m}=0} = ∂_{x_{m}}^{k}f_{α^{i}}|_{x_{m}=0} = α_{k}^{i} ⟹ D⃗f_{α^{i}} = α^{i}

⟹ (p⃗ ⊕ s⃗)(f_{α^{i}}) = ΦD⃗(f_{α^{i}}) = ΦΨ(0, …, φp_{i}u, …, 0) = (0, …, φp_{i}u, …).

Especially f_{α^{i}} can be used to compute T_{i}(φp_{i}u). Now

|T_{i}(φp_{i}u)|^{2} = |(f_{α^{i}}, g) − (Af_{α^{i}}, f)|^{2}

≤^{2} |f_{α^{i}}|_{L^{2}}^{2} · |g|_{L^{2}}^{2} + |Af_{α^{i}}|_{L^{2}}^{2} · |f|_{L^{2}}^{2} ≤^{|g|^{2}+|f|^{2}∥A∥^{2}} |f_{α^{i}}|_{H^{μ}}^{2} ≤^{2^{μ}} ∑_{k}|α_{k}^{i}(x′)v_{k}(x_{m})|_{H^{μ}(ℝ^{m})}^{2}

= ∑_{k} ∑_{j=0}^{μ} ∫_{0}^{r_{C}}|∂_{x_{m}}^{j}α_{k}^{i}(·)v_{k}(x_{m})|_{H^{μ−j}(ℝ^{m−1})}^{2}dx_{m}

≤^{sup_{j,k,xm}|∂_{xm}^{j}v_{k}(x_{m})|^{2}} ∑_{j,k} ∫_{0}^{r_{C}}|α_{k}^{i}|_{H^{μ−j}(ℝ^{m−1})}^{2}

≤^{μr_{C}} ∑_{k}|α_{k}^{i}|_{H^{μ}(ℝ^{m−1})}^{2} = |α^{i}|_{H^{μ}}^{2}

= |Ψ(0, …, φp_{i}u, …, 0)|_{H^{μ}}^{2} ≤^{∥Ψ∥^{2}} |φp_{i}u|_{H^{2μ}}^{2}

≤^{∥φ∥} |p_{i}u|_{H^{2μ}}^{2}

Note that the constants do not depend on p_{i}u. Therefore, we have shown that T_{i} ∈ H_{loc}^{2μ}. Equation (4.21) follows from (4.20) and the fact p⃗(u) = p⃗(∑_{i} u_{i}) (by construction of the u_{i}).

The regularity result allows us to derive a decomposition of L^{2} for formally self adjoint boundary value problems which is identical to the splitting we get if the operators are really adjoint.

4.22. Corollary. Make the same assumptions as in theorem 4.18. Suppose (A, p⃗) and (B, q⃗) are elliptic and formally adjoint with respect to the Greenian formula

(Ag, f) − (g, Bf) = (p⃗g, s⃗f) − (t⃗g, q⃗f) f ∈ C_{0}^{∞}; g ∈ C_{0}^{∞}.

(Here, we require that the orders of p⃗ and q⃗ are in standard form —ord p_{i} = i— but not the orders of s⃗ and t⃗. Otherwise, we adopt the notation of 3.34). Then we have an orthogonal decomposition

L^{2}(F) = \underbrace{\{f ∈ C^{∞}(F) ∩ L^{2}(F); Bf = 0, q⃗f = 0\}}_{=:ker B} ⊕\overline{A \underbrace{\{g ∈ C_{0}^{∞}(E); p⃗g = 0\}}_{=:D_{A}^{∞}}} (4.23)

If the bundles are of bounded geometry and the boundary value problems are uniformly elliptic, the following stronger result holds: Let s^{∞} = (s_{μ}, s_{μ+1}, …) be a collection of boundary differential operators so that p^{∞} := p⃗ ⊕ ⨁ s^{∞} is an infinite normal system. Then

L^{2}(F) = \{f ∈ H^{∞}(F); Bf = 0, q⃗f = 0\} ⊕ \overline{A\{g ∈ C_{0}^{∞}(E); p^{∞}g = 0\}}  (4.24)

Proof. The Greenian formula implies that ker B and A(D_{A}^{∞}) from the splitting (4.23) are orthogonal to each other: if f ∈ ker B and g ∈ D_{A}^{∞} are given, apply the formula to ψf, with ψ = 1 on supp g and compactly supported.

Suppose now that f ∈ L^{2}(F) is orthogonal to AD_{A}^{∞}. Then lemma 4.19 and theorem 4.18 show that Bf = 0 and f ∈ H_{loc}^{∞} = C^{∞}. We only have to show that q⃗f = 0: Choose an arbitrary smooth boundary value t⃗g with compact support. By lemma 3.32 we find g ∈ C_{0}^{∞}(E) with p⃗g = 0 and t⃗g as chosen. Let ψ be a compactly supported cutoff function which is 1 on supp g. Then

(t⃗g, q⃗f) = (t⃗g, q⃗(ψf)) =^{Green} (\underbrace{p⃗g}_{=0}, s⃗(ψf)) + (Ag, ψf) − (g, B(ψf)) = \underbrace{(Ag, f)}_{f⊥AD_{A}^{∞}} −(g, \underbrace{Bf}_{=0}) = 0

Here, we used the fact that all the operator are local differential operators. For (4.24), observe first ker B ⊂ H^{∞}(E) because of the a priori estimates of theorem 4.15. Since A : H^{μ}(E) → L^{2}(E) is bounded, it suffices to show that {g ∈ C_{0}^{∞}(E); p^{∞}g = 0} is H^{μ}-dense in {g ∈ C_{0}^{∞}(E); p⃗g = 0}. Then, the image of the first is L^{2}-dense in the image of the second. But H^{μ}-denseness is just the content of theorem 3.42.

We arrive now at a very important theorem for applications. Here we require bounded geometry. It gives sufficient conditions for the question, when two operators on a ∂- manifold are adjoint in the sense of unbounded operators on Hilbert spaces. Especially, this yields conditions for (essential) self adjointness.

4.25. Theorem. Let (A, p⃗) be uniformly elliptic of order μ with uniformly elliptic formal adjoint (B, q⃗) with respect to the Greenian formula

(Af, g) − (f, Bg) = (p⃗f, s⃗g) − (t⃗f, q⃗g).

(Here, we require that the orders of p⃗ and q⃗ are in standard form —ord p_{i} = i— but not the orders of s⃗ and t⃗. Else, we adopt the notation of 3.34). Let

D_{A} := {f ∈ H^{μ}(E); p⃗f = 0} and D_{A}^{∞} := {f ∈ C_{0}^{∞}(E); p⃗f = 0}.

Consider A as an unbounded operator on L^{2}(E) with domain D_{A}. Denote the same operator but with domain D_{A}^{∞} with A_{∞}. Then: The L^{2}-adjoint of A and of A_{∞} is

A^{t} = A_{∞}^{t} = B with domain D_{B} = {g ∈ H^{μ}(F); q⃗g = 0}.

Analogously, A is the adjoint of B and, therefore, the closure of A_{∞}.

Proof. We get immediately that D_{B} lies in the domain of A^{t} (and of A_{∞}^{t}) and that B and A^{t} coincide on D_{B}. Take therefore f ∈ D(A_{∞}^{t}) ⊃ D(A^{t}) with g = A_{∞}^{t}f. We have to show: f ∈ H^{μ}(F), q⃗f = 0 and g = Bf. By definition of the adjoint

(Aφ, f) = (φ, g) ∀φ ∈ C_{0}^{∞}(E) with p⃗φ = 0.

Lemma 4.19 now says f ∈ H_{loc}^{μ}(E) and Bf = g. Next we show q⃗f = 0:

0 = (Aφ, f) − (φ, g) = (Aφ, f) − (φ, Bf)

=^{formal adjoint} (p⃗φ, s⃗f) − (t⃗φ, q⃗f) =^{p⃗φ=0} −(t⃗φ, q⃗f).

Note that by lemma 3.32 we find a corresponding φ with p⃗φ = 0 to arbitrary smooth t⃗φ with compact support. Since C_{0}^{∞} is dense in L^{2} it follows q⃗f = 0. Now we apply the a priori estimate of theorem 4.15:

|f|_{H^{μ}} ≤^{C} |Bf|_{L^{2}} + ∑_{i}|q_{i}f|_{H^{μ−i−1/2}} = |Bf|_{L^{2}} < ∞.

This concludes the proof.

4.26. Remark. We will see in the next section that theorem 4.25 applies to the Laplacian with Dirichlet/Neumann boundary conditions and yields that this operator is self adjoint. Another result in this direction is the following: For complete manifolds without boundary (not necessarily of bounded geometry) Chernoff [22] shows that the Laplacian on compactly supported smooth forms is essentially self adjoint. He uses finite propagation speed.

Now, we have to recall further (well known) aspects of general Hilbert space theory.

4.27. Definition. An unbounded operator A with domain D on a Hilbert space H is called c-positive (c > 0) if

(Ax, x) ≥ c(x, x) ∀x ∈ D.

It is denoted positive if it is c-positive for some c > 0. If (Ax, x) ≥ 0 ∀x ∈ D then A is called non-negative.

4.28. Lemma. Let A : D → H be a positive self adjoint unbounded operator on a Hilbert space H. Then the natural Hilbert space norm | · |_{D} on D given by |x|_{D}^{2} = |x|_{H}^{2} + |Ax|_{H}^{2} is equivalent to | · |_{A} with |x|_{A} = |Ax|_{H}. Moreover, A : (D, | · |_{A}) → (H, | · |_{H}) is an isometric isomorphism.

Proof. First, we show that the two norms on D are equivalent: Positivity implies

(Ax, x)_{H} ≥^{c} (x, x)_{H} ⟹^{Schwartz I.E.} |Ax|_{H} ≥^{c} |x|_{H} ∀x ∈ D.

⟹ |x|_{A}^{2} = |Ax|_{H}^{2} ≤ |x|_{H}^{2} + |Ax|_{H}^{2} = |x|_{D} ≤^{1+c} |Ax|_{H}^{2} = |x|_{A}^{2} ∀x ∈ D

Especially (D, | · |_{A}) is complete because (D, | · |_{D}) is always complete for a closed operator A : D → H. Moreover,

(im A)^{⊥} = ker A^{∗} =^{A=A^{∗}} ker A =^{A positive} 0.

Therefore, im(A) is dense in H. Since A is an isometry from the Hilbert space (D, | · |_{A}) into (H, | · |_{H}), im(A) is complete and therefore closed. Together, im(A) = H and the proof is finished.

4.29. Lemma. Suppose A : D ⊂ L^{2} → L^{2} is a positive self adjoint differential operator of order μ and D ⊂ H^{μ} is closed. Then the norms | · |_{A} and | · |_{H^{μ}} on D are equivalent, where | · |_{A} was defined in 4.28.

Proof. By lemma 4.28, (D, | · |_{A}) is complete. Moreover, |ω|_{A} = |Aω|_{L^{2}} ≤^{∥A∥} |ω|_{H^{μ}} ∀ω ∈ D because A is an operator of order μ. Since (D, | · |_{H^{μ}}) is also complete the two norms are equivalent by the open mapping theorem.

4.30. Corollary. In the situation of lemma 4.29 A : D → L^{2} is an isometry if we equip D ⊂ H^{μ} with | · |_{A}, which is equivalent to the inherited Sobolev norm.

4.31. Theorem. Let (A, p⃗) be a uniformly elliptic formally self adjoint boundary value problem of order μ. Choose ϵ > 0 small enough so that (ϵ+A, p⃗) is again uniformly elliptic (compare 4.14). Define the boundary system

p^{∞} = (p⃗, p⃗ ∘ (ϵ + A), p⃗ ∘ (ϵ + A)^{2}, …)

which is normal because (ϵ + A, p⃗) is elliptic. Set

D_{A^{k}} := {f ∈ H^{kμ}(E); p⃗f = 0 = ⋯ = p⃗ ∘ (ϵ + A)^{k−1}f} k = 0, 1, 2, …

Suppose A is non-negative, considered as an unbounded operator on D_{A}. Then

(ϵ + A)^{k} : D_{A^{k+l}} → D_{A^{l}}

is an isomorphism of topological vector spaces and

D_{A^{l}} = \{f ∈ H^{∞}(E); Af = 0, p⃗f = 0\} ⊕ \overline{A \underbrace{\{f ∈ H^{∞}(E); p^{∞}f = 0\}}_{=:D_{∞}}}.

Moreover, (ϵ + A)^{k} respects the splitting. The operator (ϵ + A)^{k} with domain D_{A^{k}} is the k-th power of (ϵ + A) with domain D_{A} in the sense of unbounded operators.

Proof. Observe D_{A^{0}} = L^{2}. It suffices to prove the theorem for (ϵ + A)^{k} : D_{A^{k}} → L^{2}. Now (ϵ + A)^{k}, considered as an unbounded operator on L^{2} with domain D_{A^{k}}, is self adjoint by theorems 4.16 and 4.25. Obviously, (ϵ + A) is positive on D_{A}. Then so is its k-th power as unbounded operator, and (ϵ + A)^{k} with domain D_{A^{k}} is obviously a restriction of this operator. By elliptic regularity 4.15 they are equal. Moreover, D_{A^{k}} is the kernel in H^{kμ} of a bounded operator (namely a boundary differential operator of order kμ − 1) and hence closed in H^{kμ}. Therefore, lemma 4.29 applies and yields the isomorphism. By corollary 4.22 we have the desired splitting of L^{2}(E) = D_{A^{0}}. Via the isomorphism (ϵ + A)^{k} we get corresponding splittings of D_{A^{k}}. We have to show only that the inverse images are actually as stated. First note that (ϵ + A) is an isomorphism of {f ∈ C^{∞}(E) ∩ L^{2}(E); Af = 0, p⃗f = 0} onto itself. Consequently, the inverse image of this space is the space itself. For D_{∞} note that the boundary conditions are designed in a way that (ϵ + A)^{k} maps D_{∞} to itself. Suppose, on the other hand, that f ∈ D_{A^{k}} and (ϵ + A)^{k}f ∈ D_{∞}. Then by elliptic regularity f ∈ H^{∞}(E) (note that f ∈ D_{A^{k}} guarantees the required boundary regularity). The boundary conditions on (ϵ + A)^{k}f together with those on f just say f ∈ D_{∞}. We conclude that (ϵ + A)^{k} is an automorphism of D_{∞} and then also of its closures.

5 The Hodge decomposition

In this section we prove Hodge decomposition results. For arbitrary complete ∂-manifolds, we give the splitting of L^{2}(Λ^{p}T^{∗}M) as the direct sum of harmonic forms (which fulfill Dirichlet/Neumann boundary conditions) plus image of d plus image of δ (the domains of the latter fulfill certain boundary conditions, too). Then we derive the decomposition of higher Sobolev spaces for manifolds of bounded geometry (here, we have to use higher boundary conditions). For this, we use the ellipticity theory developed in section 4. To apply the latter, we show that the Laplacian with Dirichlet/Neumann boundary conditions is a uniformly elliptic boundary value problem. (For a Riemannian covering of a compact manifold this is clear and 5.11, 5.13 and 5.14 can be omitted.) It follows that the Laplacian on the space of forms which fulfill these boundary conditions is essentially self adjoint.

Hodge decomposition for complete ∂-manifolds

Let M be a complete oriented Riemannian manifold with ∂M = M_{1} ⨿ M_{2}. Let d be the differential on forms, δ = ± ∗ d∗ its formal adjoint where ∗ is the Hodge-∗-operator. The Laplace operator is Δ = dδ + δd.

5.1. Notation. For a differential form ω, ω|_{M_{1}} means the pullback of ω to the submanifold M_{1} and similarly for other submanifolds.

5.2. Definition. Set Ω^{p}(M) = C_{0}^{∞}(Λ^{p}T^{∗}M).

Define the boundary differential operators

p_{0} : Ω^{∗}(M) → Ω^{∗}(∂M) : ω ↦ (ω|_{M_{1}} + ∗ω|_{M_{2}})
p_{1} : Ω^{∗}(M) → Ω^{∗}(∂M) : ω ↦ (δω|_{M_{1}} + ∗dω|_{M_{2}})
p_{k} : Ω^{∗}(M) → Ω^{∗}(∂M) : ω ↦ \begin{cases}p_{0}((1 + Δ)^{k/2}ω) & if k ≥ 2 even \\ p_{1}((1 + Δ)^{(k−1)/2}ω) & if k ≥ 3 odd \end{cases}

Set

p^{k} = p_{0} ⊕ p_{1} ⊕ ⋯ ⊕ p_{k}; k = 0, 1, 2, …

With the aid of these operators we can now define our (Dirichlet/Neumann) boundary conditions (k ≥ 0)

bd(k + 1)(ω) : ⟺ p^{j}(ω) ≡ 0 ∀j ≤ k
bd(∞)(ω) : ⟺ bd(k)(ω) ∀k ∈ ℕ

By definition bd(0) shall always be true.

5.3. Remark. The idea behind these boundary conditions is the following: bd(1) are the classical absolute/relative boundary conditions for the Laplacian. Namely, (Δ, p^{1}) is formally self adjoint (3.35). bd(k) is defined so that bd(k)(ω) ⟹ bd(k − 2)(ω) and

bd(k + 1)(ω) ⟺ bd(k)(δω) and bd(k)(dω) and bd(k)(ω)

(follows from 5.4). This means that we have nice mapping properties.

Hodge decomposition for complete ∂-manifolds 43

5.4. Proposition. For ω ∈ Ω^{p}(M) we have

bd(k)(ω) ⟺ \begin{cases}ω|_{M_{1}} = 0, δω|_{M_{1}} = 0, δdω|_{M_{1}} = 0, …, \underbrace{δdδd ⋯}_{k different.} ω|_{M_{1}} = 0; \\ ∗ω|_{M_{2}} = 0, ∗dω|_{M_{2}} = 0, …, \underbrace{∗dδdδ ⋯}_{k different.} ω|_{M_{2}} = 0 \end{cases}

If M has bounded geometry, the same is true for ω ∈ H^{k+1}(Λ^{p}T^{∗}M).

Proof. This is proven by induction on k, using Δ = δd + dδ and the fact that (dω)|_{∂M} = 0 if ω|_{∂M} = 0.

5.5. Notation. Spaces of forms which fulfill boundary conditions are written in the following manner:

H^{s}(Λ^{∗}(T^{∗}M), bd(k)) := {ω ∈ H^{s}(Λ^{∗}(T^{∗}M)); bd(k)(ω)}

5.6. Proposition. On a complete oriented Riemannian manifold with boundary ∂M = M_{1} ⨿ M_{2}, the boundary value problem (Δ, p^{1}) is formally self adjoint in the sense of definition 3.34 with respect to the Greenian formula

(Δω, η)_{L^{2}(M)} − (ω, Δη)_{L^{2}(M)} = (p^{1}ω, t⃗η)_{L^{2}(∂M)} − (t⃗ω, p^{1}η)_{L^{2}(∂M)},

where t⃗ is defined als follows: Use ∗ to denote the Hodge-∗-operator of M and ˆ∗ for the Hodge-∗-operator of ∂M (note that they do not commute with pulling back). Define t⃗ by

\begin{matrix}t_{0} : Ω^{p}(M) → Ω^{∗}(∂M) : & ω ↦ ((−1)^{pm}ˆ∗(∗dω)|_{M_{1}} − ˆ∗(δω)|_{M_{2}}) \\ t_{1} : Ω^{p}(M) → Ω^{∗}(∂M) : & ω ↦ ((−1)^{m(p+1)}ˆ∗(∗ω)|_{M_{1}} − ˆ∗ω|_{M_{2}}) \end{matrix}

Proof. For ω, η ∈ Ω^{p}(M) we have

(Δω, η) = (dδω, η) + (δdω, η)
= (δω, δη) + ∫_{∂M}(δω)|_{∂M} ∧ (∗η)|_{∂M} + (dω, dη) − ∫_{∂M} η|_{∂M} ∧ (∗dω)|_{∂M}
= (ω, Δη) + (−1)^{(m−p)(p−1)} ∫_{∂M}(δω)|_{∂M} ∧ ˆ∗ˆ∗(∗η)|_{∂M} − ∫_{∂M} ∗dω|_{∂M} ∧ ˆ∗ˆ∗(η)|_{∂M}
− ∫_{∂M}(∗ω)|_{∂M} ∧ ˆ∗ˆ∗(δη)|_{∂M} + (−1)^{pm} ∫_{∂M} ω|_{∂M} ∧ ˆ∗ˆ∗(∗dη)|_{∂M}
= (ω, Δη) + (p^{1}ω, t⃗η) − (t⃗ω, p^{1}η),

which is what we had to show. However, the proof is not finished. It is essential that p^{1} ⊕t⃗ is a Dirichlet system. In view of the fact that ±ˆ∗ is a linear bundle isomorphism on ∂M we can examine d⃗: ω ↦ ω|_{∂M} ⊕ (∗ω)|_{∂M} ⊕ (δω)|_{∂M} ⊕ (∗dω)|_{∂M} instead of p^{1} ⊕ t⃗.

This is a local question. Let U ⊂ ∂M be some open set with compact closure. We use normal coordinates near the boundary: let f : U × [0, ϵ) → M be such that for each x ∈ U f_{x} : [0, ϵ) → M is a geodesic with unit speed and f′_{x}(0) is the unit inward normal vector.

For ϵ sufficiently small this is a diffeomorphism onto its image, hence a collar of U in M. By the Gauss lemma A.4 we have: f′_{x}(t) is orthogonal to f(U × {t}) =: U_{t} for every t ∈ [0, ϵ). Now denote with ˆ∗ = ˆ∗(t), d̂ = d̂(t) and δ̂ the corresponding families of operators on U_{t} ⊂ (∂M_{t})_{t}.

Via the diffeomorphism f every ω ∈ C_{0}^{∞}(Λ^{p}(T^{∗}M)) can be written uniquely as ω = ω_{1} + ω_{2} ∧ dt with ω_{1}(t), ω_{2}(t) smooth families of forms on ∂M_{t}. Then

\begin{matrix}∗ ω = ±ˆ∗ω_{1} ∧ dt ± ˆ∗ω_{2} \\ ω|_{∂M} = ω_{1}(0). \end{matrix}

The zero-order part of d⃗ is therefore just

ω ↦ ω_{1}(0) ⊕ ˆ∗ω_{2}(0)

which is clearly a bundle isomorphism. Now we have to compute the coefficient of ∂/∂t in the first order part. We get

\begin{aligned}dω & = d̂ω_{1} + \frac{∂ω_{1}}{∂t} ∧ dt + d̂ω_{2} ∧ dt \\ ⟹ (∗dω)|_{∂M} & = ±ˆ∗\frac{∂ω_{1}}{∂t} (0) ± ˆ∗d̂ω_{2}(0) \\ d(∗ω) & = ±d̂ˆ∗ω_{1} ∧ dt ± d̂ˆ∗ω_{2} ± \frac{∂(ˆ∗ω_{2})}{∂t} ∧ dt \\ ⟹ δω|_{∂M} & = ±δ̂ω_{1}(0) ± ˆ∗\frac{∂(ˆ∗ω_{2})}{∂t}(0). \end{aligned}

In view of the fact that ∂/∂t and ˆ∗ commute modulo a differential operator of order zero the ∂/∂t-part of d⃗ is therefore

ω ↦ ±ˆ∗\frac{∂ω_{1}}{∂t} (0) ⊕ ±\frac{∂ω_{2}}{∂t} (0),

and this is ∂/∂t composed with a linear bundle isomorphism.

5.7. Definition. Define the space H^{p}(M, M_{1}) of harmonic forms which fulfill absolute boundary conditions on M_{1} and relative boundary conditions on M_{2} as

H^{p}(M, M_{1}) := {ω ∈ C^{∞} ∩ L^{2}(Λ^{p}T^{∗}M); Δω = 0, bd(2)(ω)}

5.8. Proposition. If M is complete, then these harmonic forms are closed and coclosed, i.e.

H^{p}(M, M_{1}) = {ω ∈ C^{∞} ∩ L^{2}(Λ^{p}T^{∗}M); dω = 0, δω = 0, ω|_{M_{1}} = 0, ∗ω|_{M_{2}} = 0}.

Proof. Apparently, dω = 0 = δω ⟹ Δω = 0. On the other hand, assume Δω = 0 and bd(2)(ω). We extend an idea of Gromov [32, 1.1.B] to manifolds with boundary. For arbitrary compact K ⊂ M and ϵ > 0, we can find a smooth compactly supported cutoff function

Hodge decomposition for ∂-manifolds of bounded geometry 45

ψ : M → [0, 1] with ψ = 1 on K, |dψ(x)| ≤ ϵ and |dψ(x)|^{2} ≤ ϵψ(x) ∀x ∈ M (Choose a smooth and compactly supported function α : ℝ → [0, 1] with α = 1 in a neighbourhood of 0, |α′| ≤ ϵ and |α′|^{2} ≤ ϵα.) Let ψ(x) be an appropriate smooth approximation to α(d(x, K)). Let ω ∈ H^{p}(M, M_{1}).

0 = ∫_{M} ψ(Δω, ω) = ∫_{M}(d(ψδω), ω) − ∫_{M}(dψ ∧ δω, ω) + ∫_{M}(δdω, ψω)

= ∫_{M} ψ(δω, δω) ± ∫_{M_{1}} ψ \underbrace{δω|_{M_{1}}}_{=0} ∧(∗ω)|_{M_{1}} ± ∫_{M_{2}} ψ(δω)|_{M_{2}} ∧ \underbrace{∗ω|_{M_{2}}}_{=0} − ∫_{M}(dψ ∧ δω, ω)

+ ∫_{M} ψ(dω, dω) + ∫_{M}(dω, dψ ∧ ω) ± \underbrace{∫_{∂M} ψω|_{∂M} ∧ (∗dω)|_{∂M}}_{=0}

Using the pointwise Cauchy-Schwarz inequality, it follows

\begin{aligned}∫_{M} ψ(|dω|^{2} + |δω|^{2}) & ≤ ∫_{M}|dψ||δω||ω| + ∫_{M}|dω||dψ||ω| \\ & ≤ 2ϵ|ω|_{L^{2}} \sqrt{∫_{M}|ψ|(|dω|^{2} + |δω|^{2})} \end{aligned}

The last estimate is a consequence of the Cauchy-Schwarz inequality and our choice of ψ. Since ϵ may be arbitrarily small, and ψ = 1 on arbitrarily large compact sets, we conclude dω = 0 = δω.

5.9. Definition. Set Ω_{d}^{p} := {ω ∈ C_{0}^{∞}; ω|_{M_{1}} = 0} and Ω_{δ}^{p} := {ω ∈ C_{0}^{∞}; (∗ω)|_{M_{2}} = 0}.

5.10. Theorem. Let M be a complete oriented Riemannian ∂-manifold. Then we have the orthogonal decomposition (called Hodge decomposition):

L^{2}(Λ^{p}T^{∗}M) = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{d}^{p−1}} ⊕ \overline{δΩ_{δ}^{p+1}}

Proof. This is just an application of corollary 4.22. We have only to show that we can replace \overline{ΔD_{Δ}^{∞}} by \overline{dΩ_{d}}⊕\overline{δΩ_{δ}}. Now Δ = dδ+δd, therefore ΔD_{Δ}^{∞} ⊂ dΩ_{d}+δΩ_{δ}. Integration by parts like in proposition 5.6 shows that H^{p}(M, M_{1}), dΩ_{d} and δΩ_{δ} are mutually orthogonal. This concludes the proof.

Hodge decomposition for ∂-manifolds of bounded geometry

Now, we want to investigate how to check for uniform ellipticity directly from the coefficients of the boundary value problem, without explicitly constructing the fundamental solutions. We will use this criterion to show that on a manifold of bounded geometry the Dirichlet/Neumann boundary value problem for the Laplacian (Δ, p^{1}) is uniformly elliptic. We conclude that it is essentially self adjoint and derive a Hodge decomposition for higher Sobolev spaces. From now on, we assume that M has bounded geometry.

5.11. Definition. Let P = (A, p⃗) be a boundary value problem as in definition 3.27. For each point b ∈ ∂M we get normal collar coordinates and admissible bundle trivialisations. In these normal coordinates the problem is described by matrices (∑_{α} A_{α}^{(ij)}(x)D^{α})_{ij} (for the global operator) and (∑_{α} p_{α,k}^{(ij)}D^{α})_{ij} (k = 0, …, μ − 1) with corresponding symbols A(x, ξ), … (we have omitted an additional index b). In the following we inspect the exposition in Hörmander’s [36, chapter 10]. We consider only the principal parts A^{0} and p_{k}^{0}. Substituting D = ∂/∂_{t} for ξ_{m} in the principal symbol and freezing x and ξ′ = (ξ_{1}, …, ξ_{m−1}) we get an ordinary boundary value problem on ℝ_{≥0}. As our boundary value problem is elliptic we find C_{x,ξ} > 0 so that ∀v = (v_{1}, …, v_{n}) ∈ C^{∞}(ℝ, 𝕂^{n}) (compare Hörmander [36, 10.2.3]):

∑_{l} ∑_{i=0}^{m} ∫_{0}^{∞}|D^{i}v_{l}|^{2}dt + ∑_{l} ∑_{j=0}^{m−1}|D^{j}v_{l}(0)|^{2}

≤ C_{x,ξ′} (∑_{l′} ∫_{0}^{∞}|(A^{0}(x, ξ′, D)v)_{l′}|^{2}dt + ∑_{k,l′}|(p_{k}^{0}(x, ξ′, D)v)_{l′}(0)|^{2}) (5.12)

The smallest possible constant depends continuously on the coefficients of the matrices. Therefore, C_{x}^{b} < ∞ may denote the best possible constant which holds ∀|ξ′| = 1. Moreover, if λ_{s,ξ′} − 1 is a bound for the absolute value of the zeros of the polynomial p(z) = A^{0}(x, ξ′, z) then C_{x,ξ′}^{λ_{x,ξ′}} > 0 exists so that

∑_{l} ∑_{i=0}^{m} ∫_{0}^{∞} e^{2λt}|D^{j}v_{l}|^{2} dt + ∑_{l} ∑_{j=0}^{m−1}|D^{j}v_{l}(0)|^{2} ≤ C_{x,ξ′}^{λ} ∑_{l′} ∫_{0}^{∞} e^{2λt}|(A^{0}(x, ξ′, D)v)_{l′}|^{2} dt

Again, C_{x,ξ′}^{λ} depends continuously on λ and the coefficients. As long as the degree of A^{0} is fixed, λ_{x,ξ′} depends continuously on the coefficients of A^{0}, too. Of course, this is the case if we deal with elliptic boundary value problems of a given order. Let λ_{x}^{b} be the maximum of {λ_{x,ξ′}; |ξ′| = 1} and set C_{x,λ}^{b} := max_{|ξ′|=1}{C_{x,ξ′}^{λ}} (be careful to choose λ ≥ λ_{x}^{b}).

Let {c_{i}^{b}(x)}_{i} be the collection of all coefficients of the principal part of any of the differential operators in question (this is for fixed b only a finite set). Set

δ_{b} = \frac{1}{∑_{i} sup_{|x|≤r}|c_{i}(x)|}.

Let r be a Euclidian radius so that the via a normal chart transported boundary value problem is defined on B(0, r) ⊂ ℝ^{m}. On a manifold M of bounded geometry, r can be found independently of b.

5.13. Proposition. If (with notation of the preceding definition) λ > 0 and C > 0 exist so that

(C0) λ_{x}^{b} ≤ λ ∀x, b

(C1) C_{x,λ}^{b} ≤ C and C_{x}^{b} ≤ C ∀x, b

(C2) |c_{i}^{b}(x)| ≤ C ∀i, b, |x| ≤ r

then we can find ρ independently of b so that in normal coordinates around b a fundamental solution R_{b} on B(b, ρ) in the sense of definition 4.7 exists; and the norm of R_{b} : M_{0} → H^{μ}(𝕂^{n}↓ℝ_{+}^{m}) is bounded by a constant independent of b.

Proof. Close inspection of Hörmander’s construction [36, theorem 10.4.1] of the fundamental solution shows that for the domain of the fundamental solution we take a ball with radius ρ_{b} = C_{b} · δ_{b} where C_{b} depends only on C_{x}^{b}, λ and C_{x,λ}^{b}. Following Hörmander’s construction further we see that the norm of this fundamental solution is bounded by a universal expression in C_{x}^{b}, λ and C_{x,λ}^{b}. Considering the definition of δ_{b} this finishes the proof.

5.14. Proposition. On a manifold of bounded geometry the boundary value problems (Δ, p^{1}) and (1 + Δ, p^{1}) are uniformly elliptic.

Proof. It is well known that these boundary value problems are elliptic (compare Müller [46] and Schwarz [61]). The principal symbol of Δ (and 1 + Δ) on forms in normal coordinates around b ∈ M is just (∑_{i,j} g_{b}^{ij}(x)ξ_{i}ξ_{j}) · 1. To prove uniform ellipticity of the Laplacian we must show that

\frac{|∑ g_{b}^{ij}(x)ξ_{i}ξ_{j}|}{∑_{i}|ξ_{i}|^{2}}; ξ ≠ 0

is bounded from below by some positive constant. Since the matrix g = (g^{ij})_{ij} is symmetric and positive and by substituting \sqrt{g}^{−1}ξ for ξ we see that this is equivalent to finding an upper bound for

\frac{|∑ g_{ij}^{b}(x)ξ_{i}ξ_{j}|}{∑|ξ_{i}|^{2}}

where g_{ij} is the inverse of g^{ij}. But, by the definition of bounded geometry, such a bound exists independently of x and b.

It remains to check boundary uniformity. Here we will use proposition 5.13. What is the local expression for Δ and p^{1}? We will not write down the rather complicated formulae (see f.e. Schwarz [61]) but observe that the coefficients c_{i}^{b} are polynomials in g_{ij}^{b}, g_{b}^{ij} and their derivatives up to order 2. Therefore, condition (C2) of lemma 5.13 is fulfilled. Concerning (C0) and (C1) we have the map (adopting the notation of 5.13)

\begin{matrix}M × \overline{B(0, r)} × S^{m−2} & ⟶ & 𝕂^{R} & ⟶ & ℝ × ℝ × ℝ \\ (b, x, ξ′) & ⟶ & (c_{i}^{b}(x, ξ′))_{i} & ⟶ & (C_{x}^{b}, λ, C_{x,λ}^{b}) \end{matrix}

where in our specific case each of the c_{i}^{b}(x, ξ′) is a polynomial in ξ′ and derivatives of g_{ij}^{b}(x), g_{b}^{ij}(x). The map 𝕂^{R} → ℝ^{3} is only defined on the subset E ⊂ 𝕂^{R} which comes from elliptic boundary value problems and is continuous on this subset. We will show that the closure of the range of M × B(0, r) × S^{m−2} is a compact subset of E. Then we find a bound λ for λ_{x}^{b} and a bound C for C_{x}^{b} and C_{x,λ}^{b} on that compact subset and the proof is finished. For compactness, take a sequence (b_{n}, x_{n}, ξ′_{n}) ⊂ M × B × S^{m−2}. We have to produce a subsequence so that the image converges to some element of E. But we may choose a subsequence so that (after relabelling) (ξ′_{n}), x_{n}, g_{ij}^{b_{n}}(·) and g_{b_{n}}^{ij}(·) converge for each i, j (the latter in C^{2}-norm on \overline{B(0, r)} by Ascoli’s theorem and uniform boundedness (of the derivatives up to order 3 of g^{∗∗} and g_{∗∗})) to ξ′, x and g_{ij}, g^{ij} respectively. Each of the algebraic relations between the g_{ij}^{n} and g_{n}^{ij} are preserved under the limit. Especially, g_{ij} and g^{ij} form the components of the covariant and contravariant metric tensor resp. of some (twice differentiable) Riemannian metric g on B(0, r). Similarly, (c_{i}(ξ′))_{i} := lim_{n} c_{i}^{b_{n}}(x_{n}, ξ′_{n}) are just the corresponding coefficients at x for the boundary value problem (Δ, p^{1}) with Riemannian metric g. In particular, they come from an elliptic boundary value problem, i.e. they lie in E.

Now we are in the situation to apply our theory of uniformly elliptic (and formally self adjoint) boundary value problems to (Δ, p^{1}). It only remains to check

5.15. Lemma. The operator Δ is non-negative on D_{Δ} = {ω ∈ H^{2}(Λ^{p}T^{∗}M); p^{1}ω = 0}. More precisely:

(Δω, ω) = (dω, dω) + (δω, δω) ≥ 0  ∀ω ∈ D_{Δ}.  (5.16)

Proof. This is a consequence of the computations of proposition 5.6. By continuity, everything works for ω ∈ H^{2} because d and δ are bounded operators in the sense of bounded geometry and therefore bounded from H^{2} to H^{1} and from H^{1} to L^{2}.

5.17. Remark. H^{p}(M, M_{1}) ⊂ H^{∞}(Λ^{p}T^{∗}M)

Proof. This is an immediate consequence of the a priori estimates of theorem 4.15.

5.18. Definition. Set Ω_{∞}^{p} := H^{∞}(Λ^{p}(T^{∗}M), bd(∞)).

5.19. Theorem. Let M be an orientable manifold of bounded geometry with boundary. Then

D_{Δ^{k}} := H^{2k}(Λ^{p}(T^{∗}M), bd(2k)) = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{∞}^{p−1}} ⊕ \overline{δΩ_{∞}^{p+1}}  (5.20)

The closure is taken with respect to the given topology on H^{2k}, and the decomposition is orthogonal with respect to the Hilbert space structure pulled back from L^{2} via the following isometry (it induces the H^{2k}-topology):

The unbounded operator (1 + Δ)^{k} on L^{2}(Λ^{∗}(T^{∗}M)) with domain D_{Δ^{k}} is positive self adjoint and considered as an operator

(1 + Δ)^{k} : D_{Δ^{k}} → L^{2}(Λ^{∗}(T^{∗}M)),

is an isometry which respects the decomposition above.

(1 + Δ)^{k} with domain D_{Δ^{k}} is the k-th power in the sense of unbounded operators of (1 + Δ) with domain D_{Δ}.

Proof. This is just the application of 4.31 to our situation (note that the specific situation fits exactly in the general framework). We only have to show that we can replace \overline{ΔΩ_{∞}^{p}} by \overline{dΩ_{∞}^{p−1}} ⊕ \overline{δΩ_{∞}^{p+1}}. If ω ∈ Ω_{∞} then dω and δω lie in Ω^{∞}, too (this is directly clear from proposition 5.4). Consequently, ΔΩ_{∞}^{p} ⊂ dΩ_{∞}^{p−1} + δΩ_{∞}^{p+1} (Δ = dδ + δd). Straightforward integration by parts yields that dΩ_{∞}, δΩ_{∞} and H(M, M_{1}) are mutually orthogonal. Therefore, (5.20) holds. Because Δ commutes with d and δ, (1 + Δ)^{k} respects not only the decomposition of 4.31 but also of (5.20).

5.21. Example. In view of the Hodge decompostion of L^{2}(Λ^{p}T^{∗}M) one may ask if we cannot derive a similar splitting for smooth functions, then without the necessity to take closures. This is true if M is compact (compare Müller [46]) but not in general. Already M = ℝ = S̃^{1} with the standard metric can serve as an example. Here ∂M = ∅ and H^{0}(ℝ) = {f ∈ L^{2} ∩ C^{∞}; f′′ = 0} = {0}. It remains to check that C_{0}^{∞} ⊄ δΩ^{1}: Take f ∈ C_{0}^{∞}(ℝ) with ∫_{−∞}^{∞} f(t) dt = A ≠ 0. If f = δ(F dt) we know that F (x) = ∫_{0}^{x} f(t)dt+C_{F} and immediately F ∉ L^{2}(ℝ) (because F (x) ≡ C_{1} for −x large and F (x) ≡ C_{2} for x large with C_{1} ≠ C_{2}). However, we already know f ∈ \overline{δC_{0}^{∞}}. Explicitly, f = lim f_{n} where f_{n} ≡ f on supp f, f_{n} ≈ −A/n on an interval of length approximately n outside supp f.

6 Hilbert-de Rham complexes and L^{2}-cohomology

In this section we introduce two versions of the de Rham complex and show that their L^{2}- cohomology is isomorphic to H_{(2)}^{∗}(M, M_{1}) if M is oriented. We conclude Poincare duality and we will investigate invariance of this de Rham cohomology under vb-homotopy. (vbmaps are Lipschitz maps with vol(f^{−1}A) ≤ C vol(A) ∀A ⊂ N) Moreover, we compute L^{2}- de Rham cohomology for certain negatively curved spaces with boundary, using methods of Donnelly/Xavier. Hilbert complexes like our L^{2}-de Rham complexes and their cohomology are studied in a very general setting by Brüning und Lesch [16].

L^{2}-de Rham complex

The first version of our de Rham complex can be defined for arbitrary complete manifolds: Let M be a complete Riemannian manifold with boundary ∂M = M_{1} ⨿ M_{2}.

Consider the unbounded operator

d : Ω_{d} = C_{0}^{∞}(Λ^{p}(T^{∗}M); ·|M_{1} = 0) ⊂ L^{2} → L^{2}

Let A^{p} = A^{p}(M, M_{1}) be the domain of its closure and define the norm |·|_{A}^{2} := |·|_{L^{2}}^{2} +|d·|_{L^{2}}^{2} on A^{p}. With this norm A^{p} becomes a Hilbert space.

We have d(A^{p}) ⊂ ker(d : A^{p+1} → L^{2}) since for ω ∈ A^{p} we find, by definition, ω_{n} ∈

C_{0}^{∞}(Λ^{p}(T^{∗}M)) with ω_{n}|_{M_{1}} = 0 so that ω_{n} ⟶_{n→∞}^{L^{2}} ω and dω_{n} ⟶_{n→∞}^{L^{2}} dω. But then dω_{n} ∈

C_{0}^{∞}(Λ^{p+1}T^{∗}M) and ddω_{n} = 0 ⟶^{n→∞} 0 and (dω_{n})|_{M_{1}} = d(ω_{n}|_{M_{1}}) = 0.

Therefore, we get a Hilbert cochain complex

→ A^{p−1} →^{d} A^{p} →^{d} A^{p−1} →

6.1. Definition. Set

Z^{p}(M, M_{1}) := ker(d : A^{p} → A^{p+1})
B^{p}(M, M_{1}) := im(d : A^{p−1} → A^{p})
H_{dR,(2)}^{p}(M, M_{1}) := Z^{p}(M, M_{1})/\overline{B^{p}(M, M_{1})}

6.2. Theorem. The inclusion H^{p}(M, M_{1}) ↪ A^{p} induces an isomorphism

H^{p}(M, M_{1}) ≅ H_{dR,(2)}^{p}(M, M_{1})

50 6 HILBERT-DE RHAM COMPLEXES AND L^{2}-COHOMOLOGY

Proof. First note that the two norms | · |_{A} and | · |_{L^{2}} are equal on Z^{p}. For the computation of Z^{p}/\overline{B^{p}} we will therefore work with the latter. We consider the Hodge decomposition 5.10

L^{2}(Λ^{p}(T^{∗}M)) = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{d}^{p−1}} ⊕ \overline{δΩ_{δ}^{p+1}}

Clearly H^{p}(M, M_{1}) ⊕ \overline{dΩ_{d}^{p−1}} ⊂ Z^{p}. On the other hand, Z^{p} ⊥ δΩ_{δ}^{p+1} since for ω ∈ Z^{p} and δη ∈ δΩ_{δ}^{p+1} (take ω_{n} ∈ C_{0}^{∞} with ω_{n} → ω and dω_{n} → dω = 0):

(ω, δη)_{L^{2}} = lim_{n}(ω_{n}, δη) =^{ω_{n}|_{M_{1}}=0,∗η|_{M_{2}}=0} lim_{n}(dω_{n}, η) = 0

Therefore, Z^{p} = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{d}^{p−1}}.

Along similar lines one shows \overline{B^{p}} = \overline{dΩ_{d}^{p−1}}. This concludes the proof.

Using the description of L^{2}-de Rham cohomology with harmonic forms, Poincare duality is a triviality:

6.3. Theorem. (Poincaré duality) Let M be a complete orientable Riemannian manifold with boundary ∂M = M_{1} ⨿M_{2}. It is clear that the Hodge-∗-operator is an isomorphism between H^{p}(M, M_{1}) and H^{m−p}(M, M_{2}). Via the Hodge isomorphism we get the Poincaré duality isomorphism

∗ : H_{dR,(2)}^{p}(M, M_{1}) ⟶^{≅} H_{dR,(2)}^{m−p}(M, M_{2})

6.4. Definition. A map f : M → N between metric spaces is called a Lipschitz map, if C > 0 exists so that d(fx, fy) ≤ Cd(x, y) ∀x, y ∈ M. If M and N are Riemannian manifolds and f is C^{1} this is equivalent to the existence of C′ > 0 so that ∥T_{x}f∥ ≤ C′ ∀x ∈ M. f is called a bilipschitz map, if f is a homeomorphism and f as well as f^{−1} are Lipschitz.

6.5. Definition. Let M and N be complete Riemannian manifolds with boundaries ∂M = M_{1} ⨿ M_{2} and ∂N = N_{1} ⨿ N_{2} respectively. A C^{1}-map f : (M, M_{1}) → (N, N_{1}) is called a vb-map, if C > exists so that

(1) f is a Lipschitz map

(2) vol(f^{−1}(A)) ≤ C vol(A) ∀A ⊂ N measurable.

Two such maps f and g are called vb-homotopic, if a C^{1}-homotopy h : (M × I, M_{1} × I) → (N, N_{1}) from f to g exists which is a vb-map.

6.6. Proposition. Suppose f : (M, M_{1}) → (N, N_{1}) is a vb-map. Then f induces (via pullback) a bounded chain map A^{∗}(N, N_{1}) → A^{∗}(M, M_{1}). Therefore, it induces a bounded map H_{(2)}^{p}(f) : H_{dR,(2)}^{p}(N) → H_{dR,(2)}^{p}(M).

L^{2}-de Rham complex 51

Proof. Let ω_{n} ∈ Ω_{d}(N) with ω_{n} →^{L^{2}} ω, dω_{n} →^{L^{2}} dω. Apparently f^{∗}ω_{n} ∈ Ω_{d}(M) (since pullback commutes with restriction to the boundary). Moreover, f^{∗} : L^{2}(Λ^{p}T^{∗}N) → L^{2}(Λ^{p}T^{∗}M) is bounded. Granted this, f^{∗}ω_{n} →: f^{∗}ω and df^{∗}ω_{n} = f^{∗}dω_{n} →: df^{∗}ω, and this proves the claim. It remains to prove the asserted L^{2}-boundedness: take ω ∈ L^{2}(Λ^{p}T jN). Then

∫_{M}|f^{∗}ω|^{2} ≤^{|T_{x}f|≤C} ∫_{M} C^{p}|ω(f(x))|_{Λ^{p}T_{fx}N}^{2}dx ≤^{(∗)} C^{p+1} ∫_{N}|ω|^{2}

(*) follows from the definition of the integral with step functions and from the volume estimate vol(f^{−1}(A)) ≤ C vol(A).

6.7. Theorem. Suppose f, g : (M, M_{1}) → (N, N_{1}) are vb-homotopic. Then

H_{(2)}^{∗}(f) = H_{(2)}^{∗}(g) : H_{dR,(2)}^{∗}(N, N_{1}) → H_{dR,(2)}^{∗}(M, M_{1}).

Especially, L^{2}-de Rham cohomology is an invariant of the vb-homotopy type.

Proof. Let h be a vb-homotopy between f and g. We construct a bounded chain homotopy H : A^{p+1}(N, N_{1}) → A^{p}(M, M_{1}) by

H(ω)|_{x} := ∫_{0}^{1} i_{∂_{t}}(h^{∗}ω|_{(x,t)})dt.

Here i_{X} denotes contraction with the vector field X. Since pullback to the boundary commutes with all the operations involved in the definition of H, H maps Ω_{d}^{p+1}(N) to Ω_{d}^{p}(M). Next, we show that it is L^{2}-bounded.

\begin{aligned}∫_{M}|Hω|^{2} & = ∫_{M} |∫_{0}^{1} i_{∂_{t}}h^{∗}ω|_{(x,t)}dt|^{2} dx |∫_{0}^{1} f·1|≤|f|_{L^{2}}·1 ; ≤ ∫_{M} ∫_{0}^{1}|i_{∂_{t}}h^{∗}ω|_{(x,t)}^{2} dt dx \\ & ≤ D^{p+1} ∫_{M×I} |ω|_{h(x,t)}|^{2} dx dt ≤ D^{p+2} ∫_{N}|ω|^{2} \end{aligned}

The last steps are completely analogous to steps in the above proof.

The following standard calculation (using an integration by parts on I) shows

dHω = H(dω) − f^{∗}ω + g^{∗}ω ∀ω ∈ C_{0}^{∞}(Λ^{∗}T^{∗}N) :

Split h^{∗}ω as

h^{∗}ω = h_{1}(x, t) + dt ∧ h_{2}(x, t)

⟹ h^{∗}(dω) = dh^{∗}ω = d_{x}h_{1}(x, t) + dt ∧ (\frac{∂}{∂t}h_{1}(x, t) + d_{x}h_{2}(x, t))

Let i_{k} : M ↪ M × I (k = 0, 1) be the top and bottom inclusion ⟹ i_{k}^{∗}h^{∗}ω =^{i_{k}^{∗}dt=0} h_{1}(x, k). Note i_{0}^{∗}h^{∗} = f^{∗} and i_{1}^{∗}h^{∗} = g^{∗} since h is a homotopy from f to g. Therefore

H(dω) = ∫_{0}^{1} i_{∂_{t}}(h^{∗}dω)dt = ∫_{0}^{1} d_{x}h_{2}(x, t)dt + ∫_{0}^{1} \frac{∂}{∂t}h_{1}(x, t)dt
= d_{x} ∫_{0}^{1} h_{2}(x, t)dt + h_{1}(x, 1) − h_{1}(x, 0)
= d ∫_{0}^{1} i_{∂_{t}}h^{∗}ωdt + g^{∗}ω − f^{∗}ω

The relation which we have just proved implies

|dHω|_{L^{2}} ≤ |Hdω|_{L^{2}} + |f^{∗}ω|_{L^{2}} + |g^{∗}ω|_{L^{2}} ≤^{∥H∥_{B(L^{2})}+∥f^{∗}∥+∥g^{∗}∥} |dω|_{L^{2}} + |ω|_{L^{2}}

Therefore, H : A^{p+1} → A^{p} is bounded. Since dH+Hd = g^{∗}−f^{∗}, the statement follows.

6.8. Remark. Every bilipschitz map is a vb-homotopy equivalence. Consequently, L^{2}-de Rham cohomology is a bilipschitz invariant.

6.9. Example. We can apply vb-homotopy invariance in the following way: Suppose M and N are compact manifolds with the same boundary, which are homotopy equivalent iwht homotopies which fix the boundary pointwise. Then surgery on an arbitrary complete Riemannian manifold, which cuts M away and glues in N instead (with the given diffeomorphism of the boundary), yields a new manifold which is vb-homotopy equivalent to the old one. Therefore the L^{2}-de Rham cohomology is unchanged.

In some cases, one can compute the L^{2}-de Rham cohomology analytically. For example, Donnelly/Xavier [28] show that for a complete simply connected Riemannian manifold M^{m} without boundary, if for 0 ≤ ϵ < 1 the sectional curvature is pinched between −1 and −1 + ϵ, then H_{(2)}^{p}(M) vanishes whenever p < \sqrt{1 − ϵ}(m − 1)/2 or m − p < \sqrt{1 − ϵ}(m − 1)/2. We extend their methods to certain manifolds with boundary and pinched negative curvature.

First, we establish their fundamental estimate for manifolds with boundary.

6.10. Theorem. Suppose M^{m} is a complete Riemannian ∂-manifold, ω ∈ C_{0}^{∞}(Λ^{p}T^{∗}M) (p ≥ 1), and F : M → ℝ is twice continuously differentiable on supp ω. Let λ_{1}, …, λ_{m} be the (pointwise) eigenvalues of the Hessian ∇^{2}F of F. Suppose |dF | ≤ 1. If ν is the inward unit normal vector field, set B := \frac{1}{2} ∫_{∂M} ν(F)(x′)|ω(x′)| dx′. Then,

(|dω|_{L^{2}} + |δω|_{L^{2}})|ω|_{L^{2}}

≥ \begin{cases}\frac{1}{2} ∫_{M}(tr(∇^{2}F) − 2p max\{λ_{i}\})|ω(x)| dx + B & if ω|_{∂M} = 0 \\ \frac{1}{2} ∫_{M}(tr(∇^{2}F) − 2p max\{λ_{i}\})|ω(x)| dx − B & if ∗ω|_{∂M} = 0 \\ \frac{1}{2} ∫_{M}(tr(∇^{2}F)) − 2(m − p) max\{λ_{i}\}|ω(x)| dx − B & if ω|_{∂M} = 0 \\ \frac{1}{2} ∫_{M}(tr(∇^{2}F) − 2(m − p) max\{λ_{i}\})|ω(x)| dx + B & if ∗ω|_{∂M} = 0 \end{cases} (6.11)

Proof. Let (e_{1}, …, e_{m}) be a local orthonormal frame, so that e_{m} is the outward normal at the boundary. Let u_{s_{1}...s_{p}} be the components of ω with respect to this frame. We use the usual summation convention for multi-indices. Donnelly/Xavier [28, proof of 2.2] show that

\begin{matrix}(p − 1)!(|dω|_{L^{2}} + |δω|_{L^{2}}) ≥ |\frac{(p − 1)!}{2} ∫_{M} tr(∇^{2}F)|ω(x)| dx − ∫_{M} \underbrace{F_{,i,l}u_{is_{1}...s_{p−1}}u_{ls_{1}...s_{p−1}}}_{=:V} dx \\ + \underbrace{∫_{M}(F_{,i}u_{is_{1}...s_{p−1}}u_{ls_{1}...s_{p−1}})_{,l}}_{=:A} −\frac{1}{2p} \underbrace{∫_{M}(F_{,i}u_{s_{1}...s_{p}}u_{s_{1}...s_{p}})_{,i}}_{=:A′} | \end{matrix}

(The indices , i denote derivatives.) We choose, for any given point off the boundary, an orthonormal frame field so that F_{,i,l} is diagonal (note that the Hessian is symmetric because we suppose F ∈ C^{2}). Then,

V ≤ max{λ_{1}, …, λ_{m}}p!|ω(x)|

— this relation is used by Donnelly/Xavier. Redistributing the summands, we also see

\begin{aligned}V & = (p − 1)! ∑_{s_{1}<···<s_{p}}(F_{,s_{1},s_{1}} + … F_{,s_{p},s_{p}})u_{s_{1}...s_{p}}u_{s_{1}...s_{p}} \\ & ≥ (p − 1)!(tr(∇^{2}F) − (m − p) max\{λ_{1}, …, λ_{m}\})|ω(x)| \end{aligned}

By continuity, these relations hold at the boundary, too. Then, Donnelly/Xavier use Stoke’s theorem to conclude A = 0 = A′. We obtain boundary terms instead:

\begin{aligned}A & = ∫_{∂M} F_{,i}u_{is_{1}...s_{p−1}}u_{ms_{1}...s_{p−1}} \\ A′ & = ∫_{∂M} F_{,m}u_{s_{1}...s_{p}}u_{s_{1}...s_{p}} = p! ∫_{∂M}(−νF)|ω(x′)| dx′ \end{aligned}

If ω|_{∂M} = 0 then u_{ms_{1}...s_{p−1}} = 0 ⟹ A = 0. If ∗ω|_{∂M} = 0 then u_{s_{1}...s_{p}} = 0 if none of the s_{i} = m. Therefore u_{is_{1}...s_{p−1}}u_{ms_{1}...s_{p−1}} = 0 if: neither i nor s_{1}, …, s_{p−1} equals m, but also if one of s_{1}, …, s_{p−1} is m (since ω is alternating). It remains A = ∫_{∂M} F_{,m}u_{ms_{1}...s_{p−1}}u_{ms_{1}...s_{p−1}} = (p−1)! ∫_{∂M}(−νF)|ω(x′)| dx′. Using the elementary relations |a| ≥ a and |a| ≥ −a concludes the proof.

We apply this to derive our vanishing result:

6.12. Theorem. Let M be an orientable Riemannian ∂-manifold of bounded geometry so that the sectional curvature s is pinched: −1 ≤ s ≤ −1 + ϵ with 0 ≤ ϵ < 1. Suppose, a sequence x_{1}, x_{2}, … ⊂ M exists for every compact subset K ⊂ M so that d(x_{i}, K) →^{i→∞} ∞ and ∀i and ∀y ∈ K a unique distance realizing geodesic exists, and lies entirely in the interior of M (except possibly the endpoints, of course). Consider the absolute Laplacian Δ_{p}^{abs} as unbounded operator on D_{abs} := {ω ∈ H^{2}(Λ^{p}T^{∗}M); (∗ω)|_{∂M} = 0 = (∗dω)|_{∂M}} and set λ_{p}^{abs} := min spec(Δ_{p}^{abs}). Let Δ_{p}^{rel} be the unbounded operator with domain D_{rel} := {ω ∈ H^{2}(Λ^{p}T^{∗}M); ω|_{∂M} = 0 = (δω)|_{∂M}} and define λ_{p}^{rel} accordingly. Then

\begin{matrix}\sqrt{λ_{p}^{abs}} ≥ (\sqrt{1 − ϵ}(m − 1) − 2p)/2 ∀p, and \\ \sqrt{λ_{m}^{abs}} ≥ \sqrt{1 − ϵ}(m − 1)/2. \end{matrix}

Moreover, λ_{p}^{rel} = λ_{m−p}^{abs}. In particular,

H^{p}(M) = \{0\}  if p < \sqrt{1 − ϵ}(m − 1)/2 or p = m
H^{p}(M, ∂M) = \{0\}  if p > m − \sqrt{1 − ϵ}(m − 1)/2 or p = 0.

6.13. Example. The universal coverings of finite volume cusps of hyperbolic manifolds are manifolds which fulfill the conditions of theorem 6.12.

6.14. Corollary. Let M^{m} be a compact manifold with sectional curvature pinched between −1 and −1 + ϵ (0 ≤ ϵ < 1) and with convex boundary (i.e. the second fundamental of ∂M is l ≥ 0). Suppose π_{1}(M) is infinite. Then there are no absolute L^{2}-harmonic p-forms on the universal covering M̃, i.e. b_{(2)}^{p} = 0, if p < \sqrt{1 − ϵ}(m − 1)/2 or if p = m.

Proof. If the boundary is convex, then arbitrary x ∈ M − ∂M and y ∈ M may be joined by a distance realising geodesic lying entirely (besides y) in the interior of M. The same is true for M̃, and there, the geodesics are unique because M̃ is simply connected. The covering is infinite, i.e. M̃ has infinite diameter. This implies that we find the sequence (x_{i}) of the assumptions of theorem 6.12. Now apply the theorem.

6.15. Example. The conditions on the curvature are necessary. We find ∂-manifolds with constant sectional curvature −1 but with non convex boundary, so that the conclusion of the theorem does not hold. Take f.i. a closed hyperbolic manifold of dimension 8 with fundamental group Γ. Choose an embedding i : S^{5} × D^{3} ↪ M, use i_{∂} for the corresponding embedding of the boundary. Set N = M − S^{5} × D^{3} (i_{N} : N ↪ M). Note that i_{N} induces an isomorphism of the fundamental groups, while π_{1}(i_{∂}) is trivial. Now consider the weakly exact Mayer-Vietoris sequence for L^{2}-cohomology [41, 2.2] (weakly exact means that the image is not equal, but only dense in the kernel).

\begin{matrix}H_{(2)}^{2}(M, l^{2}Γ) → H_{(2)}^{2}(N, i_{N}^{∗}l^{2}Γ) ⊕ H_{(2)}^{2}(S^{5} × D^{3}, i^{∗}l^{2}Γ) \\ → H_{(2)}^{2}(S^{5} × S^{2}, i_{∂}^{∗}l^{2}Γ) → H_{(2)}^{3}(M, l^{2}Γ) \end{matrix}

Note that 6.14 implies

H_{(2)}^{2}(M, l^{2}Γ) = {0} = H_{(2)}^{3}(M, l^{2}Γ).

Since π_{1}(i) and π_{1}(i_{∂}) are trivial,

H_{(2)}^{2}(S^{5} × D^{3}, i^{∗}l^{2}Γ) = H^{2}(S^{5} × D^{3}, ℂ) ⊗ l^{2}Γ = {0}
H_{(2)}^{2}(S^{5} × S^{2}, i_{∂}^{∗}l^{2}Γ) = H^{2}(S^{5} × S^{2}, ℂ) ⊗ l^{2}Γ ≅ l^{2}Γ

Consequently, H_{(2)}^{2}(N) is weakly isomorphic to l^{2}Γ (here we use the fact that π_{1}(i_{N}) is an isomorphism), and b_{(2)}^{2}(N) ≠ 0. The corollary 6.14 implies that there is no way to embed S^{5} × D^{3} in M so that the boundary is concav everywhere (then the boundary of N would be convex).

Proof of theorem 6.12. First note that by theorem 4.25 Δ_{p}^{abs} and Δ_{p}^{rel} are self adjoint and by 5.15 non-negative and hence λ_{p}^{abs}, λ_{p}^{rel} ≥ 0 are well defined. The Hodge-∗-operator provides the Poincare duality isometry which obviously interchanges Δ_{p}^{abs} and Δ_{m−p}^{rel}. Especially the two are isospectral and λ_{p}^{abs} = λ_{m−p}^{rel}.

The proof of Donnelly/Xavier works with only minor changes. However, we will give the details here. Let ω be a smooth form with support K. Then choose the sequence x_{i} accordingly. Set F_{i}(y) := d(x_{i}, y). The assumptions guarantee that, for y ∈ K, y ↦ d(x_{i}, y) behaves exactly as the distance function on a complete simply connected manifold with identical curvature assumptions. Therefore by [28], F_{i} is smooth on K and

tr ∇^{2}F_{i} ≥ \sqrt{1 − ϵ}(m − 1) coth(\sqrt{1 − ϵ}d(x_{i}, y)),
max λ_{i} ≤ coth(d(y, x_{i})).

Moreover, |dF | ≤ 1 and νF_{i}(y) < 0 for y ∈ ∂M, where ν is the inward unit normal (the latter follows from the fact that distance realising geodesics lie in the interior of M). Now we apply theorem 6.10 and use coth(x) →^{x→∞} 1, to obtain

|dω|_{L^{2}} + |δω|_{L^{2}} ≥ \frac{\sqrt{1 − ϵ}(m − 1) − 2p}{2}|ω|_{L^{2}} ∀ω ∈ C_{0}^{∞}(Λ^{p}T^{∗}M) with ∗ ω|_{∂M} = 0
|dω|_{L^{2}} + |δω|_{L^{2}} ≥ \frac{\sqrt{1 − ϵ}(m − 1) − 2(m − p)}{2}|ω|_{L^{2}} ∀ω ∈ C_{0}^{∞}(Λ^{p}T^{∗}M) with ω|_{∂M} = 0

(6.16)

We have to require p ≥ 1. If p = m then the second estimate holds also if ∗ω|_{∂M} = 0, because this condition implies that the boundary integral in (6.11) (which has the wrong sign) vanishes identically.

A standard cutoff argument (like the one in the proof of 5.10) shows that estimate (6.16) holds for arbitrary ω ∈ H^{2}(Λ^{p}T^{∗}M) which fulfills the boundary conditions.

Let λ_{m} be either λ_{m}^{abs} or λ_{m}^{rel}, # = abs or # = rel. For α > 0, let χ = χ_{α} be the characteristic function of the interval [λ_{m}, λ_{m} + α]. By definition of λ_{m} = min spec(Δ_{m}^{#}), an η ∈ L^{2} exists so that ω := χ(Δ_{m}^{#})η ≠ 0. By the spectral theorem, ω ∈ D_{#} ⟹ the estimate (6.16) holds for ω. (Here we need p = m for the absolute case). Therefore (Δω, ω) = (δω, δω) (by (5.16), since ω ∈ D_{#} and ω is top-dimensional). The spectral theorem tells us

(λ_{m} + α)|ω|_{L^{2}}^{2} ≥ (Δω, ω)_{L^{2}} = |δω|_{L^{2}}^{2}

⟹^{(6.16)} \sqrt{λ_{m} + α} ≥ \frac{1}{2}\sqrt{1 − ϵ}(m − 1)

Since α can be arbitrarily small, this proves the assertion.

We proceed with the estimates of λ_{p}^{abs} by induction on p. p = 0 is done by Poincare duality and the estimate for λ_{m}^{rel}. Assume, the statement is true for p−1 but not for p. Then α > 0 exitsts so that λ_{p} + α < λ_{p−1}. Just as above we construct 0 ≠ ω = χ(Δ)η ∈ D_{abs} with

(λ_{p} + α)|ω|_{L^{2}}^{2} ≥ (Δω, ω)_{L^{2}} = (dω, dω)_{L^{2}} + (δω, δω)_{L^{2}}

(the latter again by (5.16)). The spectral theorem implies δχ(Δ_{p}) = χ(Δ_{p−1})δ (because δΔ_{p} = Δ_{p−1}δ). By assumption

χ(Δ_{p−1}) = 0 ⟹ δω = δχ(Δ)η = 0.

Together with the above equation:

⟹^{(6.16)} \sqrt{λ_{p} + α} ≥ \frac{\sqrt{1 − ϵ}(m − 1) − 2p}{2}

Since α can be arbitrarily small, this contradicts our assumption

⟹ \sqrt{λ_{p}} ≥ \frac{\sqrt{1 − ϵ}(m − 1) − 2p}{2}

Sobolev-de Rham complex

The second complex we want to introduce is a Sobolev de Rham complex. To define this complex, we assume that M is a manifold of bounded geometry:

6.17. Definition. Consider the cochain complex

→ H^{2k+2}(Λ^{p−1}(T^{∗}M), bd(2k + 2)) →^{d} H^{2k}(Λ^{p}(T^{∗}M), bd(2k)) →^{d} H^{2k−2}(⋯)

Set

Z^{k,p}(M, M_{1}) := ker(d : H^{2k}(Λ^{p}T^{∗}M, bd(2k)) → H^{2k−2}(Λ^{p+1}T^{∗}M, bd(2k − 2)))
B^{k,p}(M, M_{1}) := im(d : H^{2k+2}(Λ^{p−1}T^{∗}M, bd(2k + 2)) → H^{2k}(Λ^{p}T^{∗}M, bd(2k)))
H_{(2)}^{k,p}(M, M_{1}) := Z^{k,p}(M, M_{1})/\overline{B^{k,p}(M, M_{1})}

6.18. Theorem. Suppose M is oriented. Then H^{p}(M, M_{1}) ↪ H^{2k}(Λ^{p}(T^{∗}M), bd(2k)) induces an isomorphism

H^{p}(M, M_{1}) ≅ H_{(2)}^{k,p}(M, M_{1}) ∀k ≥ 1

Proof. We have the Hodge decomposition 5.19

H^{2k}(Λ^{p}(T^{∗}M), bd(2k)) = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{∞}^{p−1}} ⊕ \overline{δΩ_{∞}^{p+1}}

On the one hand, H^{p}(M, M_{1})⊕\overline{dΩ_{∞}^{p−1}} ⊂ Z^{k,p} = ker d. On the other hand, Z^{k,p} ⊥_{H^{2k}} δΩ_{∞}^{p+1}, because if ω ∈ Z^{k,p} and δη ∈ δΩ_{∞}^{p+1} then

(ω, δη)_{H^{2k}} = ((1 + Δ)^{k}ω, δ(1 + Δ)^{k}η)_{L^{2}} =^{bd(2k)(ω),bd(∞)(η)} (d(1 + Δ)^{k}ω, (1 + Δ)^{k}η)_{L^{2}} =^{dω=0} 0.

Consequently, Z^{k,p} = H^{p}(M, M_{1}) ⊕ \overline{dΩ_{∞}^{p−1}}.

Next, we show \overline{B^{k,p}} = \overline{dΩ_{∞}^{p−1}}. Obviously, dΩ_{∞}^{p−1} ⊂ B^{k,p}. Integration by parts shows B^{k,p} ⊥ H^{p}(M, M_{1}) and B^{k,p} ⊥ δΩ_{∞}^{p+1}. The Hodge decomposition 5.19 now implies the rest.

Of course, Poincare duality holds in this setting, too:

6.19. Theorem. (Poincaré duality) Suppose M is orientable. It is obvious that the Hodge-∗-operator is an isomorphism between H^{p}(M, M_{1}) and H^{m−p}(M, M_{2}). Via the Hodge isomorphism we get the Poincaré duality isomorphism

∗ : H_{(2)}^{k,p}(M, M_{1}) ⟶^{≅} H_{(2)}^{k,m−p}(M, M_{2})

L^{2}-de Rham cohomology and coverings 57

L^{2}-de Rham cohomology and coverings

Using coverings, the preceding definitions yield interesting invariants for compact manifolds.

6.20. Theorem. Suppose M = Ñ is the normal covering of a compact manifold N with covering group Γ. Then the Hodge decomposition of 5.19 is Γ-invariant and the isometry (1 + Δ)^{k} between the different Sobolev spaces is Γ-equivariant.

Proof. This is clear regarding the fact that d, ∗, δ, Δ and all the boundary operators involved are lifts from the base and, therefore, Γ-equivariant.

6.21. Theorem. If M is the normal covering of a compact manifold with covering group Γ and with induced Riemannian metric, then the isomorphisms in 6.18 and 6.2 are Γequivariant.

Proof. The inclusion is Γ-equivariant and, therefore, the induced map is a Γ-isomorphism.

6.22. Theorem. Let M be a compact manifold with boundary ∂M = M_{1} ⨿ M_{2}. If M̃ is a normal covering of M (with lifted metric and with covering group Γ), we can assign the L^{2}-de Rham cohomology H_{dR,(2)}^{∗}(M, M_{1}, l^{2}(Γ)) := H_{dR,(2)}^{∗}(M̃, M̃_{1}) to M. Obviously, this depends on the diffeomorphism type of the base manifold only and not on the metric, because every diffeomorphism on compact manifolds is bilipschitz, and its lift to a covering has the same property. Using the specialisation of the L^{2}-de Rham theorem to coverings 1.7, we see that the Hilbert module H_{dR,(2)}^{∗}(M, M_{1}, l^{2}(Γ)) is a homotopy invariant of M.

Proof. We have to check that the isomorphic L^{2}-cohomology of the lifted triangulation is a homotopy invariant. By 1.7 we see that we may choose any triangulation we want. Therefore, without loss of generality, we work with simplicial maps. We have to lift every step of the standard proof for homotopy invariance of simplicial cohomology to conclude. This result is well known and follows from Lott/Lück [41, section 5].

The L^{2}-index theorem for ∂-manifolds shows how to compute the Euler characteristic via the L^{2}-Betti numbers of the L^{2}-de Rham cohomology of normal coverings. This is explained in detail in 13.5

7 L^{2}-cohomology and curvature

In this section we examine the relations between L^{2}-de Rham cohomology and curvature. This is based on the Hodge isomorphism and the Weizenböck formula Δ = ∇^{∗}∇ − R^{W} with its integrated consequence (for ∂-manifolds)

|∇ω|_{L^{2}}^{2} = |dω|_{L^{2}}^{2} + |δω|_{L^{2}}^{2} + (R^{W} ω, ω)_{L^{2}} + ∫_{∂M}(Sω, ω)_{x} dx.

58 7 L^{2}-COHOMOLOGY AND CURVATURE

We derive algebraic properties of the operators S and R^{W} in this formula, especially sufficient conditions for negativity. With this at hand, it is an easy task to get vanishing results for square integrable harmonic forms. Next, we specialize to manifolds of bounded geometry. We take the point of view that it makes sense to vary Riemannian metrics only in a given bilipschitz class. We show, that for metrics which are bilipschitz to Riemannian metrics with groups of isometries which are infinite and discrete, Bochner type vanishing results hold even if the curvature assumptions are fulfilled only outside a compact set. An example shows that this is not true without this isometry group assumption. Note that infinite normal coverings always admit the isometry groups required in the statement. As a corollary, we answer a question of Roe concerning existence of metrics with Ric ≥ 0 outside a compact set. However, we also show that such manifolds have polynomial growth, if ∂M = ∅.

For the preliminary algebraic considerations, to some extent we follow the exposition of Schwarz [61] (he deals only with ordinary cohomology).

7.1. Definition. The (symmetric) Weizenböck endomorphism R_{k}^{W} is defined by Δ_{k} = ∇^{∗}∇ − R_{k}^{W}, where ∇ is the Levi-Civita connection with formal adjoint ∇^{∗} and Δ_{k} is the Laplacian on k-forms. The (symmetric) endomorphism S_{k} ∈ End(Λ^{k}(M)|_{∂M}) is defined by the identity

(Sω, ω)_{Λ^{k}} = −(ω, i_{ν}dω)_{Λ^{k}} + (δω, i_{ν}ω)_{Λ^{k−1}} − 1/2\frac{∂}{∂ν} (ω, ω)_{Λ^{k}},

where ν is the unit inward normal field, and i_{X} denotes contraction of a form with the vector field X. By Schwarz [61, (2.1.13)], this is explicitly given in terms of the second fundamental form l.

7.2. Remark. The most important property of R^{W} and S (besides the Weizenböck formula) is the following formula for ω, η ∈ C_{0}^{∞}(Λ^{p}T^{∗}M) with ω|_{M_{1}} = 0 = η|_{M_{1}} and ∗ω|_{M_{2}} = 0 = ∗η|_{M_{2}} (Schwarz [61, 2.1.5 and 2.1.7] — 2.1.7 applies because ∗R^{W} = R^{W} ∗. Schwarz states the equation only for ω = η, but because R^{W} and S are symmetric, using polarisation it holds in the more general form):

\begin{aligned}(∇ω, ∇η)_{L^{2}} & = (R^{W} ω, η) + (dω, dη) + (δω, δη) + \\ & ∫_{M_{1}}(Sω, η)_{Λ^{∗}(T_{x}^{∗}M)} dx + ∫_{M_{2}}(S∗ω, ∗η)_{Λ^{∗}(T_{x}^{∗}M)} dx. \end{aligned}  (7.3)

7.4. Definition. We call the boundary of a Riemannian manifold (M, g) conformally geodesic near x ∈ ∂M, if a neighbourhood U of x exists so that g|_{U} = e^{2σ} · g′ (σ : U → ℝ), and the second fundamental form of ∂M ∩ U with respect to g′ vanishes identically, i.e. ∂M ∩ U is embedded geodesically in (U, g′).

7.5. Proposition. Let M be a Riemannian manifold, R_{p}^{W} the Weizenböck tensor and S_{p} the boundary tensor of definition 7.1.

(1) R_{1}^{W} = − Ric (Ric the Ricci operator).

(2) S_{p}|_{x} ≤ 0 iff the sum of the m − p smallest eigenvalues of l|_{x} is non-negative (l the second fundamental form operator with respect to the inward unit normal). Especially, if trace l ≥ 0, then S_{1} ≤ 0; and if the second fundamental form is ≥ 0 then S_{p} ≤ 0 ∀p.

(3) If the curvature operator is ≥ 0 then R_{p}^{W} ≤ 0 ∀p.

The (symmetric) curvature operator on Λ^{∗}T M is defined via the symmetric bilinear form ρ(x ∧ y, z ∧ t) = R(x, y, z, t).

(4) If ∂M is conformally geodesic near x and tr l|_{x} ≥ 0 then S_{p}|_{x} ≤ 0 ∀p.

(5) If (M, g) is conformally flat near x ∈ M and (m − 2p) Ric +r(p − 1)m/(m − 1) ≥ 0 at x (r denotes the scalar curvature), then R_{p}^{W} |_{x} ≤ 0.

Proof. For (1) and (3), compare Lawson and Michelsohn [39, II.8.3 and II.8.6]. (5) is a result of Goldberg [30, 3.9.11]. For (2) we explicitly perform the computations: The second fundamental form l : T ∂M × T ∂M → ℝ is defined by l(v, w)ν = (∇_{v}w)^{⊥} with ν the inward unit normal (·^{⊥} simply means projection onto this direction). Using the metric, we can consider l as a symmetric fiberwise operator l : T ∂M → T ∂M. Let e_{1}, …, e_{m−1} be an orthonormal base for T_{b}∂M of eigenvectors of l with eigenvalues λ_{i}, i.e. l(e_{i}, e_{j}) = δ_{ij}λ_{i}. Then ν, e_{1}, …, e_{m−1} is an orthonormal base for T_{b}M at b ∈ ∂M. Let ω be the p-form

ω_{b} = ∑_{i_{1}<···<i_{p}} α_{i_{1}...i_{p}}e_{i_{1}} ∧ ⋯ ∧ e_{i_{p}} + ∑ β_{i_{1}...i_{p−1}}ν ∧ e_{i_{1}} ∧ ⋯ ∧ e_{i_{p−1}}.

Then the explicit formulae of Schwarz [61, (2.1.13)] say that the coefficient in Sω_{b} of e_{i_{1}} ∧ ⋯ ∧ e_{i_{p}} is 0 and the coefficient γ_{i_{1}...i_{p−1}} of ν ∧ e_{i_{1}} ∧ ⋯ ∧ e_{i_{p−1}} is

γ_{i_{1}...i_{p−1}} = (Sω)_{b}(ν, e_{i_{1}}, …, e_{i_{p−1}}) = − ∑_{r}(l^{Λ}(e_{r})ω)(e_{r}, e_{i_{1}}, …, e_{i_{p−1}})

with

(l^{Λ}(e_{r})ω)(e_{r}, e_{i_{1}}, …, e_{i_{p−1}}) := ω(\underbrace{l(e_{r}, e_{r})}_{λ_{r}ν}, e_{i_{1}}, …, e_{i_{p−1}}) + ∑_{l=1}^{p−1} ω(e_{r}, …, \underbrace{l(e_{r}, e_{i_{l}})}_{0 if r ≠ i_{l}}, …)

= \begin{cases}0 & ; if r ∈ \{i_{1}, …, i_{p−1}\} \\ λ_{r}β_{i_{1},...,i_{p−1}} & ; else \end{cases} .

Therefore,

γ_{i_{1},...,i_{p−1}} = −( ∑_{r∉{i_{1},...,i_{p−1}}} λ_{r})β_{i_{1},...,i_{p−1}}.

Hence (S_{p}ω, ω)_{b} = − ∑_{i_{1}<···<i_{p−1}}(∑_{r∉{i_{1},...,i_{p−1}}})|β_{i_{1},...,i_{p−1}}|^{2}, and S_{p}|_{b} ≤ 0 iff the sum of the smallest (m−1)−(p−1) = m−p of the eigenvalues {λ_{r}} of l|_{b} is non-negative. For p = 1, this is the trace.

Last, we prove (4): Let g be the geodesic metric, and g^{∗} = e^{2σ}g a conformal metric. Then for the Christoffel symbols with respect to a chart we have (compare Gallot/Hulin/Lafontaine [29, solution of 2.55 c])

Γ^{∗}_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i}∂_{k}σ + δ_{k}^{i}∂_{j}σ − ∑_{l} g_{jk}g^{il}∂_{l}σ

(δ_{j}^{i} = 0 if i ≠ j and δ_{i}^{i} = 1). If the chart is a boundary chart we may assume that the first m − 1 coordinates are the coordinates of the boundary and x_{m} is orthogonal to the boundary (normal boundary chart). Then the components of the second fundamental form are related as follows:

l_{ij}^{∗} = Γ^{∗}_{ij}^{m} = l_{ij} − g_{ij}∂_{m}σ ∀i, j < m

We may use a Gaussian chart of the boundary, so that at one point we can require g_{ij} = δ_{ij}. ∂M geodesic in the metric g means l_{ij} = 0 ∀i, j. It follows that the eigenvalues of l^{∗}|_{x} are all equal (i.e. the principal curvatures of the boundary at a given point are all equal), and their common sign is determined by the sign of the trace and (2) applies.

7.6. Remark. The property of the second fundamental form to be non-negative at a point is related to convexity of the manifold (at its boundary). If trace l is non-negative, then the boundary is convex ”in the mean”.

7.7. Theorem. (L^{2}-Bochner) Let M^{m} be a complete orientable manifold with boundary. Suppose M has infinite volume.

• If the Weizenböck endomorphism R^{W} and the fundamental form endomorphism S are both negative semidefinite on k-forms, then H_{dR,(2)}^{k}(M, ∂M) vanishes. If S_{m−p} ≤ 0 instead of S_{p}, then the absolute L^{2}-de Rham cohomology H_{dR,(2)}^{k}(M) = {0}.

• If the Ricci tensor of M and the trace of the second fundamental form l of ∂M ⊂ M are both ≥ 0, then the first L^{2}-de Rham cohomology group H_{dR,(2)}^{1}(M, ∂M) vanishes.

• If the curvature operator and the second fundamental form are ≥ 0 then all L^{2}-de Rham cohomology groups H_{dR,(2)}^{∗}(M, ∂M) and H_{dR,(2)}^{∗}(M) vanish.

7.8. Remark. If M is compact, then L^{2}-de Rham cohomology and ordinary de Rham cohomology coincide. Then Bochner’s theorem reads as follows (and this is well known): replacing ”≥ 0” by ”> 0 at least at one point”, one gets the same vanishing results as above. If one sticks to the condition ≥ 0 one gets bounds on the dimension of the corresponding cohomology groups.

Proof of L^{2}-Bochner. First note that the second and the third statement are direct consequences of the first by proposition 7.5. We have to establish the first one. From the Hodge theorem 6.2 we see that it suffices to prove the non-existence of harmonic L^{2}-forms.

Remember equation (7.3) for a form ω ∈ C_{0}^{∞}(Λ^{p}T^{∗}M) with ω|_{∂M} = 0 (here ∂M = M_{1} and M_{2} = ∅):

|∇ω|_{L^{2}}^{2} = (R^{W} ω, ω)_{L^{2}} + |dω|_{L^{2}}^{2} + |δω|_{L^{2}}^{2} + ∫_{∂M}(Sω, ω)_{x} dx.

Suppose ω ∈ H^{p}(M, ∂M). By 5.8 dω = 0 = δω. To a given compact set K ⊂ M and to ϵ > 0 choose φ : M → [0, 1] with φ = 1 on K and |∇φ(x)| ≤ ϵ ∀x ∈ M. Then

|∇(φω)|_{L^{2}}^{2} = \underbrace{(R^{W} φω, φω)_{L^{2}}}_{≤0} +|d(φω)|_{L^{2}}^{2} + |δ(φω)|_{L^{2}}^{2} + ∫_{∂M} \underbrace{(Sφω, φω)}_{≤0}

≤ |φdω|_{L^{2}}^{2} + 2(dφ ∧ ω, φdω)_{L^{2}} + |dφ ∧ ω|_{L^{2}}^{2}

+ |φδω|_{L^{2}}^{2} ± 2(∗dφ ∧ ∗ω, φδω)_{L^{2}} + |dφ ∧ ∗ω|_{L^{2}}^{2}

=^{dω=0=δω} |dφ ∧ ω|_{L^{2}}^{2} + |dφ ∧ ∗ω|_{L^{2}}^{2}

≤^{dφ=∇φ} 2ϵ^{2}|ω|^{2}

Now assume that |∇ω|_{L^{2}} = C > 0. Choose φ and ϵ so that ϵ|ω|_{L^{2}} is sufficiently small and |φ∇ω|_{L^{2}} is close to C. Then

\begin{aligned}|∇(φω)|_{L^{2}}^{2} & = (∇φ ⊗ ω + φ∇ω, ∇φ ⊗ ω + φ∇ω)_{L^{2}} \\ & ≥^{|∇φ|≤ϵ} |φ∇ω|_{L^{2}}^{2} − ϵ^{2}|ω|_{L^{2}}^{2} − 2ϵ|ω|_{L^{2}}|φ∇ω|_{L^{2}} \end{aligned}

is close to C^{2}. This is a contradiction to the fact |∇(φω)|_{L^{2}}^{2} ≤ ϵ^{2}|ω|_{L^{2}}^{2}. Consequently, ∇ω = 0. Especially, the fiberwise norm of ω is constant. Since vol(M) = ∞ and ω is L^{2}, only ω = 0 is possible, therefore no nontrivial relative harmonic L^{2}-forms exist. For absolute harmonic forms ((∗ω)|_{∂M} = 0) identical arguments apply with ∫_{∂M}(Sω, ω) in equation (7.3) replaced by ∫_{∂M}(S ∗ ω, ∗ω), i.e. S_{p} replaced by S_{m−p}.

(Dodziuk proves the vanishing of square integrable harmonic forms on manifolds of infinite volume without boundary in [27] along similar lines.)

7.9. Remark. If vol(M) < ∞, but the other conditions of 7.7 hold, then dim H_{dR,(2)}^{p} ≤ dim Λ^{p}T_{x}^{∗}M. This is a direct consequence of the proof, which shows that all L^{2}-harmonic forms are parallel.

Curvature, coverings and symmetry

7.10. Example. Particularly interesting is the case where M is an infinite covering of a compact manifold N. The conclusion is: If the base manifold carries a metric which fulfills the positivity conditions in the L^{2}- Bochner theorem, then the corresponding L^{2}-cohomology groups H_{(2)}^{∗}(M, M_{1}) (with coefficients in an infinite quotient group) of the fundamental group vanish.

Take f.e. ℝP^{3}#ℝP^{3}#ℝP^{3} and (ℝP^{3}#ℝP^{3}#ℝP^{3}) − D^{3}. An explicit calculation of Lott and Lück [41, theorem 0.1] shows that the first L^{2}-cohomology of both of them is not zero. Therefore they can not carry a metric with non-negative Ricci curvature and (in the presence of boundary) non-negative trace of the second fundamental form. Since the ordinary first Betti number is zero, this result does not follow from the original theorem of Bochner.

More generally, for arbitrary connected sums M#N of closed manifolds (if not one of the fundamental groups is trivial or both are ℤ/2), the first L^{2}-Betti number is nonzero (compare Lott/Lück) [41, 3.5 and 3.7]) (note that the first L^{2}-Betti number depends on the fundamental group only). Via connected sum with suitable boundary manifolds (f.i. the ball D^{m}), and using the weakly exact Mayer-Vietoris sequence and pair sequence for L^{2}-cohomology ([20, 34] and [41, 2.2]), we can also produce numerous examples of boundary manifolds with non-vanishing first absolute and relative L^{2}-Betti number. We know now that all of these manifolds do not carry a metric with non-negative Ricci curvature and at the same time positive trace of the second fundamental form (if the boundary is non-void).

Cheeger and Gromoll [18] and [19] show that a closed manifold M with infinite fundamental group and non-negative Ricci curvature has a fundamental group π with the following structure: it is an extension 1 → Γ_{1} → π → K → 1, where Γ_{1} is finite and K is a Bieberach group, i.e. K fits itself into an extension 1 → ℤ^{k} → K → Γ_{2} → 1 with finite Γ_{2}. Of course, such a group π can never be a direct product (except π ≅ ℤ/2 ∗ ℤ/2).

Actually, Cheeger and Gromoll also show that M is finitely covered by a product with S^{1}. But a manifold with this structure has trivial L^{2}-cohomology (compare Lück [42]). Therefore, for closed manifolds, our result is also a consequence of the classification theorem of Cheeger and Gromoll.

In the case of normal coverings of compact manifolds or for other manifolds of bounded geometry with sufficient symmetry, we can give an even more general condition for the vanishing of L^{2}-cohomology groups. Now, we switch our point of view back to the noncompact object (f.i. the covering manifold). On such a manifold, the topology does not restrict the geometry very much. For example, ℝ^{m} can be equipped with the flat Euclidian metric or with the hyperbolic metric with constant negative curvature, which have quite different qualities.

Therefore, it is natural to restrict the metrics to consider. We will look for metrics in the bilipschitz class of a given metric only. This is proposed in Roe [57]. A similar problem occurs in the ”‘type problem”’ question to distinguish the conformal structure of the plane and the disc (Sario/Nakai [58] introduce bilipschitz equivalence in this connection). Recall the following:

7.11. Definition. Let (M, g) and (N, h) be two Riemannian manifolds, f : M → N a diffeomorphism. f is called bilipschitz, if C > 0 exists so that for the norms the following holds: ∥T_{x}f∥ ≤ C ∀x ∈ M and ∥T_{y}f^{−1}∥ ≤ C ∀y ∈ N.

Note that all the metrics on a compact manifold are bilipschitz. The same is true for the lifts to any covering, therefore, there is a natural bilipschitz class of metrics on a covering.

7.12. Remark. If (M, g) and (M, g′) are bilipschitz, then L^{2}(Λ^{p}T^{∗}M, g) and L^{2}(Λ^{p}T^{∗}M, g′) coincide, and the two norms are equivalent.

We will prove the following theorem:

7.13. Theorem. Let (M, g_{1}) be a oriented manifold of bounded geometry (possibly with boundary), so that R_{p}^{W} ≤ 0 outside a compact set and additionally S_{p} ≤ 0 outside a compact set. Let g be a metric on M which is bilipschitz to g_{1} and so that (M, g) admits an infinite discrete group Γ of isometries (quotient not necessarily compact).

Then H_{(2)}^{p}(M, ∂M) = {0}. If the condition S_{p} ≤ 0 is replaced by the condition S_{m−p} ≤ 0 outside a compact set, then H_{(2)}^{p}(M) = {0}.

7.14. Remark. If (M, g) is an infinite normal covering of a compact manifold, it obviously fulfills the conditions of the theorem.

The new and surprising feature (compared to theorem 7.7, which follows classical lines) is the existence of a compact subset, where the curvature assumption is not required to hold. This is due to the uniformity of the metric (infinite discrete group of isometries) and does not hold for arbitrary manifolds of bounded geometry, as the following example shows (this example was communicated to me by John Lott):

7.15. Example. Let X be a 2-dimensional infinite paraboloid. It has obviously positive curvature everywhere (and therefore, by theorem 7.7, no L^{2}-harmonic 1-forms exists on X). Now attach a 1-handle to X to obtain the manifold W. We disturb the manifold only on a compact set. Therefore, it retains positive curvature outside a compact. Nonetheless, the first L^{2}-cohomology of W is nonzero: We find compactly supported closed one-forms on W, which are not even the boundary of arbitrary (not necessarily square integrable) functions (here the metric does not matter). These forms represent non-trivial classes in L^{2}-cohomology.

Proof. We have to show why such a form is not in closure of the image of d. Let K_{n} ⊂ K_{n+1} be a sequence of compact submanifolds with boundary of codimension 0, so that W = ⋃_{n} K_{n}. Assume, ω ∈ \overline{dC_{0}^{∞}(W)}, then ω|_{K_{n}} ∈ \overline{dC_{0}^{∞}(K_{n})} (note that the absolute boundary conditions for functions are empty). On compact manifolds, L^{2}-cohomology and ordinary cohomology coincide, i.e. ω|_{K_{n}} = df_{n} for suitable f_{n} ∈ C^{∞}(K_{n}). Note that the function f_{n} is well defined up to an additive constant. Consequently, a compatible choice of f_{n} is possible in the sense that f_{n}|_{K_{n−1}} = f_{n−1}. They patch together to an f ∈ C^{∞}(W) with df = ω, which is a contradiction. Note that f is not necessarily square integrable.

For the proof of theorem 7.13 we need the following proposition which is essentially due to Roe [57, 1.11] if ∂M = ∅. We denote with Δ the unbounded operator on L^{2} with domain D_{Δ} := {ω ∈ H^{2}(Λ^{∗}T^{∗}M); ω|_{∂M} = 0 = δω|_{∂M}} (compare 5.2). Let P be the orthogonal projection onto its kernel H^{p}(M, ∂M). It has a smooth integral kernel p(x, y) by lemma 13.6.

7.16. Proposition. Suppose M is a ∂-manifold of bounded geometry. If R_{p}^{W} and S_{p} are non-positive outside a compact subset K of M, then ∀ϵ > 0 a compact subset L ⊂ M exists so that

sup_{x∈M−L}{|P s(x)|} ≤ ϵ|s|_{L^{2}} ∀s ∈ L^{2}(Λ^{p}T^{∗}M)

Proof. Suppose s ∈ L^{2}. Define s_{t} := e^{−tΔ}s, then s_{t} →^{t→∞} P s in L^{2} by the spectral theorem, and, obviously, s_{t} ∈ D_{Δ}. Even better: s_{t} ∈ D_{Δ^{k}}, the domain of the k-th iterate of (1+Δ)^{k}, ∀k ∈ ℕ. This is the case because (1 + x)^{k}e^{−tx} is a bounded function for x ≥ 0 if t > 0. Especially s_{t}|_{∂M} = 0. (Here we use the fact that Δ is self adjoint on a manifold of bounded geometry (5.14 and 4.25)). Let φ : M → [0, 1] be a smooth function with compact support L ⊂ M so that φ = 1 on K and |∇φ| < ϵ (take f.i. an approximation of the Lipschitz function x ↦ max{0, 1 − ϵ dist(x, K)}). Set ψ = 1 − φ. Observe that we find M-universal constants C_{k,l} > 0 so that for t > 1

\begin{matrix}|Δ^{l}s_{t}|_{H^{k}} =^{5.19} |(1 + Δ)^{k}Δ^{l}s_{t}|_{L^{2}} ≤ C_{k,l}t^{−l}|s|_{L^{2}} \\ |Δ^{l}s_{t}|_{C_{b}^{k}} ≤ C_{k,l}t^{−l}|s|_{L^{2}} \end{matrix}  (7.17)

The first estimate follows from the spectral theorem, and the second one from the first and the Sobolev embedding theorem. We arrive at the estimate

∫_{M−L}|∇s_{t}|^{2} ≤ ∫_{M} ψ(∇s_{t}, ∇s_{t}) =^{∇(ψs_{t})=∇ψ⊗s_{t}+ψ∇s_{t}} −(∇s_{t}, s_{t} ⊗ ∇ψ)_{L^{2}} + (∇s_{t}, ∇(ψs_{t}))_{L^{2}}

= −(∇s_{t}, s_{t} ⊗ ∇ψ)_{L^{2}} + \underbrace{(R^{W} s_{t}, ψs_{t})_{L^{2}}}_{≤0} + \underbrace{∫_{∂M}(Ss_{t}, ψs_{t})}_{≤0}

+ (ds_{t}, d(ψs_{t}))_{L^{2}} + (δs_{t}, δ(ψs_{t}))_{L^{2}}

( Apply (7.3). Note (ψs_{t})|_{∂M} = 0. Moreover, R^{W} ≤ 0 and S ≤ 0 on supp ψ and they are bundle endomorphisms. For the next inequality use Cauchy-Schwartz and |∇ψ| ≤ ϵ:)

≤|∇s_{t}|_{L^{2}} · ϵ · |s_{t}|_{L^{2}} + |(ds_{t}, dψ ∧ s_{t})_{L^{2}}| + |ds_{t}|_{L^{2}}^{2} + |δs_{t}|_{L^{2}}^{2} + |(δs_{t}, ∗(dψ ∧ ∗s_{t}))_{L^{2}}|
≤^{C_{1,0}}ϵ|s|_{L^{2}}^{2} + \underbrace{|ds_{t}|}_{≤∥d∥_{B(H^{1},L^{2})}|s_{t}|_{H^{1}}}|s_{t}|_{L^{2}} sup_{x∈M} \underbrace{|dψ(x)|}_{=|∇ψ|≤ϵ} + (Δs_{t}, s_{t})_{L^{2}} + |δs_{t}|_{L^{2}}|s_{t}|_{L^{2}} · ϵ

(First |∇s_{t}| ≤ |s_{t}|_{H^{1}} ≤ C_{1,0}|s|_{L^{2}} (by (7.17)), then Cauchy-Schwartz and (Δu, u) = (du, du) + (δu, δu) for u ∈ D_{Δ} by (5.16))

≤ϵ|s|_{L^{2}}^{2} + |Δs_{t}|_{L^{2}}|s_{t}|_{L^{2}} + ϵ · (∥d∥ + ∥δ∥)C_{1,0}|s|_{L^{2}}^{2}  by (7.17)
≤^{C}ϵ|s|_{L^{2}}^{2} + t^{−1}|s|_{L^{2}}^{2}  again by (7.17)

The constant C in the estimates is an M-universal constant involving ∥d∥_{B(H^{1},L^{2})}, ∥δ∥_{B(H^{1},L^{2})} and C_{k,l} of (7.17) for k, l ≤ 1. From now on, we proceed exactly as Roe does in [57, 1.11] to conclude

|s_{t}(y)| ≤ ϵ_{1}(ϵ)|s|_{L^{2}} ∀y ∉ K_{ϵ}  (7.18)

for t sufficiently large, where ϵ_{1}(ϵ) →^{ϵ→0} 0 and K_{ϵ} is a compact set depending only on ϵ. His analysis depends on the following facts only:

• the estimates (7.17)

• the existence of a function V_{0}(r) with vol(B(x, r)) ≥ V_{0}(r) ∀x ∈ M, with V_{0} monotonous and V_{0}(r) → ∞ if r → ∞. (We constructed such a function in lemma 3.20).

As a last step, we note from the spectral theorem that (1+Δ)^{k}e^{−tΔ}s →^{t→∞} (1+Δ)^{k}P s = P s in L^{2}-norm ∀k. Then, theorem 5.19 shows that s_{t} →^{t→∞} P s in the Sobolev space H^{k} ∀k. The Sobolev embedding theorem implies together with the estimate (7.18)

|P s(y)| ≤ ϵ_{1}(ϵ)|s|_{L^{2}} ∀y ∉ K_{ϵ}

Proof of theorem 7.13. Let D > 0 be a constant so that

D^{−1}|ω|_{L^{2}(g_{1})} ≤ |ω|_{L^{2}(g)} ≤ D|ω|_{L^{2}(g_{1})} and D^{−1}|ω(x)|_{Λ^{∗}(g_{1})} ≤ |ω(x)|_{Λ^{∗}(g)} ≤ D|ω(x)|_{Λ^{∗}(g_{1})}

Note that L^{2}(M, g) and L^{2}(M, g_{1}) are equal as topological vector spaces since g and g_{1} are bilipschitz. By 6.7 even the spaces of harmonic forms are isomorphic. The proof of lemma 8.12 shows that the restricted orthogonal projections

\begin{matrix}P_{1}| : & L^{2}(M, g_{1}) ⊃ H^{p}(M, ∂M, g) → H^{p}(M, ∂M, g_{1}) \\ P_{2}| : & L^{2}(M, g) ⊃ H^{p}(M, ∂M, g_{1}) → H^{p}(M, ∂M, g) \end{matrix}

are bounded and inverse to each other. Suppose now 0 ≠ s ∈ H^{p}(M, ∂M, g) exists. Take |s|_{L^{2}(g)} = 1, and fix x_{0} ∈ M with |s(x_{0})| = s_{0} > 0. Let p(x, y) be the (smooth) integral kernel of the projector P_{2}. Note that Q(y) = p(x_{0}, y) is a smooth L^{2}-section of the bundle {Hom(Λ^{p}T_{y}^{∗}M, Λ^{p}T_{x_{0}}^{∗}M)} over M. Choose K compact so that

∫_{K}∥Q(y)∥^{2} dy ≥ (1 − ϵ)|Q|_{L^{2}}^{2} with 0 < \sqrt{ϵ} < \frac{s_{0}}{2|Q|_{L^{2}}∥P_{1}∥}.

Because K is compact, vol(K) < ∞. Choose

0 < δ < \frac{s_{0}}{(2|Q|_{L^{2}}D^{2}\sqrt{vol(K)})}.

Accordingly to proposition 7.16 we find a compact set L ⊂ M so that

|P_{1}f(x)| ≤ δ|f|_{L^{2}} ∀x ∈ M − L ∀f ∈ L^{2}

Since the isometry group Γ is infinite and discrete, we find a γ ∈ Γ so that γ^{−1}(K)∩L = ∅. Replace now s by γ^{∗}s, x_{0} by γ^{−1}(x_{0}), Q(y) by p(γ^{−1}x_{0}, y) and K by γ^{−1}K. Because γ acts isometric, for the new data holds

s ∈ H^{p}(M, ∂M, g), |s(x_{0})| = s_{0}, ∫_{K}∥Q(y)∥^{2}dy ≥ (1 − ϵ)|Q|_{L^{2}}^{2}.

But now, by construction K ∩ L = ∅. Then (in terms of the metric g),

\begin{matrix}s_{0}^{2} = & |s(x_{0})|^{2} = |P_{2}(P_{1})s(x_{0})|^{2} = |∫_{M} p(x_{0}, y)(P_{1}s)(y) dy|^{2} \\ ≤ & 2 (|∫_{K} Q(y)P_{1}s(y) dy|^{2} + |∫_{M−K} Q(y)P_{1}s(y) dy|^{2}) \\ ≤ & 2 ∫_{K}∥Q(y)∥^{2} dy ∫_{K} \underbrace{|P_{1}s(y)|^{2}}_{≤δD|s|_{L^{2}(g_{1})}} dy + 2 ∫_{M−K}∥Q(y)^{2} dy∥ \underbrace{∫_{M−K} |P_{1}s(y)|^{2} dy}_{≤|P_{1}s|_{L^{2}}^{/}} \\ ≤ & 2|Q|_{L^{2}}^{2} ∫_{K} D^{2}δ^{2} \underbrace{|s|_{L^{2}(g_{1})}^{2}}_{≤D^{2}} +2ϵ|Q|_{L^{2}}^{2}∥P_{1}∥^{2} \\ ≤ & 2|Q|_{L^{2}}^{2}δ^{2}D^{4} vol(K) + 2ϵ|Q|_{L^{2}}^{2}∥P_{1}∥^{2} < \frac{s_{0}^{2}}{2} + \frac{s_{0}^{2}}{2} = s_{0}^{2} \end{matrix}

This is a contradiction, therefore no non-trivial relative L^{2}-harmonic forms exist. For the absolute case, the proof is completely analog. We only have to replace S by S∗ from equation (7.3). This also yields an absolute version of proposition 7.16.

7.19. Corollary. Using proposition 7.5, we can apply theorem 7.13 to more concrete situations in the following way. Let M be a compact manifold (possibly with boundary). Let M̃ be an orientable infinite normal covering of M. Here, we assume that all the Riemannian manifolds have bounded geometry.

• If the first relative L^{2}-Betti number b_{(2)}^{1}(M̃, ∂M̃) ≠ 0, then there is no metric g_{1} on M̃ which lies in the natural bilipschitz class, so that Ric ≥ 0 outside a compact set, and tr l ≥ 0 outside a compact set.

• If dim M = m and b_{(2)}^{m/2}(M̃) ≠ 0, then in the natural bilipschitz class of M̃ no metric g_{1} exists which is locally conformally flat outside a compact set, has nonnegative scalar curvature there, and whose boundary is locally conformally geodesic with mean curvature tr l ≥ 0 outside a compact set. Note that we allow for different and local conformal changes without boundedness or bilipschitz conditions.

• If the relative Euler characteristic χ(M, ∂M) ≠ 0, then in the natural bilipschitz class on the universal covering M̃ no metric exists so that the curvature operator and the second fundamental form operator are both ≥ 0 outside a compact set.

• If M is a 2-manifold or if M is a 4-manifold without boundary, and if the Euler characteristic χ(M) < 0, then b_{(2)}^{1}(M̃, ∂M̃) ≠ 0 and the first statement holds.

• If M is a closed hyperbolic manifold of even dimension, or more generally if its sectional curvature k is pinched: −1 ≤ k ≤ C < −(m−2)^{2}/(m−1)^{2} then the second statement applies.

Proof. These are direct consequences of the algebraic results in proposition 7.5, as soon as we note that because of the L^{2}-index theorem for coverings of ∂-manifolds χ(M) = ∑_{p}(−1)^{p}b_{(2)}^{p}(M̃) and χ(M, ∂M) = ∑_{p}(−1)^{p}b_{(2)}^{p}(M̃, ∂M̃) (compare proposition 13.5). Moreover, if M is even-dimensional then because of Poincare duality χ(M) = χ(M, ∂M). Also, the statement concerning closed 4-manifold follows from Poincare duality: b_{(2)}^{1}(M̃) = b_{(2)}^{3}(M̃). For the last statement we refer to Donnelly/Xavier [28, 4.1] (compare 6.14), who prove the vanishing of b_{(2)}^{i}(M̃) for i ≠ m/2, combined with a result of Bourguignon/Karcher [14, pp. 78–79, 89] which says χ(M) ≠ 0.

7.20. Remark. John Roe [57] proves a corresponding theorem, but only for amenable coverings of manifolds without boundary. A covering is called amenable, if it is normal and if the covering group is amenable. (Also, he replaces bilipschitz by the stronger assumption ”strict quasi isometry”). His question, if the theorem holds for arbitrary coverings, are answered affirmative by our result. It is interesting that to some extent we follow Roe’s proposal on how to prove this: first, we observe that the harmonic forms are bilipschitz equivalent (this replaces an index theory for non-amenable coverings). Secondly, we find a way to use the symmetry of the covering, answering questions about arbitrary metrics. Roe’s index theoretic methods yield a similar statement [57, 3.3] for scalar curvature, relating the non-vanishing of Â(M) to the non-existence of a metric of positive scalar curvature on amenable coverings. This does not generalize to non-amenable coverings, as show counterexamples of Block and Weinberger [12] for arbitrary non-amenable covering groups. In some sense this shows that no index theorem for arbitrary non-amenable spaces can exist. This fits into our framework, because there is no ”invariance of harmonic spinors” under bilipschitz perturbations.

The following theorem shows that for manifolds without boundary the Ricci-curvature statements follow already from Roe’s theorem. We even obtain the much stronger result of polynomial growth of balls, using Gromov’s proof of Bishop’s inequality. This shows that Ric ≥ 0 even if it is required only outside a compact set, is a very rare property. Note that polynomial growth is a quasi-isometry invariant of Riemannian manifolds in Gromov’s [33] sense. The theorem is probably well known, but I could not find a reference in the literature.

7.21. Theorem. Suppose (M^{m}, g) is a complete Riemannian manifold without boundary. Suppose Ric ≥ 0 outside a compact set K. Then M has polynomial growth. More precisely, for every x ∈ M we find C_{1}, C_{2} > 0 so that

vol(B(x, r)) ≤ C_{1} + C_{2}r^{m} ∀r ≥ 0.

Then (M, g) is amenable in the following sense: for every ϵ > 0 and R > 0 we find a compact set K so that

vol(U_{R}(K)) ≤ (1 + ϵ) vol(K) (U_{R}(K) := {x ∈ M; d(x, K) ≤ R})

Suppose (M, g′) is bilipschitz to another metric (M, g′), and this metric admits a discrete group of isometries Γ. Then (M, g′) has also polynomial growth with exponent at most m. The proof of Milnor [45] does work without change and shows that every finitely generated subgroup of Γ has polynomial growth with exponent ≤ m. It follows from Gromov [31] that Γ is virtually nilpotent, especially amenable. Good examples for the manifolds (M, g′) are normal coverings of compact manifolds.

Proof. Let x ∈ M. Since M is complete, the exponential map exp : T_{x}M → M is surjective. Since K is compact, it is contained in a ball of radius R around x. Let −k < 0 be an lower bound for the Ricci curvature of M (Ric ≥ −(m − 1)kg) — this exists since Ric ≥ 0 outside K. Compare M via the exponential map with the rotationally symmetric Riemannian manifold (ℝ^{m} = T_{x}M, g) with metric ds^{2} = dr^{2}+f(r)^{2}dψ^{2} (in polar coordinates, dψ^{2} the metric of the standard S^{m−1}). Choose f so that Ric ≤ −(m − 1)kg on B(0, R) and Ric ≤ 0 everywhere. Then we can apply Gromov’s proof of the Bishop inequality (Gallot/Hulin/Lafontaine [29, proof of 4.19]) to obtain the following estimate:

vol(B(x, r) ⊂ M) ≤ vol(B(0, r) ⊂ (ℝ^{m}, g)) ∀r.

(originally, the authors compare with a complete simply connected space of constant curvature −k).

It remains to construct an f(r) so that g fulfills the curvature conditions mentioned above and so that f(r) = ar + b for r ≥ R_{1}. Then we have polynomial growth:

vol(B(0, r) ⊂ (ℝ^{m}, g)) = vol(S^{m−1}) ∫_{0}^{r} f(r)^{m−1}dr

68 8 L^{2}-HODGE-DE RHAM THEOREM

≤ vol(B(0, R_{1})) + vol(S^{m−1})(am)^{−1}(ar + b)^{m} − vol(S^{m−1})(am)^{−1}b^{m} The curvature conditions are fulfilled with the following f:

f(r) = \frac{1}{k} sinh(kx) if 0 ≤ r ≤ R

(this is a standard model for the hyperbolic space of constant curvature −k). Then extend f to a smooth map with f′′ ≥ 0 (then automatically f′ ≥ 1) so that eventually for r ≥ R_{1} > R: f′′ = 0. These conditions ensure that the curvature operator of the rotationally symmetric manifold is ≤ 0 (compare the computation of the curvature of a warped product in O’Neill [47, 7.42]), and the proof is complete.

It is interesting to ask how far this theorem holds for manifolds with boundary. If we have a complete manifold with convex boundary, the method of the proof above does go through without change, because the exponential map remains surjective in this case, which is sufficient to get upper bounds on the volume. If we weaken the condition at the boundary to ”convex in the mean”, i.e. only the trace of the second fundamental form is ≥ 0 and not longer the second fundamental form itself, we can construct counterexamples:

7.22. Example. Start with the unit ball D^{3} in ℝ^{3}. Drill two perpendicular tubes through the ball which meet near the center. Smoothen the corners to get a complete Riemannian manifold with flat interior. The manifold is homotopy equivalent to S^{1} ∨ S^{1} ∨ S^{1}. Consequently, the universal covering has exponential growth of balls.

So far, the boundary of the drilled cylinders is concave. But instead of straight cylinders we can drill parabolic holes which are narrowst at the center and widen in direction off the center. If this widening is sufficiently fast, the trace of the second fundamental form of the paraboloids is ≥ 0. The smoothing of the corners does not contribute boundary which has tr l ≤ 0, because all the corners are convex. The universal covering is then a flat manifold with boundary with tr l ≥ 0 but with exponential growth of balls (because we did not change the fundamental group). Note that the fundamental group is certainly not amenable.

8 L^{2}-Hodge-de Rham theorem

In this section we prove the L^{2}-de Rham theorem. The strategy is the following: first prove the theorem if ∂M = ∅ (this was done by Dodziuk [26]). Then use this result and a standard doubling-trick to prove the result if the metric is a product near the boundary. To achieve this, we repeat some facts from representation theory of finite groups. The last step consists in the comparison of an arbitrary metric with a suitable product metric near the boundary (constructed in section 3). Important is, that the isomorphism between the harmonic forms with respect to the two different metrics commutes with the de Rham integration map. Here we use the fact that all the Sobolev spaces of the new metric are equal to the Sobolev spaces of the old metric. For two bilipschitz metrics, we get at least an abstract isomorphism. However, in many cases, f.i. in the definition of analytic torsion, we must use the concrete de Rham integration map.

Let (M, g) be a manifold of bounded geometry with boundary ∂M = M_{1} ⨿ M_{2}. Either M_{1} or M_{2} or both may be empty. Let (K, K_{1}, K_{2}) be a g-bounded smooth triangulation of (M, M_{1}, M_{2}) (compare definition 1.5).

L^{2}-de Rham theorem if ∂M = ∅ 69

8.1. Definition. The L^{2}-cochain complex C_{(2)}^{∗}(K, K_{1}) is defined as the subcomplex of the cochain complex C^{∗}(K, K_{1}) consisting of those chains whose coefficients are square summable. This is in the obvious way a separable Hilbert space. Because K is g-bounded this is actually a subcomplex and the differential d is L^{2}-bounded (compare Dodziuk [26]). Its L^{2}-cohomology are the Hilbert spaces

H_{(2)}^{p}(K, K_{1}) = ker d_{p}/\overline{im d_{p−1}}

We repeat the L^{2}-Hodge-de Rham theorem we want to prove:

8.2. Theorem. Let H^{∗}(M, M_{1}) be the space of smooth harmonic square integrable forms on M which fulfill absolute boundary conditions on M_{2} and relative boundary conditions on M_{1} (defined in 5.7). Then, the de Rham map (integration of forms over simplexes)

A : H^{∗}(M, M_{1}) → H_{(2)}^{∗}(K, K_{1})

yields an isomorphism between this space and the relative L^{2}-cohomology H_{(2)}^{∗}(K, K_{1}) of the triangulation.

L^{2}-de Rham theorem if ∂M = ∅

8.3. Theorem (Dodziuk 1981). If M has bounded geometry and if ∂M = ∅, then Hodge-de Rham-theorem 8.2 is true .

Proof. This is the result of [26] (compare also [24]). We repeat the argument here for convenience. First, note that the Hodge decomposition 5.19, which is relatively easy to prove if ∂M = ∅, implies with d_{k,p} : H^{2k}(Λ^{p}T^{∗}M) → H^{2k−2}(Λ^{p+1}T^{∗}M)

H^{p}(M) = ker(d_{k,p})/\overline{im(d_{k+1,p−1})}.

We extended this result to non-void boundary in section 6.

For k sufficiently large the de Rham map

A : H^{2k}(Λ^{p}T^{∗}M) → C_{(2)}^{p}(K)

is defined by the Sobolev lemma. It is easy to see that it is bounded and commutes with d. We are therefore tempted to construct a chain inverse. Following an idea of Whitney [69] one can actually construct a bounded map

W : C_{(2)}^{p}(K) → H^{2k}(Λ^{p}T^{∗}M) ∀k ∈ ℕ

so that A ∘ W = 1, and W commutes with d. We should note that W depends on the triangulation K.

Combined with the Hodge decomposition we see directly that our original map A : H^{∗}(M) → H_{(2)}^{∗}(K) is surjective.

For injectivity we will use the following auxiliary lemma: for every ω ∈ H^{2k}(Λ^{p}T^{∗}M) and for every ϵ > 0 exists a g-bounded subdivision K′ of K so that |W_{K′}Aω − ω|_{L^{2}} < ϵ. This can be shown using the inequality

|ω(x) − W_{K′}Aω(x)| ≤ C · diam τ · sup_{y∈|τ|}|∇ω(y)| for x ∈ |τ|

where τ is a simplex of maximal dimension of K. Note that the integral over the square of the former is just |W_{K′}Aω − ω|_{L^{2}}. In this way, it is not hard to see that a triangulation with sufficiently small simplexes will yield the desired result. For the details compare Dodziuks [26, 24, 3.9].

Now we are ready to prove injectivity (we follow [24, 3.10]): take ω ∈ H^{p}(M) with [Aω] = 0 ∈ H_{(2)}^{p}. Fix ϵ > 0 and choose a subdivision K_{ϵ} such that |ω − W_{K_{ϵ}}Aω|_{L^{2}} < ϵ.

Now, since Aω ∈ \overline{dC_{(2)}^{p−1}(M)}, we find for ϵ_{1} > 0 an f_{ϵ_{1}} ∈ C_{(2)}^{p−1} with |Aω − df_{ϵ_{1}}| < ϵ_{1}. Therefore,

|ω − dW_{K_{ϵ}}f_{ϵ_{1}}| ≤^{dW =W d} |ω − W_{K_{ϵ}}Aω| + |W_{K_{ϵ}}Aω − W_{K_{ϵ}}df_{ϵ_{1}}| ≤ ϵ + ||W_{K_{ϵ}}|| · ϵ_{1}

Since ϵ_{1} can be chosen independently of W_{K_{ϵ}} we see ω ∈ \overline{dH^{∞}} ⟹^{H^{p}⊥dH^{∞}} ω = 0.

L^{2}-de Rham theorem for product metrics

Representation theory

In the following suppose that G is a finite group. We will deal with an additional action of G on our spaces. Therefore, we will start with some preliminaries:

Remember from the real or complex representation theory of finite groups that there is only a finite number of different finite dimensional irreducible representations of G. Call them V_{1}, …, V_{r}.

8.4. Lemma. Let X be an infinite dimensional G-space. Then

X = X_{1} ⊕ ⋯ ⊕ X_{r} with G-spaces X_{i}

The X_{i} correspond to V_{i} in the following sense:

x ∈ X_{i} ⟹ ℂG · x ≅ ⨁_{α=1}^{n_{x}} V_{i}

Here ℂG · x is the G-space generated by x.

Proof. In fact, we have a direct sum X_{1} ⊕ ⋯ ⊕ X_{r} ⊂ X. Take on the other hand x ∈ X. Then ℂGx is a finite dimensional ℂ vector space. Therefore,

ℂGx = ⨁ W_{i} with W_{i} ≅_{G} ⨁_{α+1}^{r_{i}} V_{i}.

This decomposition is unique. Accordingly, write x = ∑ w_{i} with w_{i} ∈ W_{i}. Then w_{i} ∈ X_{i}. If not, there would be j ≠ i with V_{j} ⊂ ℂGw_{i} ∈ ⨁ V_{i} in contradiction to finite dimensional representation theory.

It follows similarly that each X_{i} is a G-space, i.e. GX_{i} ⊂ X_{i}.

8.5. Remark. If G acts isometrically on a Hilbert space X, the above decomposition is orthogonal.

L^{2}-de Rham theorem for product metrics 71

8.6. Lemma. Suppose X and Y are G-representations for a finite group G. Suppose we have an isomorphism f : X → Y which is a G-map. Let X = X_{1} ⊕ ⋯ ⊕ X_{r} and Y = Y_{1} ⊕ ⋯ ⊕ Y_{r} be the G-decompositions. Then f(X_{i}) ⊂ Y_{i} and

f_{i} = f|_{X_{i}} : X_{i} → Y_{i} is an isomorphism

Proof. Since f is a G-map, f(X_{i}) has the same G-type as X_{i} and Y_{i} ⟹ f(X_{i}) ⊂ Y_{i}. Since f is surjective f_{i} has to be surjective, too. Injectivity is clear.

8.7. Example. Take G = V_{4} = ℤ/2 × ℤ/2 = ⟨τ_{1}, τ_{2}⟩. Then there are four irreducible V_{4}-representations, all of which are one dimensional, namely the trivial one 𝕂^{++}, and additionally 𝕂^{+−}, 𝕂^{−+}, 𝕂^{−−}. The first and second sign resp. say that τ_{1} resp. τ_{2} acts by multiplication with this sign.

For an arbitrary V_{4}-space X we get the decomposition

X = X^{++} ⊕ X^{+−} ⊕ X^{−+} ⊕ X^{−−} : x = x_{++} + x_{+−} + x_{−+} + x_{−−}

Here x_{++} = (x + τ_{1}x + τ_{2}x + τ_{1}τ_{2}x)/4, x_{+−} = (x + τ_{1}x − τ_{2}x − τ_{2}τ_{1}x)/4, . . ..

Product metrics

Now we will apply this theory to our geometric situation. Let (M, g) be an oriented manifold of bounded geometry with boundary ∂M = M_{1} ⨿ M_{2}. Suppose the Riemannian metric is a product near the boundary. Construct the following “quadrupel” (respecting this product structure):

W = (M ∪_{M_{1}} −M) ∪_{M_{2}⨿−M_{2}} −(M ∪_{M_{1}} −M)

We have an obvious V_{4} = ℤ/2 × ℤ/2-action on W generated by flips τ_{1} along M_{1} ⨿ −M_{1} and τ_{2} along M_{2} ⨿ −M_{2}.

The Riemannian metric on M can be extended to a metric ḡ on W so that V_{4} acts isometrically. This induces a unitary V_{4} action on H(W). Use the g-bounded triangulation K of M to get a V_{4}-invariant triangulation K̄ of W. Obviously, K̄ is ḡ-bounded. Then we get a unitary action V_{4} on H_{(2)}^{(∗)}(W). Since these operations come from diffeomorphism of the underlying manifold the de Rham map A is V_{4}-equivariant by its geometric definition. Dodziuks theorem says that A is an isomorphism

A : H^{∗}(W) → H_{(2)}^{∗}(K̄)

The general theory yields

8.8. Lemma. The de Rham map restricted to the (−+)-eigenspaces is an isomorphism

A : H^{−+}(W) → H_{(2)}^{−+}(K̄)

Here X^{−+} is characterized by τ_{1}x = −x and τ_{2}x = x for x ∈ X.

Now we have to identify these eigenspaces.

8.9. Lemma. Denote the inclusion of one copy of M in W by i : M ↪ W. Then we get an isomorphism

i^{∗} : H^{−+}(W) → H(M, M_{1}).

Proof. We simply write down the inverse map: take ω ∈ H(M, M_{1}) ⟹ ω|_{M_{1}} = 0 = (∗ω)|_{M_{2}}. Define ω̄ ∈ L^{2}(W) by i^{∗}ω̄ = ω, i^{∗}τ_{1}^{∗}ω̄ = −ω, i^{∗}τ_{2}^{∗}ω̄ = ω, i^{∗}τ_{1}^{∗}τ_{2}^{∗}ω̄ = −ω. Obviously, V_{4} acts in the correct way on ω̄. We have to check that ω̄ is not only in L^{2} but actually in H. It suffices to show that Δω̄ = 0 in the weak sense. Then by elliptic regularity (4.15 and 5.14) ω̄ ∈ H^{∞} and Δω = 0 in the smooth sense.

Take an arbitrary η ∈ H^{∞}(W). (In the following we omit i^{∗} where appropriate).

(ω̄, Δη) = ∫_{i(M)∪−τ_{1}i(M)∪−τ_{2}i(M)∪τ_{1}τ_{2}i(M)} ω̄ ∧ ∗Δη

= ∫_{i(M)} ω̄ ∧ ∗Δη − τ_{1}^{∗}(ω̄ ∧ ∗Δη) − τ_{2}^{∗}(ω̄ ∧ ∗Δη) + (τ_{1}τ_{2})^{∗}(ω̄ ∧ ∗Δη)

= ∫_{M} ω ∧ ∗Δi^{∗}η − ω ∧ ∗Δi^{∗}τ_{1}^{∗}η + ω ∧ ∗Δi^{∗}τ_{2}^{∗}η − ω ∧ ∗Δi^{∗}τ_{1}^{∗}τ_{2}^{∗}η

=^{ω,η∈H^{∞}} ∫_{M} \underbrace{dω}_{=0} ∧ ⋯ + ∫_{i(M)} \underbrace{δω}_{=0} ∧ ⋯

± ∫_{∂M} ω ∧ (∗d(η − τ_{1}^{∗}η + τ_{2}^{∗}η − τ_{1}^{∗}τ_{2}^{∗}η)) ± ∫_{∂M} δ(η − τ_{1}^{∗}η + τ_{2}^{∗}η − τ_{1}^{∗}τ_{2}^{∗}η) ∧ (∗ω)

=^{\begin{matrix}ω|_{M_{1}}=0 \\ (∗ω)|_{M_{2}}=0 \end{matrix}} ± ∫_{M_{2}} ω ∧ ∗d \underbrace{(η − τ_{1}^{∗}η − η + τ_{1}^{∗}η)}_{=0} ± ∫_{M_{1}} δ \underbrace{(η − η + τ_{2}^{∗}η − τ_{2}^{∗}η)}_{=0} ∧ ∗ ω

= 0

Note that here ∫_{∂M} ω ∧ ∗η means ∫_{∂M} b^{∗}ω ∧ b^{∗}(∗η) if b : ∂M ↪ M. The Hodge-∗- operator remains the one of Λ^{∗}T^{∗}M. Furthermore note that τ_{1,2}^{∗}∗ = − ∗ τ_{1,2}^{∗} since τ_{1,2} are orientation reversing isometries, and τ_{1,2}^{∗} commute with d, δ, Δ. Moreover, τ_{1}η|_{M_{1}} = η|_{M_{1}} and τ_{2}η|_{M_{2}} = η|_{M_{2}}.

8.10. Lemma. The inclusion i : M ↪ W induces an inclusion i : K ↪ K̄ and this an isomorphism

i^{∗} : H_{(2)}^{−+}(K̄) → H_{(2)}(K, K_{1}).

Proof. The existence of an inverse map on the level of L^{2}-cochains is, given the boundary conditions, clear. It induces the inverse map on cohomology.

This concludes the proof of the Hodge-de Rham theorem 8.2 if the metric is a product near the boundary.

De Rham isomorphism: general case

Now we will prove the general case of the Hodge-de Rham theorem 8.2. We have the following strategy: We know already that the result is true if the metric is a product near the boundary. If g is arbitrary, so that(M, g) has bounded geometry, the doubling trick does not work, since the metric would no longer be smooth. However, we constructed in theorem 3.26 a metric g′ which is a product near the boundary and which is closely related to g. Especially (M, g′) has bounded geometry and the Sobolev spaces are equal. The latter property can be used show that there is an isomorphism of the spaces of harmonic forms which commutes with the de Rham map. Therefore, the positive result for product metrics implies the general result. If the two metrics are only bilipschitz the L^{2}-de Rham cohomologies are abstractly isomorphic. This is due to the fact that the de Rham complex uses only the operator d and restriction of forms to the boundary, which do not depend on the metric at all and this is proved first.

To compare the spaces of harmonic forms (and the de Rham map) for two different metrics we use the following functional analytical lemma:

8.11. Lemma. Suppose X is a Banach space, U, V and A are closed subspaces with

U ∩ A = {0} = V ∩ A; U + A = X = V + A

Then the projection p_{U} onto U along A

p_{U} : X = U + A → U : u + a ↦ u

restricted to V yields an isomorphism of topological vector spaces with inverse p_{V}.

Proof. The inverse is given by p_{V} |_{U} because for u ∈ U we have u = v + a with v ∈ V and a ∈ A unique. Apparently v = u − a is the decomposition of v. Then p_{V} (u) = v and p_{U}(v) = u. The projections are bounded because we deal with closed subspaces of Banach spaces.

Now suppose that (M, g_{1}) is a complete oriented Riemannian ∂-manifold and g_{2} is another metric on M bilipschitz to g_{1}. Operators and spaces depending on the metric will be decorated with subscripts 1 or 2 accordingly. Note that the topological vector space L^{2}(Λ^{p}(T^{∗}M)) does not depend on the metric.

8.12. Lemma. The g_{2}-orthogonal projection

pr_{2} : L^{2}(Λ^{p}(T^{∗}M)) → H^{p}(M, M_{1}; g_{2})

restricted to H(M, M_{1}; g_{1}) is a bounded isomorphism with inverse the corresponding restriction pr_{1}|.

Proof. We want to apply lemma 8.11. Therefore, we prove

H^{p}(M, M_{1}; g_{1}) ⊕ \overline{dΩ_{d}^{p−1}} = H^{p}(M, M_{1}; g_{2}) ⊕ \overline{dΩ_{d}^{p−1}}.

Note that \overline{dC_{0}^{∞}(Λ^{p−1}(T^{∗}M); ·|_{M_{1}} = 0)} = \overline{dΩ_{d}^{p−1}} does not depend on the metric. Consequently, we have only to show: ω ∈ H^{p}(M, M_{1}; g_{1}) ⟹ ω ∈ H^{p}(M, M_{1}; g_{2})⊕\overline{dΩ_{d}^{p−1}}. Take such an ω. By the Hodge decomposition, we must prove ω ⊥_{2} δ_{2}Ω_{δ,2}^{p+1}. If η ∈ Ω_{δ,2}^{p+1} then

(ω, δ_{2}η)_{2} = (\underbrace{dω}_{=0}, η)_{2} ± ∫_{M_{1}} \underbrace{ω|_{M_{1}}}_{=0} ∧ ∗_{2} η|_{M_{1}} ± ∫_{M_{2}} ω|_{M_{2}} ∧ \underbrace{∗_{2}η|_{M_{2}}}_{=0} = 0

8.13. Lemma. Suppose we are in the situation of lemma 8.12 and, additionally, (M, g_{1}) and (M, g_{2}) are manifolds of bounded geometry, such that the identity on smooth forms with compact support induces a topological isomorphism of all the Sobolev spaces (i.e. they do not depend on the metric). Let K be a g_{1}-bounded triangulation of M. Then the projections pr_{i}| commute with the de Rham map A.

Proof. Write h ∈ H^{p}(M, M_{1}; g_{1}) as h = h′ + x with h′ ∈ H^{p}(M, M_{1}; g_{2}) and x ∈ \overline{dΩ_{∞}^{p−1}} (use the Hodge decomposition 5.19). h, h′ ∈ H^{∞} ⟹ x ∈ H^{∞} and x|_{M_{1}} = 0 since h|_{M_{1}} = 0 = h′|_{M_{1}}. Moreover, dx = 0. Therefore, the following statement concludes the proof.

8.14. Lemma. Suppose x ∈ H^{∞} ∩ \overline{dΩ_{∞}^{p−1}} and x|_{M_{1}} = 0 and dx = 0. Then the de Rham map A is defined on x and [Ax] = 0 ∈ H_{(2)}^{p}(M, M_{1}).

Proof. The de Rham map

A : H^{2k}(Λ^{p}(T^{∗}M)) → C_{(2)}^{p}(K)

is defined and bounded for k sufficiently large. (Here C_{(2)}^{p} denotes the space of square summable cochains of the triangulation K. The proof of Dodziuk [24, 26] remains valid without changes in our situation: ∂M ≠ ∅ and possibly complex coefficients)). It induces a bounded map

A : {ω ∈ H^{2k}(Λ^{p}(T^{∗}M)); ω|_{M_{1}} = 0, dω = 0} → H_{(2)}^{p}(M, M_{1})

Since x ∈ H^{∞}: Ax ∈ H_{(2)}^{p}(M, M_{1}) is defined.

By theorem 5.19, x = (1 + Δ)^{k}y for a unique y ∈ \overline{dΩ_{∞}^{p−1}} (from elliptic regularity we know that actually y ∈ H^{∞}). Remember that here the closure is taken in the topology of H^{2k}. We find

y_{n} ∈ H^{∞}(Λ^{p+1}(T^{∗}M), bd(∞)) with y = lim_{n} dy_{n}.

Especially,

dy =^{d continuous on H^{2k}} lim_{n} ddy_{n} = 0.

Next observe (1 + Δ)^{k} = ∑_{i=0}^{k} \binom{k}{i}Δ^{i} and

Δ^{i} = (dδ + δd)^{i} =^{dd=0=δδ} (dδ)^{i} + (δd)^{i} for i > 0

Altogether

x = (1 + Δ)^{k}y = y + ∑_{i=1}^{k} \binom{k}{i} ((dδ)^{i}y + \underbrace{(δd)^{i}y}_{=0}) = y + d (\underbrace{δ ∑_{i=1}^{k} \binom{k}{i}(dδ)^{i−1}y}_{=:z})

Note that z ∈ H^{∞} since y ∈ H^{∞}. Moreover, by proposition 5.4 and H^{2k}-boundedness of the boundary differential operators z|_{M_{1}} = 0 ⟹ A(dz) = d(Az) = 0 ∈ H_{(2)}^{p}(M, M_{1}). Therefore,

Ax = Ay + A(dz) = Ay.  Now
Ay = A(lim_{n} dy_{n}) =^{A is H^{2k}-continuous} lim_{n} A(dy_{n})

and A(dy_{n}) = dA(y_{n}) = 0 ∈ H_{(2)}^{p}(M, M_{1}).

The metric g′ constructed in lemma 3.26 fulfills all the conditions of the lemma and has product structure near the boundary. For this by step 2 the de Rham map is an isomorphism. From the results above we see that it must be an isomorphism for arbitrary metrics and therefore the Hodge-De Rham theorem 1.6 is proved.

8.15. Corollary. Suppose M is a normal covering of a compact manifold N with covering group Γ and K is the lift of a triangulation of N. Then, by its geometric construction, the de Rham isomorphism

A : H^{p}(M, M̃) → H_{(2)}^{p}(K, K_{1})

commutes with the action of Γ and, therefore, is an isomorphism of N (Γ)-modules. This proves 1.7.

8.16. Remark. Let g be a metric on M bilipschitz to a bounded geometry metric g_{1}. In the proof of 8.14 we saw that projection to H^{∗}(M, M_{1}, g_{1}) composed with A does not depend on the specific metric g_{1}. Therefore, on (M, g) a natural de Rham isomorphism is given, but it may not coincide with integration over simplexes.

8.17. Remark. For closed manifolds, instead of explicitly constructing the de Rham map and it’s inverse, what we have done here, one can abstractly show that de Rham cohomology and simplicial cohomology are isomorphic, using a spectral sequence argument. This does also work for L^{2}-cohomology (at least in the covering case), as is indicated by Gromov [33] and Pansu [49], and possibly one can extend the argument to the case of manifolds with boundary. However, in many cases it is desirable to have an explicit isomorphism, which is provided by our theorem.

Part II

L^{2}-index theorem for ∂-manifolds

9 Traces for A-module morphisms

76 9 TRACES FOR A-MODULE MORPHISMS

In this section, we will define a regularized trace Sp_{φ} for A-module endomorphism of an arbitrary Hilbert A-module V. Here, A is a von Neumann algebra with finite trace φ. If A = ℂ and V is simply a Hilbert space, it is well known what to do: Sp(B) = ∑_{i}(Be_{i}, e_{i}) for an orthonormal base {e_{i}} of V (here, B should be positive or of trace class). Also, we know what to do if V = l^{2}(A), then the endomorphisms can be identified with A and we can apply the given trace φ.

The strategy will be the following: for operators on l^{2}(A) ⊗ H, we combine the Hilbert space trace Sp and the trace on A in a rather obvious manner. An arbitrary Hilbert A-module V embeds isometrically into l^{2}(A) ⊗ H for a suitable Hilbert space H. Using this embedding, we can extend Sp_{φ} to arbitrary A-module endomorphisms on V.

In a last step, we define the φ-trace class property for A-module morphisms between different Hilbert A-modules. It is impossible to define Sp_{φ} for morphisms which are not endomorphisms. Nevertheless, the greater flexibility is often useful, especially the relation

Sp_{φ}(fg) = Sp_{φ}(gf) if f : V_{1} → V_{2} and g : V_{2} → V_{1}

and either of the operators is of φ-trace class.

9.1. Convention. On B(H) we use the usual Hilbert space trace Sp(B) = ∑_{i}(Bh_{i}, h_{i}), where {h_{i}} is an orthonormal base of H and B a positive or trace class operator on H.

We use the usual tensor product symbol to denote the Hilbert space completion of the algebraic tensor product of two Hilbert spaces and the von Neumann algebra generated by the algebraic tensor product of two von Neumann algebras, respectively.

We use the term ”trace class operator” to denote operators in a trace class ideal.

9.2. Definition. A Hilbert A-module is a Hilbert space V with continuous left A-action so that an isometric embedding V ↪ l^{2}(A) ⊗ H exists which is compatible with the Aaction on l^{2}(A) ⊗ H. The embedding is not part of the structure. If V, W are two Hilbert A-modules, a bounded linear map f : V → W which is compatible with the A-action is called an A-module morphism.

Recall the following facts:

9.3. Proposition. On an arbitrary von Neumann algebra B with trace φ we have the φ-trace class ideal B_{φ} with positive part B_{φ}^{+} = {a ∈ B; a ≥ 0, φ(a) < ∞}. Additionally, there is the φ-Hilbert Schmidt ideal n = {a ∈ B; φ(a^{∗}a) < ∞}. It is well known that B_{φ} = n^{2}, even B_{φ} = {fg; f, g ∈ n} and that B_{φ} is the linear span of its positive part (compare Dixmier [23, I.1 and I.6]). The trace φ is defined on B_{φ}.

The trace is called semifinite, if every positive operator majorizes an operator with finite trace. It is called normal if the trace of the least upper bound of a net of positive operators is equal to the supremum of the traces. And it is called faithful, if φ(a) = 0 and a positive implies a = 0.

We will introduce the φ-trace for A-modul morphisms, based on a finite faithful normal trace φ on A.

We start with the following more general observations. (This will yield our first goal: a normalized trace on l^{2}(A) ⊗ H.)

9.4. Lemma. Let X be a Hilbert A-module, H_{1}, H_{2} be two Hilbert spaces. Let {h_{i}}_{i∈I} and {g_{j}}_{j∈J} be orthonormal bases of H_{1} and H_{2} respectively (not necessarily countable). These induce isometric embeddings

U_{i} : X → X ⊗ H_{1} : x ↦ x ⊗ h_{i} and V_{j} : X → X ⊗ H_{2} : x ↦ x ⊗ g_{j}.

Then: f : X ⊗ H_{1} → X ⊗ H_{2} commutes with the action of A if and only if

(f_{ji} := V_{j}^{∗}fU_{i} : X → X) ∈ A′ ∀i, j.

A′ is the commutant of A on the Hilbert space X.

9.5. Definition. We denote this set of operators with A′ ⊗ B(H_{1}, H_{2}).

This is just a symbol, but its formal implications are correct:

9.6. Lemma. The set A′ ⊗ B(H, H) of definition 9.5 is actually a von Neumann algebra, namely just A′ ⊗ B(H).

Proof. Dixmier [23, I.6, prop. 14]).

Proof of lemma 9.4. Let a ∈ A. Obviously, left multiplication with a commutes with U_{i}, U_{i}^{∗}, V_{j}, and V_{j}^{∗}. Therefore, af = fa implies af_{ji} = f_{ji}a ∀i, j. The other implication follows from the equation f = ∑_{ij} V_{j}f_{ji}U_{i}^{∗} (weak convergence: ∑_{i} U_{i}U_{i}^{∗} = 1_{H_{1}} weakly).

We proceed with a lemma which we need in the next theorem but also at a later stage. It will be essential to prove well definedness of the trace and the trace property Sp_{φ}(fg) = Sp_{φ}(gf).

9.7. Lemma. Adopt the situation of lemma 9.4. Let φ be a normal trace on A′. Suppose f ∈ A′ ⊗ B(H_{1}, H_{2}). Then

∑_{i∈I} φ(U_{i}^{∗}f^{∗}fU_{i}) = ∑_{j∈J} φ(V_{j}^{∗}ff^{∗}V_{j}).

Proof.

∑_{i} φ(U_{i}^{∗}f^{∗} \underbrace{(∑_{j} V_{j}V_{j}^{∗})}_{=1_{X⊗H_{2}}} fU_{i}) =^{φ normal} ∑_{i,j} φ((V_{j}^{∗}fU_{i})^{∗}V_{j}^{∗}fU_{i})

=^{φ trace} ∑_{i,j} φ(V_{j}^{∗}fU_{i}U_{i}^{∗}f^{∗}V_{j}) =^{φ normal} ∑_{j} φ(V_{j}^{∗}f \underbrace{(∑_{i} U_{i}U_{i}^{∗})}_{=1_{X⊗H_{1}}} f^{∗}V_{j})

Note that all the operators in question are non-negative, so that the expressions are defined (with values in [0, ∞]) and the order of summation is irrelevant.

9.8. Theorem. Let A, A′ be von Neumann algebras on X as in lemma 9.4, H a Hilbert space (not necessarily separable). Suppose A′ has a normal trace φ. Let {h_{i}}_{i∈I} be an orthonormal base (ONB) of H. This leads to isometric embeddings U_{i} : X → X ⊗ H : x ↦ x ⊗ h_{i}. For a positive f ∈ A′ ⊗ B(H) define

Sp_{φ}(f) := ∑_{i∈I} φ(U_{i}^{∗}fU_{i}) ∈ [0, ∞].  (9.9)

Then Sp_{φ} is a normal trace on the von Neumann algebra A′ ⊗B(H) which does not depend on the chosen ONB. In particular, we have the corresponding trace class ideal. Moreover: If f ∈ A′⊗B(H) is of Sp_{φ}-trace class then formula (9.9) holds. If φ is faithful or semifinite resp., Sp_{φ} has the same properties. If a ∈ A′ and B ∈ B(H) are positive then a ⊗ B is also positive and Sp_{φ}(a ⊗ B) = φ(a) Sp(B); with the convention 0 · ∞ = ∞ · 0 = 0.

If f ∈ A′ ⊗ B(H_{1}, H_{2}) then

Sp_{φ}(f^{∗}f) = Sp_{φ}(ff^{∗}).

Proof. Lemma 9.7 shows that Sp_{φ} does not depend on the ONB and that it is a trace (by Dixmier [23, I.6, coroll. 1 of prop. 1], because Sp_{φ}(λf + g) = λ Sp_{φ}(f) + Sp_{φ}(g) for λ ∈ ℝ_{≥0}, f, g positive operators). For normality let {f_{α}} be a monotonously increasing net with least upper bound f. Then f is the strong (therefore also weak) limit of the net (Dixmier [23, App. II]). Then U_{i}^{∗}fU_{i} is an upper bound for the increasing net U_{i}^{∗}f_{α}U_{i}. We show that it is the weak limit, by Dixmier [23, App. II] it follows that it is the least upper bound.

\begin{matrix}(U_{i}^{∗}f_{α}U_{i}x, y) = (f_{α}U_{i}x, U_{i}y) ⟶^{f_{α}→f weakly} (fU_{i}x, U_{i}y) = (U_{i}^{∗}fU_{i}x, y) ∀x, y ∈ X \\ ⟹ φ(U_{i}^{∗}f_{α}U_{i}) → φ(U_{i}^{∗}fU_{i}) ∀i \end{matrix}

The latter convergence is monotone. Since each of the terms is positive, this implies

Sp_{φ}(f_{α}) = ∑_{i} φ(U_{i}^{∗}f_{α}U_{i}) →^{α} ∑_{i} φ(U_{i}^{∗}fU_{i}) = Sp_{φ}(f).

Suppose now φ is faithful and Sp_{φ}(f) = 0 for f ≥ 0, f ∈ A′ ⊗ B(H). Then φ(U_{i}^{∗}fU_{i}) =

0 ∀i ⟹^{φ faithful} U_{i}^{∗}fU_{i} = 0 ∀i. For i ∈ I

0 = (U_{i}^{∗}fU_{i}x, x) = (\sqrt{f}U_{i}x, \sqrt{f}U_{i}x) ∀x ∈ X

⟹ \sqrt{f}U_{i} = 0 ⟹ U_{i}U_{i}^{∗}\sqrt{f} = 0

⟹ 0 = ∑_{i} U_{i}U_{i}^{∗}\sqrt{f} = \sqrt{f} ⟹ f = 0.

Therefore, Sp_{φ} is faithful, too. Let a ∈ A′ and B ∈ B(H) be positive. Then a ⊗ B = (\sqrt{a} ⊗ \sqrt{B})^{∗}(\sqrt{a} ⊗ \sqrt{B}) is also positive and

Sp_{φ}(a ⊗ B) = ∑_{i} φ(U_{i}^{∗}(a ⊗ B)U_{i}) = ∑_{i} φ(a) · (Bh_{i}, h_{i}) = φ(a) Sp(B).

It remains to check formula (9.9) for arbitrary Sp_{φ}-trace class operators. If f is one of these, it is a finite linear combination of positive ones with finite Sp_{φ}-trace (compare Dixmier [23, I.6 proposition 9]). Equation (9.9) for f follows immediately.

Suppose now φ is semifinite (or finite). This is equivalent to the fact that 1 is the least upper bound of {p = p^{∗} = p^{2} ∈ A′; φ(p) < ∞} ([65, 2.4.8] or [23, I.6 corol. 2 of prop. 1]). Choose therefore projections p_{α} ∈ A′ with p_{α} ↗ 1_{X}. Let P_{J} be the orthogonal projection onto the subspace of H spanned by h_{i}; i ∈ J for J ⊂ I, |J| < ∞. Then Sp(P_{J}) = |J| < ∞ and 1_{H} = l.u.b.{P_{J}; |J| < ∞}. By the preceding arguments Sp_{φ}(p_{α} ⊗ P_{J}) = φ(p_{α}) Sp(P_{J}) < ∞. It is easy to see that 1_{X⊗H} is the l.u.b. of {p_{α} ⊗ P_{J}} and, consequently, Sp_{φ} is semifinite.

The property Sp_{φ}(f^{∗}f) = Sp_{φ}(ff^{∗}) is a direct consequence of lemma 9.7.

Now we specialize to morphisms of Hilbert modules over a finite von Neumann algebra. We start working with arbitrary Hilbert A-modules (and not only l^{2}(A) ⊗ H). Adopt the following situation:

9.10. Situation. Let A be a von Neumann algebra with a finite faithful normal trace φ. Then l^{2}(A) is the Hilbert space completion of A with respect ot the inner product φ(a^{∗}b). We consider A as von Neumann algebra with underlying space l^{2}(A) (A embeds into the bounded operators of l^{2}(A) via multiplication from the left). Its commutant is A^{op} (with multiplication from the right), also equipped with the trace φ. A reference for this is Dixmier [23, I.6 theorème 2].

To gain greater flexibility, we define φ-trace class morphisms between different Hilbert A-modules. However, the complex number Sp_{φ}(f) can be defined only for endomorphisms f : V → V.

9.11. Lemma. Suppose f : V → W and g : W → U are Hilbert A-module morphisms. Let f = u|f| be the polar decomposition of f with u : V → W a partial isometry. Then f^{∗}, gf, u, and |f| are Hilbert A-module morphisms. Moreover, the Hilbert A-module endomorphisms on V constitute a von Neumann algebra.

Proof. The claim is clear for the composition. If x, y ∈ W and a ∈ A then

(x, af^{∗}y) = (a^{∗}x, f^{∗}y) = (fa^{∗}x, y) = (a^{∗}fx, y) = (x, f^{∗}ay)

Therefore, f^{∗} commutes with A and is a Hilbert A-module morphism.

The Hilbert A-module endomorphisms of V are by the very definition a commutant and, consequently, a von Neumann algebra. With f^{∗}f, every continuous function of f^{∗}f lies in this von Neumann algebra. Especially, |f| is a Hilbert A-module morphism.

Concerning u we note that u is characterized by [63, prop. A.3.4] ux = 0 ∀x ⊥ im|f| and u(|f|x) = fx ∀x ∈ V. This implies ua = au ∀a ∈ A.

9.12. Definition. Let V_{k} be Hilbert A-modules with isometric A-embeddings i_{k} : V_{k} ↪ l^{2}(A) ⊗ H_{k}. Let p_{k} := i_{k}^{∗} be the corresponding projections (k = 1, 2). Let f : V_{1} → V_{2} be a Hilbert A-module morphism.

Then f : V_{1} → V_{2} is called φ-Hilbert Schmidt (φ-HS) iff Sp_{φ}(i_{1}f^{∗}fp_{1}) < ∞. We call f φ-trace class (φ-tr) iff Hilbert A-module morphisms f_{1} : V_{1} → V_{3} and f_{2} : V_{3} → V_{2} exist which are φ-HS, so that f = f_{2}f_{1} .

If V_{1} = V_{2} and f is φ-tr we set

Sp_{φ}(f) := Sp_{φ}(i_{1}fp_{1}).

This obviously generalizes previously defined notions: If V_{1} = V_{2} = l^{2}(A) ⊗ H then f is φ-HS and φ-tr resp. if and only if f is Sp_{φ}-HS and Sp_{φ}-trace class respectively. All of the above specializes to the ordinary trace Sp on Hilbert spaces if A = ℂ is the trivial von Neumann algebra on X = ℂ.

Now we formulate the theorem which includes all the basic properties of a trace.

9.13. Theorem. Let f : V_{1} → V_{2}, g : V_{2} → V_{3}, e : V_{0} → V_{1} be Hilbert A-module morphisms. Then (1) The properties in 9.12 are well defined, i.e. depend not on the isometric embeddings

i_{k}. Nor does the value Sp_{φ}(f).

(2) f φ-tr ⟺ f^{∗} φ-tr and f φ-HS ⟺ f^{∗} φ-HS

(3) f φ-HS ⟹ gf, fe φ-HS

(4) f φ-tr ⟹ gf, fe φ-tr

(5) f φ-tr ⟺ |f| φ-tr

(6) f φ-tr and V_{1} = V_{3} ⟹ g ↦ Sp_{φ}(gf) is ultra weakly continuous.

(7) f φ-tr and V_{1} = V_{3} ⟹ Sp_{φ}(gf) = Sp_{φ}(fg)

(8) f, g φ-HS and V_{1} = V_{3} ⟹ Sp_{φ}(gf) = Sp_{φ}(fg)

(9) If V_{1,2} = l^{2}(A) ⊗ H_{1,2}, a ∈ A^{op} is φ-HS and B ∈ B(H_{1}, H_{2}) is HS, then f = a ⊗ B is φ-HS. If, moreover, H_{1} = H_{2}, a is φ-tr and B is trace class, then f is φ-tr with Sp_{φ}(f) = φ(a) Sp(B).

Proof. First remember that Sp_{φ} is a trace on A′ ⊗ B(H) with all the properties of a trace. We will rely heavily on these properties. The strategy of the proof is the following: First we prove the assertions for V_{i} = l^{2}(A) ⊗ H_{i} with i_{V_{i}} = 1. Then we show that the special case implies that the properties hold in general and, in particular, that the expressions in 9.12 are well defined. For the latter, suppose that, additionally, we have embeddings i′_{k} : V_{k} ↪ l^{2}(A) ⊗ H′_{k} with corresponding projections p′_{k} = i′_{k}^{∗}.

Note that i_{k}p_{k} = p_{k} and p_{k}i_{k} = 1. Set f̄:= i_{2}fp_{1}, f̄′ := i′_{2}fp′_{1}, and so on. Then by definition: f is φ-HS iff f̄ is φ-HS and f is φ-tr iff f̄ is φ-tr. Moreover Sp_{φ}(f) = Sp_{φ}(f̄), \overline{fg} = f̄ḡ and f̄^{∗} = f̄^{∗}. This makes it clear that (2), (3), (4), (7) and (8) hold in general if they hold in the special case.

So, first, suppose V_{i} = l^{2}(A) ⊗ H_{i}.

(2) f φ-HS ⟺ ∞ > Sp_{φ}(f^{∗}f) =^{(9.7)} Sp_{φ}(ff^{∗})

(3) If f φ-HS and g ∈ A′ ⊗ B(H_{2}) then g is the sum of four unitary operators in the von Neumann algebra A′ ⊗ B(H_{2}) [65, 0.4.7]. We may, therefore, suppose g is unitary. Then Sp_{φ}((gf)^{∗}gf) = Sp_{φ}(f^{∗}g^{∗}gf) = Sp_{φ}(f^{∗}f) < ∞. If H_{3} ≠ H_{2} use the polar decomposition g = u|g| and observe

Sp_{φ}(f^{∗}g^{∗}gf) = Sp_{φ}(f^{∗}|g|u^{∗}u|g|f) = Sp_{φ}(f^{∗}|g||g|f) < ∞

by the result of the first step.

Concerning fe, observe: f^{∗} φ-HS ⟹ e^{∗}f^{∗} φ-HS ⟹ fe φ-HS.

(4) f φ-tr ⟹ f = f_{2}f_{1} with f_{i} φ-HS ⟹ gf = (gf_{2})f_{1} with gf_{2} Sp_{φ}-HS by (3). Similarly for fe.

(7) The statement is clear if H_{1} = H_{2} = H_{3} (this is simply the trace property). We reduce the case H_{1} ≠ H_{2} to the known case. First suppose g is injective with polar decomposition g = u|g|. Then u is also injective.

Sp_{φ}(gf) = Sp_{φ}(u|g|f) = Sp_{φ}(uu^{∗}u|g|f)

=^{Sp_{φ} trace!} Sp_{φ}(u|g|fuu^{∗}) =^{(∗)} Sp_{φ}(|g|fu) =^{Sp_{φ} trace} Sp_{φ}(fu|g|)

It remains to prove (∗), namely that ”conjugation” with an injective partial isometry does not change the trace: Since arbitrary trace class operators are linear combinations of positive operators, assume that f is positive and u ∈ A′ ⊗ B(H_{1}, H_{2}) is an injective partial isometry (this implies u^{∗}u = 1). Then

Sp_{φ}(ufu^{∗}) = Sp_{φ}(u\sqrt{f}(u\sqrt{f})^{∗}) =^{9.7} Sp_{φ}((u\sqrt{f})^{∗}u\sqrt{f})

= Sp_{φ}(\sqrt{f} \underbrace{u^{∗}u}_{=1} \sqrt{f}) = Sp_{φ}(f)

Now suppose g is surjective. Then g^{∗} is injective and

Sp_{φ}(gf) =^{Sp_{φ} trace} Sp_{φ}(f^{∗}g^{∗}) = Sp_{φ}(g^{∗}f^{∗}) = Sp_{φ}(fg)

If g is arbitrary, decompose g as follows:

g = p_{2} ∘ (1 ⊕ g) : X ⊗ H_{2} → (X ⊗ H_{2}) ⊕ (X ⊗ H_{2}) → X ⊗ H_{2}

Sp_{φ}(gf) = Sp_{φ}(p_{2}(1 ⊕ g)f) =^{p_{2} surjective} Sp_{φ}((1 ⊕ g)fp_{2})
=^{(1 ⊕ g) injective} Sp_{φ}(fp_{2}(1 ⊕ g)) = Sp_{φ}(fg)

(8) Suppose f, g are φ-HS and g = u|g|. Then

Sp_{φ}(gf) = Sp_{φ}(u \underbrace{|g|f}_{φ−tr}) = Sp_{φ}(|g|fu) =^{Sp_{φ} trace} Sp_{φ}(fu|g|) = Sp_{φ}(fg)

(9) f^{∗}f = a^{∗}a ⊗ B^{∗}B and by assumption a^{∗}a is φ-tr and B^{∗}B is Sp-tr. Therefore by theorem 9.8 Sp_{φ}(f^{∗}f) = φ(a^{∗}a) Sp(B^{∗}B) < ∞. If a is φ-trace class and B is Sp-trace class then we find a = λ_{1}a_{1} ⋯ + λ_{k}a_{k} with λ_{i} ∈ ℂ, a_{i} ∈ A′, a_{i} ≥ 0, φ(a_{i}) < ∞ and similarly B = μ_{1}B_{1} + ⋯ + μ_{l}B_{l}. Then a ⊗ B = ∑_{ij} λ_{i}μ_{j}a_{i} ⊗ B_{j} with a_{i} ⊗ B_{j} ≥ 0 and Sp_{φ}(a_{i} ⊗ B_{j}) = φ(a_{i}) Sp(B_{j}) < ∞ is Sp_{φ}-trace class and, by theorem 9.8

Sp_{φ}(a ⊗ B) = ∑_{ij} λ_{i}μ_{j} Sp_{φ}(a_{i} ⊗ B_{j}) = φ(a) Sp(B).

Now we drop the assumption that V_{i} = l^{2}(A) ⊗ H_{i}.

(1) Consider f̄ and f̄′. Note that p ∘ i = 1. Therefore

f̄′ = i′_{2}p_{2}i_{2}fp_{1}i_{1}p′_{1} = i′_{2}p_{2}f̄i_{1}p′_{1}

and f̄ = i_{2}p′_{2}f̄′i′_{1}p_{1}. Since ip′ is bounded and by the special case of (4) and (3) we have: f̄ φ-tr ⟺ f̄′ φ-tr, and f̄ φ-HS ⟺ f̄′ φ-HS. For the computation of Sp_{φ}(f) if V_{1} = V_{2} =: V observe

Sp_{φ}(f̄′) = Sp_{φ}(i′pf̄ip′) =^{Sp_{φ} trace} Sp_{φ}(f̄i \underbrace{p′i′}_{1_{V′}} p) = Sp_{φ}(f̄ip) = Sp_{φ}(if \underbrace{pi}_{1_{V}} p) = Sp_{φ}(f̄)

(5) Polar decomposition f = u|f| and |f| = u^{∗}f. Since u is bounded: f φ-tr ⟺ |f| φ-tr by (4).

(6) Suppose g_{α} →^{α} g ultra weakly. Let fp_{1} = u|fp_{1}| be the polar decomposition of fp_{1}. Then g_{α}u → gu ultra weakly in A′ ⊗ B(H_{1}). Since φ is normal, Sp_{φ} is normal, too. By Dixmier [23, I.6 prop. 1],

Sp_{φ}(g_{α}f) = Sp_{φ}((i_{1}g_{α}u)|fp_{1}|) →^{α} Sp_{φ}((i_{1}gu)|fp_{1}|) = Sp_{φ}(gf)

The next theorem shows that our trace does not depend on the inner product but only on the equivalence class of norms on V.

9.14. Proposition. Suppose V_{1} = (V, (·, ·)_{1}) is a Hilbert A-module and f : V → V is φtrace class. Suppose V can be equipped with another Hilbert space structure V_{2} = (V, (·, ·)_{2}) so that it is again a Hilbert A-module and so that the induced norms are equivalent. Then f is φ-tr also with respect to the new Hilbert space structure and Sp_{φ}(f) does not change.

Proof. Clearly 1 : V = V_{1} → V_{2} = V is a Hilbert A-module morphism with bounded inverse 1^{−1}. Then

Sp_{φ}(f : V_{1} → V_{1}) = Sp_{φ}(1^{−1}f1 : V_{1} → V_{1}) = Sp_{φ}(f : V_{2} → V_{2})

9.15. Definition. Let i : V ↪ W be an A-inclusion of Hilbert A-modules, p : W → W the orthogonal projection onto V. Then

dim_{φ}(V) := Sp_{φ}(p) ∈ [0, ∞].

If V ⊂ W is A-invariant but not closed, set dim_{φ}(V) = dim_{φ}(V̄).

Proof that dim_{φ} is well defined. It suffices to show Sp_{φ}(1_{V}) = Sp_{φ}(p). But 1_{V} = i^{∗}i = i^{∗}pi and p = ii^{∗}p. Consequently

Sp_{φ}(1_{V}) = Sp_{φ}(i^{∗}pi) = Sp_{φ}(ii^{∗}p) = Sp_{φ}(p)

We now come to an important result, which is essentially proved in Atiyah’s paper [1]. He does not state it in full generality, and for convenience we will give the proof here.

9.16. Proposition. Suppose V, W are Hilbert A-modules and φ is, as before, a normal trace on A′. Let T_{0} : V → V and T_{1} : W → W be bounded Hilbert A-module morphisms which are φ-tr. Let D : V → W be a closed operator with domain D(D) which commutes with the action of A. Especially, we require that D(D) is A-invariant and dense. Suppose

DT_{0} = T_{1}D; ker D ⊂ ker T_{0}; ker D^{∗} ⊂ ker T_{1}^{∗}.

This implies

Sp_{φ}(T_{0}) = Sp_{φ}(T_{1})

Proof. If D is bounded with bounded inverse D^{−1}, then the assertion is clear from Sp_{φ}(T_{0}) = Sp_{φ}(D^{−1}T_{1}D) = Sp_{φ}(DD^{−1}T_{1}). The idea is now to replace an arbitrary D by bounded operators with bounded inverse which are sufficiently close to D. First assume that D = A : V → V is positive self adjoint with ker A = 0. Set

P_{n} := χ_{[1/n,n]}(A); A_{n} := P_{n}AP_{n} + (1 − P_{n})

Then (note that A, P_{n} and A_{n} all commute)

\begin{aligned}A_{n}P_{n}T_{0}P_{n} & = P_{n}AP_{n}T_{0}P_{n} + \underbrace{(1 − P_{n})P_{n}}_{=0} T_{0}P_{n} \\ & = P_{n}T_{1}AP_{n} = P_{n}T_{1}P_{n}P_{n}AP_{n} + P_{n}T_{1} \underbrace{P_{n}(1 − P_{n})}_{=0} \\ & = P_{n}T_{1}P_{n}A_{n} \end{aligned}  (9.17)

By construction A_{n} is invertible. Therefore,

Sp_{φ}(T_{0}P_{n}) =^{P_{n}=P_{n}^{2}} Sp_{φ}(P_{n}T_{0}P_{n}) = Sp_{φ}(A_{n}P_{n}T_{0}P_{n}A_{n}^{−1}) =^{(9.17)} Sp_{φ}(P_{n}T_{1}P_{n}) = Sp_{φ}(T_{1}P_{n})

Now P_{n} →^{n→∞} 1 strongly (since ker A = 0). Therefore, T_{0}P_{n} → T_{0} weakly. The continuity property (6) in theorem 9.13 implies

Sp_{φ}(T_{0}) = lim_{n} Sp_{φ}(T_{0}P_{n}) = lim_{n} Sp_{φ}(T_{1}P_{n}) = Sp_{φ}(T_{1}).

For arbitrary D, consider the inclusions i_{1} : ker D^{⊥} ↪ V, i_{2} : \overline{im D} ↪ W and the orthogonal projections p_{k} := i_{k}^{∗}. We have the polar decomposition p_{2}Di_{1} = UA with isometry U : ker D^{⊥} → \overline{im D} and positive A : ker D^{⊥} → ker D^{⊥}. By the assumptions on the kernels and cokernels:

Sp_{φ}(T_{1}) = Sp_{φ}(i_{2}p_{2}T_{1}) = Sp_{φ}(p_{2}T_{1}i_{2}) = Sp_{φ}(UU^{∗}p_{2}T_{1}i_{2}) = Sp_{φ}(U^{∗}p_{2}T_{1}i_{2}U)
Sp_{φ}(T_{0}) = Sp_{φ}(p_{1}T_{0}i_{1})

For the composed operators we have

\begin{aligned}A(p_{1}T_{0}i_{1}) & = U^{∗}UAp_{1}T_{0}i_{1} = U^{∗}p_{2}Di_{1}p_{1}T_{0}i_{1} \\ & = U^{∗}p_{2}T_{1}Di_{1} = (U^{∗}p_{2}T_{1}i_{2}U)A \end{aligned}

Consequently, the special case applies and gives the desired result.

84 10 L^{2}-RELLICH LEMMA

10 L^{2}-Rellich lemma

Let M be a compact m-dimensional manifold with boundary ∂M (possibly empty). Let M̃ be a normal covering of M with covering group Γ (acting by isometries). Let E↓M be a vector bundle with pullback Ẽ ↓M̃. Then M̃ is a manifold of bounded geometry with boundary in the sense of section 3 and Ẽ is a bundle of bounded geometry. We can specifically define the Sobolev spaces H^{s}(Ẽ) (s ∈ ℝ). We will show in this section that H^{s}(Ẽ) is a Hilbert N (Γ)-module and that the inclusion H^{s+r}(Ẽ) ↪ H^{s}(Ẽ) is Γ-HS for r ≥ r(m) sufficiently large.

10.1. Definition. Here and from now on φ_{Γ} is the canonical trace for N (Γ) operating on l^{2}(Γ): φ_{Γ}(f) := (f(e), e) (e the neutral element of Γ). Instead of φ_{Γ}-HS we will use the expression Γ-HS, ….

Let W be the double of M with reflection fl : W → W. Let X↓W be the double of E. The reflection fl extends as a bundle map to X. Construct similarly W̃ and X̃. Then W̃ is a normal covering of W with covering group Γ. Again, we denote the reflection fl.

Suppose U ⊂ M̃ ⊂ W̃ is a fundamental domain for the covering p : M̃ → M. This means that U is open, p|_{U} is injective and M − p(U) is a set of measure zero. Choose U so that its closure is compact. Apply the following proposition 10.2 to obtain a compact submanifold T of codimension zero with boundary, so that U ∪ fl(U) ⊂ T.

10.2. Proposition. Let U be a precompact subset of a manifold N without boundary. Then there exists a compact submanifold T of codimension zero with boundary so that U ⊂ T.

Proof. Embed N as closed subset in a large Euclidian space. Equip it with the induced metric d. Choose x_{0} ∈ U and define the function d(x) := d(x, x_{0}). Choose a Morse function f close to d (so that |f(x) − d(x)| < 1). Set T_{r} := f^{−1}(−∞, r]. Then T_{r} ⊂ {x ∈ N; d(x, x_{0}) < r + 1} is a closed bounded subset, therefore is compact. Especially, it contains only a finite number of critical points of f. Moreover, T_{−2} = ∅. Classical Morse theory implies that T_{r} is a compact submanifold with boundary for all but a discrete set of values r. Clearly, N = ⋃_{r∈ℝ} T_{r} and the proof is complete.

10.3. Lemma. Fix s ∈ ℝ. There exists a bounded Γ-equivariant extension map e : H^{s}(M̃) → H^{s}(W̃), i.e. e(f)|M̃ = f ∀f ∈ H^{s}(M̃). The corresponding statement holds for Ẽ.

Proof. The proof can be found in [59]. But it is obvious how to construct e using a Γinvariant covering of M̃ by charts with a subordinate Γ-invariant partition of unity and using the corresponding well known extension map on Euclidian space (compare Taylor [66, I.5.1]).

10.4. Lemma. Suppose s ∈ ℕ. The map p defined by the composition

H^{s}(M̃) ⟶^{e} H^{s}(W̃) ⟶^{p̄} l^{2}(Γ) ⊗ H^{s}(T)
∈  ∈  ∈
f ⟶ ef ⟶ ∑_{g∈Γ} g ⊗ \underbrace{g^{−1}(ef)}_{=g^{∗}(ef)} |_{T}

is Γ-equivariant, and there exist C_{1,2} > 0 so that

|f|_{H^{s}(M̃)} ≤^{C_{1}} |pf|_{l^{2}(Γ)⊗H^{s}(T)} ≤^{C_{2}} |f|_{H^{s}(M̃)}.

Pull back the inner product on l^{2}(Γ)⊗H^{s}(T) via p. The new norm on H^{s}(M̃) is equivalent to the original one and p becomes an isometric Γ-equivariant embedding. Especially H^{s}(M̃) is a Hilbert N (Γ)-module. The corresponding statement holds for Ẽ.

Proof. By lemma 10.3, e has the required properties. It remains to consider p̄. Obviously, p̄ is Γ-equivariant. Because Γ is discrete and T is compact, it meets only finitely many, say N, of its translates {gT }_{g∈Γ}. By definition |∑ g ⊗ f_{g}|_{l^{2}(Γ)⊗H^{s}(T)}^{2} = ∑|f_{g}|_{H^{s}(T)}^{2}. Let ∇ be the Levi-Civita connection of any lifted metric on W̃. Now we show that p̄ is bounded. If f ∈ H^{s}(W̃)

\begin{aligned}|p̄f|_{l^{2}(Γ)⊗H^{s}(T)}^{2} & = ∑_{i=0}^{s} ∑_{g∈Γ} ∫_{T} |∇^{i}(g^{∗}f)|^{2} = ∑_{i} ∑_{g} ∫_{g(T)}|∇^{i}(f)|^{2} \\ & ≤^{N} ∑_{i=0}^{s} ∫W̃ |∇^{i}f|^{2} = |f|_{H^{s}(W̃)}^{2} \end{aligned}

On the other hand, for f ∈ H^{s}(M̃)

|f|_{H^{s}(M̃)}^{2} = ∑_{i} ∑_{g} ∫_{g(U)}|∇^{i}f|^{2} ≤ ∑_{i} ∑_{g} ∫_{g(T)}|∇^{i}f|^{2} = |pf|_{l^{2}(Γ)⊗H^{s}(T)}^{2}

The computations for Ẽ are similar, but notationally more complicated.

10.5. Theorem. Suppose s, r ∈ ℕ and r > m/2. Then the inclusion ĩ : H^{s+r}(Ẽ) → H^{s}(Ẽ) is Γ-HS.

10.6. Corollary. If s, r ∈ ℕ and r > m, then the inclusion ĩ : H^{s+r}(Ẽ) → H^{s}(Ẽ) is Γ-tr.

Proof of theorem 10.5. The idea of the proof is to compare ĩ with 1⊗i : l^{2}(Γ)⊗H^{s+r}(T) ↪ l^{2}(Γ) ⊗ H^{s}(T). Let X↓W be the double of E. The following diagram commutes by the geometric definition of p:

H^{s+r}(Ẽ) ⟶^{p_{s+r}} l^{2}(Γ) ⊗ H^{s+r}(X̃|_{T})
ĩ↓  ↓^{1⊗i}
H^{s}(Ẽ) ⟶^{p_{s}} l^{2}(Γ) ⊗ H^{s}(X̃|_{T})

Remember that we have equipped H^{s}(Ẽ) with the Hilbert space structure which makes p an isometric embedding, therefore p^{∗}p = 1. This yields

ĩ = p_{s}^{∗}p_{s}ĩ = p_{s}^{∗}(1 ⊗ i)p_{s+r}

Now we apply property (3) and (9) of theorem 9.13, together with the classical result that for bundles over compact manifolds (such as T) the inclusion H^{s+r} ↪ H^{s} is HS if r > m/2. For the application of 9.13 note that 1 : l^{2}Γ → l^{2}Γ has the finite trace φ_{Γ}(1) = 1.

11 Boutet de Monvel calculus

The Boutet de Monvel (BdM) calculus is a tool to deal with boundary value problems. It generalizes the calculus of pseudodifferential operators on manifolds without boundary. Because the BdM calculus is technically complicated, we will not go into the details, but only cite the results which are essential for our applications. More can be found in Schrohe/Schulze [60] or in Rempel/Schulze [53]. We use the definition of order and type of [60], which differs from the old one. An elementary overview is given by Booß [13, II.8.B].

The main point of the Boutet de Monvel calculus is the introduction of an algebra of operators which includes the boundary value problems we want to study and also their inverses. The first observation we have to make is that we naturally have to consider matrices of operators (compare below): The operators A and T come from the boundary value problems. The potential operator K is well known as a solution operator, and the Green operator G had to be introduced to obtain an algebra (the simplest example is the composition of a trace operator and a potential operator).

For us, the main points of the theory are the following: every elliptic boundary value problem has an inverse (modulo smoothing operators), a so called parametrix. And, every BdM operator is up to a smoothing operator nearly local.

11.1. Definition. Let M be a manifold with boundary ∂M. Let E, F ↓ M be vector bundles over M, X, Y ↓∂M bundles over the boundary. A BdM operator P has the shape

P = (\begin{matrix}A + G & K \\ T & p \end{matrix}) : \begin{matrix}C_{0}^{∞}(E) \\ ⊕ \\ C_{0}^{∞}(X) \end{matrix} → \begin{matrix}C^{∞}(F) \\ ⊕ \\ C^{∞}(Y) \end{matrix}

where A is a pseudodifferential operator (pdo) with the transmission property on M, p is a pdo on ∂M. T : C^{∞}(E) → C^{∞}(Y) is a trace operator, K : C_{0}^{∞}(X) → C^{∞}(F) a potential operator and G : C_{0}^{∞}(E) → C^{∞}(F) a Green operator. Note that the splitting A + G is not canonical.

11.2. Remark. Every BdM operator has an order μ ∈ [−∞, ∞) and a type d ∈ ℕ_{0}. The order is a generalisation of the order of a (pseudo)differential operator, the type is determined by the trace and Green operator and says “how much restriction to the boundary” is involved.

BdM operators are locally defined (except the smoothing operators, see definition 11.4): P is BdM, iff for all cutoff functions φ and ψ (ψ = 1 on supp φ) with support in a chart the Euclidian operator φPψ is BdM and if φP(1 − ψ) is a BdM operator of order −∞ and type zero. Their behavior under coordinate transformations is checked in [53].

For the rest of the section, we adopt the situation of the previous definition.

11.3. Proposition. Suppose M is compact. Let P : C^{∞}(E)⊕C^{∞}(X) → C^{∞}(F)⊕C^{∞}(Y) and Q : C^{∞}(F)⊕C^{∞}(Y) → C^{∞}(G)⊕C^{∞}(Z) be BdM operators of order μ and type d and μ′, d′ respectively. Then the composition QP is a BdM operator of order μ + μ′ and type max{d′, d + μ′}.

The BdM operator P of order μ ≥ 0 and type d ≤ μ is elliptic iff there exists a BdM operator Q : C^{∞}(F) ⊕ C^{∞}(Y) → C^{∞}(E) ⊕ C^{∞}(X) of order −μ and type zero so that

S_{0} := QP − 1  and  S_{1} := PQ − 1

are of order −∞ and S_{0} is of type μ, S_{1} of type zero. Q is called a parametrix of P. Two parametrices differ by an operator of order −∞.

Every differential boundary value problem P = (A, T) : C_{0}^{∞}(E) → C_{0}^{∞}(F) ⊕ C_{0}^{∞}(Y) is a Boutet de Monvel operator. If it is elliptic in the Lopatinsky-Shapiro sense, it is also elliptic in the sense of the BdM algebra.

Proof. Compare Schrohe/Schulze [60] and Rempel/Schulze [53].

In the study of elliptic boundary value problems, the BdM operators of order −∞ are important. Simply stated: These are operators with smooth integral kernels. More precisely (but a little bit lenghty):

11.4. Definition. We call P a smoothing Boutet de Monvel operator (an operator of order −∞ and type d ≥ 0), iff there exist smooth integral kernels

a, g_{0} ∈ C^{∞}(Hom(p_{2}^{∗}E, p_{1}^{∗}F)↓M × M),
g_{1}, …, g_{d} ∈ C^{∞}(Hom(p_{2}^{∗}E|_{∂M}, p_{1}^{∗}F)↓M × ∂M),
k ∈ C^{∞}(Hom(p_{2}^{∗}X, p_{1}^{∗}F)↓M × ∂M),
p ∈ C^{∞}(Hom(p_{2}^{∗}X, p_{1}^{∗}Y)↓∂M × ∂M),
t_{1}, …, t_{d} ∈ C^{∞}(Hom(p_{2}^{∗}E|_{∂M}, p_{1}^{∗}Y)↓∂M × ∂M),
t_{0} ∈ C^{∞}(Hom(p_{2}^{∗}E, p_{1}^{∗}Y)↓∂M × M),

so that for F ∈ C_{0}^{∞}(E) and f ∈ C_{0}^{∞}(X) we have

\begin{aligned}AF (x) & = ∫_{M} a(x, y)F (y)dy \\ pf(x′) & = ∫_{∂M} p(x′, y′)f(y′)dy′ \\ GF (x) & = ∑_{i=1}^{d} ∫_{∂M} g_{i}(x, y′)(∂_{ν})^{i−1}F (y′)dy′ + ∫_{M} g_{0}(x, y)F (y)dy \\ Kf(x) & = ∫_{∂M} k(x, y′)f(y′)dy′ \\ T F (x′) & = ∑_{i=1}^{d} ∫_{∂M} t_{i}(x′, y′)(∂_{ν})^{i−1}F (y′)dy′ + ∫_{M} t_{0}(x′, y)F (y)dy \end{aligned}

∂_{ν} denotes differentiation in inward normal direction.

11.5. Definition. (Pseudolocal operators) Suppose M is equipped with a Riemannian metric. Let ϵ > 0. An operator P : C_{0}^{∞}(E) ⊕ C_{0}^{∞}(X) → C^{∞}(F) ⊕ C^{∞}(Y) is called ϵ-local, iff

supp(Pf) ⊂ {x ∈ M; d(x, supp f) < ϵ} ∀f ∈ C_{0}^{∞}

The following proposition will be important for our purposes.

11.6. Proposition. Suppose M is a compact Riemannian manifold and ϵ > 0 is given. Every BdM operator P is the sum of an ϵ-local operator and a smoothing BdM operator of type zero.

Proof. Choose a finite covering of M by balls {U_{i}} of radius ϵ/2. Let {φ_{i}} be a subordinate partition of unity and ψ_{i} cutoff functions with ψ_{i} = 1 on supp φ_{i} and supp ψ_{i} ⊂ U_{i}. Set

P_{1} := ∑_{i} φ_{i}Pψ_{i}, P_{2} := P − P_{1} = ∑_{i} φ_{i}P(1 − ψ_{i}).

Then P_{2} is a smoothing BdM operator of type zero by definition of BdM operators and P_{1} is by construction ϵ-local. Moreover, P_{1} is a BdM operator of same order and type as P: it is a fact that multiplication with a smooth function is a BdM operator of order and type zero, and the composition with such an operator yields a BdM operator with unchanged order and type (Rempel/Schulze [53, 2.3.3.2. Theorem 1]).

11.7. Proposition. Suppose M is compact. Let P be a BdM operator of order μ and type d. If s > d − 1/2, then P extends to a continuous operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s−μ}(F) ⊕ H^{s−μ}(Y).

Proof. Compare Schrohe/Schulze [60, 2.2.19].

11.8. Proposition. Let M̃ ↓M be a normal Riemannian covering of Riemannian manifolds with covering group Γ, where M is compact. Suppose, the covering is trivial over balls of radius 2ϵ. Suppose

P : C^{∞}(E) ⊕ C^{∞}(X) → C^{∞}(F) ⊕ C^{∞}(Y)

is an ϵ-local operator which extends to a bounded operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s−μ}(F) ⊕ H^{s−μ}(Y).

Then P lifts to an operator

P̃ : C^{∞}(Ẽ) ⊕ C^{∞}(X̃) → C^{∞}(F̃) ⊕ C^{∞}(Ỹ),

which has a bounded extension

P̃ : H^{s}(Ẽ) ⊕ H^{s}(X̃) → H^{s−μ}(F̃) ⊕ H^{s−μ}(Ỹ)

Proof. (The method of proof is typical for the situation of coverings and Γ-equivariance.) Let {U_{i}}_{i=1,...,N} be a covering of M by balls of radius ϵ, let V_{i} be the corresponding balls of radius 2ϵ. Let φ_{i} be a subordinate covering of unity. This induces a Γ-invariant covering {U_{i,γ}}_{γ∈Γ} of M̃ with subordinate Γ-invariant partition of unity φ_{i,γ}. It is clear how to lift P. It remains to check boundedness. Let F = (F, f) ∈ C_{0}^{∞}(Ẽ) ⊕ C_{0}^{∞}(X̃) be given. Then (we use |a + b|^{2} ≤ 3(|a|^{2} + |b|^{2}))

|P̃F|_{H^{s−μ}}^{2} = |P̃ ∑_{i,γ} φ_{i,γ}F|_{H^{s−μ}}^{2} ≤^{3^{N}} ∑_{i=1}^{N}|P̃ ∑_{γ} φ_{i,γ}F|_{H^{s−μ}}^{2}

=^{(∗)} ∑_{i,γ}|P̃φ_{i,γ}F|_{H^{s−μ}}^{2} ≤^{∥P∥^{2}} ∑_{i,γ}|φ_{i,γ}F|_{H^{s}}^{2}

=^{Def} |F|_{H^{s}}^{2}

Here (∗) holds because the supports of φ_{i,γ} and φ_{i,γ′} are disjoint for γ ≠ γ′.

We will apply this proposition to BdM operators which are pseudolocal.

Next we compute the trace of sufficiently regularizing BdM operators. Most important is the fact that the Γ-trace of a lift equals the trace of the operator on the base.

11.9. Theorem. Let P : C^{∞}(E) ⊕ C^{∞}(X) → C^{∞}(E) ⊕ C^{∞}(X) be a BdM operator of order −μ < −4m and type d. Here m is the dimension of the underlying manifold. Then, for s > d − 1/2, P extends to a bounded trace class operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s}(E) ⊕ H^{s}(X).

The value of the trace is independent of s.

If P is ϵ-local then its lift P̃ : H^{s}(Ẽ) ⊕ H^{s}(X̃) → H^{s}(Ẽ) ⊕ H^{s}(X̃) (defined for s > d − 1/2) is Γ-tr and

Sp_{Γ}(P̃) = Sp(P).

If −μ = −∞ and P is given by the integral kernels of definition 11.4 then we have explicitly

Sp(P) = ∫_{M} Sp_{E_{x}} a(x, x)dx + ∫_{∂M} Sp_{X′_{x}} p(x′, x′)dx′

+ ∫_{M} Sp_{E_{x}} g_{0}(x, x)dx + ∑_{i=1}^{d} ∫_{∂M} Sp_{E_{x′}} ∂_{ν,x}^{i−1}p_{i}(x, y)|_{x=x′=y}dx′

(Sp_{F} denotes the trace on the finite dimensional vector space F; ∂_{ν} is differentiation in normal direction).

Proof. The inclusion H^{s+μ} ↪ H^{s} is of trace class (by theorem 10.5). Therefore P : H^{s} →^{P} H^{s+μ} ↪ H^{s} is the composition of a bounded operator and a trace class operator, therefore of trace class. Moreover, if μ − 4m > s′ − s > 0 then

Sp(P : H^{s′} → H^{s′}) = Sp(H^{s′} ↪ H^{s} →^{P} H^{s+μ} ↪ H^{s′})

= Sp(H^{s} →^{P} H^{s+μ} ↪ H^{s′} ↪ H^{s}) = Sp(P : H^{s} → H^{s})

Here we used the trace property, noting that H^{s+μ} ↪ H^{s} is trace class. Inductively, the trace is independent of s for arbitrary s.

Identical arguments apply to the lift P̃, replacing trace by Γ-trace and using theorem 10.5.

Now we come to the explicit computations. Consequently, suppose μ = −∞. Observe (with the notion of 11.1) Sp(P) = Sp(A) + Sp(G) + Sp(p). Note that A and p are actually defined on L^{2}. The above argument applies to show that Sp(A : H^{s} → H^{s}) = Sp(A : L^{2} → L^{2}). However, A is an integral operator with a smooth kernel. It is well known (compare Hörmander [37]) that the trace of such an operator on L^{2} has the following value:

Sp(A) = ∫_{M} Sp_{E_{x}} a(x, x)dx.

Similarly Sp(p) = ∫_{∂M} Sp_{X_{x′}} p(x′, x′)dx′. For the obvious splitting G = G_{0} + G_{1} + ⋯ + G_{d}, note that each summand is trace class. G_{0} behaves exactly as A does. For i > 0, the operator G_{i} is a composition

H^{s}(E) →^{∂_{ν}^{i−1}} H^{s−i+1}(E) →^{res} H^{s−i+1/2}(E|_{∂M}) →^{K_{i}} H^{∞}(E) ↪^{i} H^{s}(E)

90 12 PROOF OF THE L^{2}-INDEX THEOREM

Each of the operators is bounded and the inclusion is trace class (res denotes the restriction to the boundary and K_{i} is the obvious integral operator with smooth kernel from E|_{∂M} → E). Using the trace property and the fact that inclusions of Sobolev spaces commute with differentiation and restriction to the boundary, we see

Sp(G_{i}) = Sp(i ∘ \underbrace{res ∘∂_{ν}^{i−1} ∘ K_{i}}_{=:P_{i}})

Now P_{i} is an integral operator with smooth kernel on ∂M, namely

P_{i}f(x′) = ∫_{∂M}(∂_{ν,x}^{i−1}g_{i})(x′, y′)f(y′)dy′.

Therefore it extends to a trace class operator on L^{2}(E|_{∂M}) with

Sp(G_{i}) = Sp(P_{i}) = ∫_{∂M} Sp_{E_{x′}}(∂_{ν,x}^{i−1}g_{i}(x, y))|_{x=x′=y}dx′

This establishes the formula for Sp(P).

Identical arguments apply to the lift P̃ as far as follows:

Sp_{Γ}(P̃) = Sp_{Γ}(Ã) + Sp_{Γ}(p̃) + Sp_{Γ}(G̃_{0}) + ∑_{i=1}^{d} Sp_{Γ}(P̃_{i})

where each summand is the lift of an integral operator with smooth kernel on L^{2}(E), L^{2}(X) and L^{2}(E|_{∂M}), respectively.

Therefore, it remains to show that for this type of operator the Γ-trace of the lift coincides with the trace on the base. But this is easy: choose a fundamental domain U ⊂ M̃ for the covering. Then L^{2}(Ẽ) → l^{2}(Γ)⊗L^{2}(E|U) is an isometric N (Γ)-isomorphism and Ã = 1 ⊗ A (by definition of a lift). By theorem 9.13 (9),

Sp_{Γ}(Ã) = φ_{Γ}(1) · Sp(A) = Sp(A),

and similarly for the other operators.

12 Proof of the L^{2}-index theorem

12.1. Situation. M is a compact manifold with boundary ∂M. Let M̃↓M be a normal covering with covering group Γ = π_{1}(M)/π_{1}(M̃). Let P = (A, T) : C_{0}^{∞}(E) → C_{0}^{∞}(F) ⊕ C_{0}^{∞}(Y) be an elliptic differential boundary value problem on M. Denote its lift to M̃ with P̃ : C_{0}^{∞}(Ẽ) → C_{0}^{∞}(F̃) ⊕ C_{0}^{∞}(Ỹ). Suppose P has order μ ≥ 0 and type d ≤ μ.

We have the extension P : H^{μ}(E) → L^{2}(F) ⊕ L^{2}(Y). We consider this as a bounded operator.

Let H_{0} : L^{2}(E) → ker(P) be the orthogonal projection onto the kernel, H_{1} : L^{2}(F) ⊕ L^{2}(Y) → im(P)^{⊥} the orthogonal projection onto the cokernel of P. Similarly, let H̄_{0} and H̄_{1} be the projections onto kernel and cokernel of P̃.

We want to prove the L^{2}-index theorem for manifolds with boundary 1.14:

12.2. Theorem. The numbers dim_{Γ} ker P̃ = Sp_{Γ}(H̄_{0}) and dim_{Γ} coker P̃ = Sp_{Γ}(H̄_{1}) are finite, and ind_{Γ}(P̃) := Sp_{Γ}(H̄_{0}) − Sp_{Γ}(H̄_{1}) equals ind(P) = Sp(H_{0}) − Sp(H_{1}).

The idea of the proof is the following: H_{i} and H̄_{i} are highly non-local and have in general nothing to do with each other (this is the reason why I did not use a tilde for the operators on M̃). But suppose we could find a bounded ”inverse” Q to P (from the image to the orthogonal complement of the kernel). If additionally, Q was a differential operator (what is impossible), then the equations

PQ = 1 − H_{1} QP = 1 − H_{0}

could be lifted and we could compare the trace of H_{i} and H̄_{i} directly. Of course, this does not work, but we can find a parametrix:

Let Q be an ϵ-local parametrix of P (existence is proved below) so that

PQ = 1 − S_{1}, QP = 1 − S_{0}  ⟹  P̃Q̃ = 1 − S̃_{1}, Q̃P̃ = 1 − S̃_{0}. (12.3)

Automatically, S_{0} = 1 − QP and S_{1} are ϵ-local since the right hand side is. The operators with a tilde are the lifts. Note that S_{0} and S̃_{0} are operators of order −∞ and type μ, whereas S_{1} and S̃_{1} have order −∞ and type zero.

We know already that Sp_{Γ} S̃_{i} = Sp S_{i} (theorem 11.9). It remains to show that we can compute the index also in terms of the S_{i}, namely

Sp S_{0} − Sp S_{1} = Sp H_{0} − Sp H_{1}  (12.4)

(and similarly on M̃). This will be achieved with Atiyah’s theorem 9.16. We start with two preparatory results:

12.5. Proposition. The boundary value problem P has for any ϵ > 0 an ϵ-local parametrix.

Proof. Start with an arbitrary parametrix Q_{0}. Make it ϵ-local with the aid of proposition 11.6: Q_{0} = Q + S. S is a smoothing operator of type zero. Claim: Q is also a parametrix of P:

QP = Q_{0}P − SP = 1 + smoothing operator

Here we use the fact that the composition of a smoothing operator with an arbitrary operator remains smoothing, as well as the sum of two smoothing operators. The argument works also for PQ.

12.6. Theorem. The image of the projection H_{0} : L^{2}(E) → L^{2}(E) (i.e. the kernel of P) is contained in H^{∞}(E) and H_{0} restricts to a bounded operator H_{0} : H^{s}(E) → H^{s+t}(E) for arbitrary s, t ≥ 0. Especially H_{0} : H^{s} → H^{s} is trace class for every s ≥ 0 and the trace is independent of s.

The same holds for H̄_{0} if we replace tr by tr_{Γ}.

Proof. Elliptic regularity implies that the kernels of P and P̃ are contained in every Sobolev space H^{s}(E) and H^{s}(Ẽ) resp., and that the Sobolev norms on this subspace are equivalent to the L^{2}-norm (4.15 — use the correponding a priori estimate). This implies all of the statements, if we remember that H_{0} can be viewed as composition of the bounded operator H_{0} : H^{s} → H^{s+4m} with the trace class operator i : H^{s+4m} ↪ H^{s} (and similarly for H̄_{0}). That the trace is independent of s is also immediate from this decomposition.

Now we can prove equation 12.4. The following computations are formulated only for the lifted operators. They are valid also on the base, by the following (formal) changes: omit tilde and bar, and replace Γ by the trivial group.

Multiplying the equations in (12.3) with H̄_{1} from the left and with H̄_{0} from the right, we get

H̄_{1} = H̄_{1}S̃_{1}  H̄_{0} = S̃_{0}H̄_{0},  (12.7)

where the equation for H̄_{0} is valid on H^{μ} and the one for H̄_{1} is valid on all of L^{2} (from the domains of definition of operators of type d). By multiplication of (12.3) with P̃ we get on H^{μ}

P̃S̃_{0} = S̃_{1}P̃.

Following Atiyah [1] we now define T̄_{i} := (1 − H̄_{i})S̃_{i}(1 − H̄_{i}) (i = 0, 1). From the ideal property T̄_{0} is a Γ-tr operator on the Hilbert N (Γ)-module H^{μ} and T̄_{1} is a Γ-tr operator on the Hilbert N (Γ)-module L^{2}. Because H̄_{i} are projectors

Sp_{Γ} T̄_{0} = Sp_{Γ}(S̃_{0}(1 − H̄_{0})) = Sp_{Γ} S̃_{0} − Sp_{Γ} H̄_{0} (use (12.7))
Sp_{Γ} T̄_{1} = Sp_{Γ}((1 − H̄_{1})S̃_{1}) = Sp_{Γ} S̃_{1} − Sp_{Γ} H̄_{1} (use (12.7))

Therefore,

Sp_{Γ} T̄_{0} = Sp_{Γ} T̄_{1} ⟺ Sp_{Γ} S̃_{0} − Sp_{Γ} S̃_{1} = Sp_{Γ} H̄_{0} − Sp_{Γ} H̄_{1}.

Next observe

ker P̃ ⊂ ker T̄_{0} ker P̃^{∗} ⊂ ker T̄_{1}^{∗}

P̃T̄_{0} = P̃S̃_{0} − \underbrace{P̃H̄_{0}}_{=0} S̃_{0} − P̃ \underbrace{S̃_{0}H̄_{0}}_{=H̄_{0}} + \underbrace{P̃H̄_{0}}_{=0} S̃_{0}H̄_{0} = S̃_{1}P̃ = ⋯ = T̄_{1}P̃

Application of proposition 9.16 yields Sp_{Γ} T̄_{0} = Sp_{Γ} T̄_{1}, i.e. ind_{Γ} P̃ = Sp_{Γ} S̃_{0} − Sp_{Γ} S̃_{1}. Similarly, ind P = Sp S_{0} − Sp S_{1}. Now theorem 11.9 appplied to the ϵ-local smoothing operators S_{0}, S_{1} finishes the proof of theorem 1.14.

13 Explicit expressions for the analytical index

Index and adjoint boundary value problems

The purpose of this section is a simplification of the index formula. We replace the not very convenient cokernel of an elliptic boundary value problem by the kernel of the adjoint boundary value problem. For the concept of adjoint boundary value problem compare section 3.

13.1. Theorem. Let E, F ↓M, X, Y ↓∂M be Riemannian vector bundles, P := (A, p) : C_{0}^{∞}(E) → C_{0}^{∞}(F) ⊕ C_{0}^{∞}(Y) be an elliptic differential boundary value problem. Suppose Q := (B, q) : C_{0}^{∞}(F) → C_{0}^{∞}(E) ⊕ C_{0}^{∞}(X) is adjoint to (A, p) with respect to the Greenian formula

(Ae, f)_{L^{2}(F)} − (e, Bf)_{L^{2}(E)} = (pe, sf)_{L^{2}(Y)} − (te, qf)_{L^{2}(X)} ∀e ∈ C_{0}^{∞}(E), f ∈ C_{0}^{∞}(F) (13.2)

(Here t, s are auxiliary differential operators). Then

L^{2}(F) ⊕ L^{2}(Y) ⊃ im(P)^{⊥} →^{p_{1}} L^{2}(F) : (f, y) ↦ f

is an isomorphism onto ker(Q) with inverse

α : ker(Q) → im(P)^{⊥} : f ↦ (f, −sf).

Proof. First, we have to prove that the maps have range as stated. Take (f, y) ∈ im(P)^{⊥}. In particular, f ⊥ A(D_{A} = {f; pf = 0}). The orthogonal splitting L^{2}(F) = ker(Q) ⊕ A(D_{A}) of theorem 4.22 implies directly that f ∈ ker(Q). That α has the correct image follows immediately from the Greenian formula. It remains to check α ∘ p_{1} = 1_{im P^{⊥}}. If (f, y) ∈ im(P)^{⊥}, then for arbitrary e ∈ C_{0}^{∞}(E)

(pe, y) =^{(f,y)⊥im(P)} −(Ae, f) =^{(13.2)} −(e, Bf) − (pe, sf) + (te, qf) =^{f∈ker Q} −(pe, sf).

From ellipticity it follows that p is normal, consequently, im(p) is dense in L^{2}(Y) and y = −sf. This shows that the two maps are actually inverse to each other.

If we are in the situation of the L^{2}-index theorem 1.14, then the isomorphism of theorem 13.1 is clearly equivariant under the group operation. Therefore,

13.3. Corollary. For lifts with covering group Γ, coker(P̃) is Γ-isomorphic to ker(Q̃) and especially

ind_{Γ}(P̃) = dim_{Γ}(ker(P̃)) − dim_{Γ}(ker(Q̃)).

The index theorem then has the following form:

13.4. Theorem. Suppose M is a compact boundary manifold with normal covering M̃ and covering group Γ. Let P := (A, T) be an elliptic differential boundary value problem on M with lift P̃. Let Q := (B, S) be an adjoint boundary value problem with lift Q̃. Then

ind(P) = dim(ker P) − dim(ker Q) = ind_{Γ}(P̃) = dim_{Γ}(ker P̃) − dim_{Γ}(ker Q̃)

We will apply this to the computation of the Euler characteristic of a manifold with boundary. Lott/Lück [41] derive the same result, using different methods. They use the elementary fact that the simplicial L^{2}-Euler characteristic is equal to the ordinary Euler characteristic.

13.5. Theorem. Suppose M is a compact manifold, whose boundary ∂M = M_{1} ⨿ M_{2} is a disjoint union (M_{1} and/or M_{2} may be empty). Let M̃ be a normal covering of M with covering group Γ. Then the relative Euler characteristic computes as follows:

χ(M, M_{1}) = ∑_{p}(−1)^{p} dim_{Γ} H_{(2)}^{p}(M̃, M̃_{1})

with H_{(2)}^{p}(M̃, M̃_{1}) = {ω ∈ C^{∞}(Alt^{p}T M̃); |ω|_{L^{2}} < ∞, dω = 0 = δω, b_{1}^{∗}(ω) = 0 = b_{2}^{∗}(∗ω)}. (b_{i} : M̃_{i} ↪ M̃ are the inclusions).

Proof. To keep notation simple suppose M_{1} = ∅. From Hodge theory we known that χ(M, M_{1}) = χ(M) = ∑_{p}(−1)^{p} dim H^{p}(M) = dim ker(P^{ev}) − dim ker(P^{odd}), where P^{ev} is the boundary value problem

P^{ev} = (d + δ, b_{2}^{∗} ∘ ∗) : C^{∞}(Alt^{ev}T M) → C^{∞}(Alt^{odd}T M) ⊕ C^{∞}(Alt^{∗}T ∂M).

We have the following Greenian formula for this boundary value problem

((d + δ)ω, η)_{L^{2}(M)} = (ω, (δ + d)η)_{L^{2}(M)} ± ∫_{∂M} b^{∗}ω ∧ b^{∗}(∗η) ± ∫_{∂M} b^{∗}(η) ∧ b^{∗}(∗ω).

Theorem 13.1 yields then

dim ker(P^{ev}) − dim ker(P^{odd}) = ind(P^{ev})

Now 13.4 tells us

χ(M) = ind_{Γ}(P̃^{ev}) = dim_{Γ} ker(P̃^{ev}) − dim_{Γ} ker(P̃^{odd})

In view of elliptic regularity this is just the claim.

One can use this theorem to get information about the sign of χ(M), if the geometry implies that in most dimensions no L^{2}-harmonic forms exists. This is done by Donelly/Xavier [28] (compare section 6, where we review their computations). Their result implies the following: if M^{m} is a closed manifold with sectional curvature K pinched: −1 ≤ K ≤ C < −1 + (m − 2)^{2}/(m − 1)^{2} and m is even, then (−1)^{m/2}χ(M) ≥ 0.

A formula for the Γ-dimension of ker P̃

In this section we will give an explicit expression for the Γ-dimension of kernels of lifted elliptic boundary value problems in terms of elements of the kernel. We start with an auxiliary result. For this statement, we have to define Banach spaces of r-times continuously differential sections. To do this, choose a covering {U_{i}} of M by charts and corresponding trivialisations of the bundle. Choose compact set K_{i} ⊂ U_{i} so that ⋃_{i} K_{i} = M. Via the charts and trivialisations we transport sections s to Euclidian space. Then the Banach norm is |s|_{r} := sup_{i} sup_{x∈K_{i}; |α|≤r}|∂^{α}s(x)|, and the Banach space consists by definition of those sections for which this norm is finite. Note that for M not compact this space depends on the choices.

13.6. Lemma. Let E↓M be a Riemannian vector bundle over a not neccesarily compact Riemannian boundary manifold M. Let A be a bounded self adjoint operator on L^{2}(E) which maps L^{2}(E) continuously to the Banach space C_{b}^{r+1}(E) of (r+1)-times continuously differentiable sections. Then A^{2} has an integral kernel k(x, y) s. t.

A^{2}f(x) = ∫_{M} k(x, y)f(y)dy

The kernel is of class C^{r} up to the corners, with |k(·, ·)|_{C^{r}} ≤ ∥A∥_{B(L^{2},L^{2})}∥A∥_{B(L^{2},C^{r+1})}, and with k(x, ·) ∈ L^{2}(Hom(E, E_{x})) ∀x with uniform L^{2}-bound |k(x, ·)|_{L^{2}} ≤ C.

A formula for the Γ-dimension of ker P̃ 95

Proof. The proof is given f.i. in Roe’s book [55, (5.6)]. We only have to keep track of the constants and norms, which is not neccesary in his case, since he deals with compact manifolds.

If we deal with covering manifolds, the Banach spaces in lemma 13.6 are just the Banach spaces for bundles of bounded geometry (definition 3.15). They are built with respect to the following data: one starts with covering, charts, trivialisations on the base and lifts everything to the covering (as much lifts as there are leafs). This yields data which is compatible with deck transformations.

13.7. Proposition. Let P be an elliptic BdM operator on M with lift P̃ to M̃. Let H̄_{0} be the orthogonal projection onto ker(P̃) in L^{2}. Then, H̄_{0} has a integral kernel k(x, y) on M̃ × M̃, smooth up to the corners, so that every derivative is uniformly bounded and k(x, ·) is square integrable for every x ∈ M̃.

Let F be a fundamental domain for M in M̃. We have Sp_{Γ}(H̄_{0}) = ∫_{F} SpẼ_{x} k(x, x)dx. Moreover, if f_{i}(x) is an orthonormal basis of ker P̃ then for x, y in the interior of M̃

k(x, y) = ∑_{i} f_{i}(y)f_{i}^{∗}(x).

The sum converges in the following sense: arbitrary derivatives converge uniformly on compact subsets of M̃^{∘} × M̃^{∘}. Moreover,

dim_{Γ}(ker P̃) = ∑_{i} ∫_{F} SpẼ_{x} f_{i}(x)f_{i}^{∗}(x)dx

13.8. Remark. The series is perfectly feasible to get lower bounds for dim_{Γ}, since the operators f_{i}(x)f_{i}^{∗}(x) are non-negative and, consequently, their traces are ≥ 0.

Proof of proposition 13.7. The existence of the smooth kernel is just the content of lemma 13.6, because H̄_{0} = H̄_{0}^{2} = H̄_{0}^{∗} by definition of orthogonal projections. Moreover, im H̄_{0} ⊂ H^{∞}(Ẽ) by elliptic regularity 4.15, and H̄_{0}, considered as an operator from L^{2} to H^{k} is bounded for every k by the corresponding a priori estimates. Atiyah [1, section 2] proves the same result for manifolds without boundary. His method works perfectly well in our situation, if we consider only interior points. It remains to prove the formula for dim_{Γ}. Since M is compact, we may assume without loss of generality that the fundamental domain F is compact. Let K be a compact subset of the interior M̃^{∘}. Then uniform convergence tells us

∫_{F∩K} Sp_{E_{x}} k(x, x) = ∑_{i} ∫_{F∩K} Sp_{E_{x}} f_{i}(x)f_{i}^{∗}(x).

Because F is compact and the boundary is a set of measure zero, if K_{1} ⊂ K_{2} ⊂ ⋯ is a sequence of compact sets whose union contains the interior of F:

dim_{Γ}(ker P̃) = ∫_{F} Sp_{x} k(x, x) = lim_{j} ∫_{K_{j}∩F} Sp_{x} k(x, x)

= lim_{j} ∑_{i} ∫_{K_{j}∩F} Sp_{x} f_{i}(x)f_{i}^{∗}(x) = ∑_{i} lim_{j} ∫_{K_{j}∩F} Sp_{x} f_{i}(x)f_{i}^{∗}(x)

= ∑_{i} ∫_{F} Sp_{x} f_{i}(x)f_{i}^{∗}(x)

We can interchange integration and summation because f_{i}(x)f_{i}^{∗}(x) is a positive operator, therefore we have monotonous increasing positive sequences in j.

Representations of Lie groups

In this section we show how one can use our index theorem 1.14 to construct non-trivial representations of semisimple Lie groups in spaces of functions on manifolds with boundary. This is in the spirit of the work of Atiyah/Schmid [7, 8]. Atiyah/Schmid work with manifolds without boundary. They construct all of the discrete series of a real semisimple Lie group with finite center in terms of kernels of Dirac operators. The geometric realisation allows very precise information about the irreducible representations and their characters. This generalizes the Peter-Weyl theorem for compact Lie groups with the Hermann-Weyl formula for the characters and the geometrical construction of irreducible representations due to Borel and Weil. Geometrical simply means that the representation shows up in a natural way in a geometrical situation, for instance as kernel of a (twisted) Dirac operator.

13.9. Proposition. Let G be a semisimple Lie group, Γ a torsion free discrete cocompact subgroup and K be a compact subgroup with the following properties: as a smooth manifold K = ∂W and the action of K on ∂W by left multiplication is extended to the an isometric action on the compact manifold W (the prototype is K = S^{1} and W = D^{2}). Define X := G ×_{K} W. This is a manifold with boundary ∂X = G, and G acts on X from the left. The corresponding action of Γ on X is properly discontinuous with compact quotient M = Γ\X. If χ(M) ≠ 0 then H_{(2)}^{∗}(X) is a non-trivial geometric G-representation.

Proof. The last statement is a direct consequence of our L^{2}-index theorem 1.14. We have to check that we can apply this theorem, i.e. that the operation of Γ on X is properly discontinuous. This is the case if Γ ∩ K = 1, since Γ is discrete and K compact. Now Γ ∩ K is not only discrete and compact, but also a torsion free subgroup of G, therefore it is the trivial group.

13.10. Example. Let G = P SL(2, ℝ), K = S^{1} = ∂D^{2}. Then X = G ×_{S^{1}} D^{2} is the disc bundle of the S^{1}-bundle S^{1} → G → G\S^{1}. Consequently, if Γ is as above, then M = Γ\X is the disc bundle of the S^{1}-bundle S^{1} → Γ/G → Γ\G/S^{1}. In particular, χ(M) = χ(Γ\G/S^{1}). Raymond/Vasquez [51] show that we find torsion free Γ so that Γ\G/S^{1} is a Riemannian surface of genus g > 1 and Γ\G is the sphere bundle of the tangent bundle of G/S^{1}. Then χ(M) ≠ 0 and H_{(2)}^{∗} is an infinite dimensional G-representation. More concretely, H_{(2)}^{1} ≠ {0} since χ(M) < 0.

A Bounded geometry: Proofs

In this appendix, we will give proofs of elementary properties of ∂-manifolds with bounded geometry. Especially we study the relation between curvature/second fundamental form, the coefficients of the metric tensors in normal coordinates, and the mappings of the change of normal coordinates. For manifolds without boundary, there are related statements in the literature (Cheeger/Gromov/Taylor [21], Roe [56], Shubin [64]), but without any reference for a proof. So this appendix fills this gap also for manifolds without boundary.

Metric tensor in normal coordinates

We start with the following theorem:

A.1. Theorem. Let M be a Riemannian manifold with boundary ∂M. Suppose M has bounded geometry, i.e. if R is the curvature tensor and l the second fundamental form tensor on ∂M with respect to the inward unit normal, then |∇^{j}R| ≤ C and | ¯∇^{j}l| ≤ C for j ≤ k. Here ∇ is the Levi-Civita connection of M and ¯∇ the one on ∂M. Suppose r_{i}, r_{i}^{C}, r_{C} > 0 (and sufficiently small, depending on C). Then there exists M-universal D > 0 so that in normal coordinates |D^{α}g_{ij}(x)| ≤ D and |D^{α}g^{ij}(x)| ≤ D for every partial derivative with |α| ≤ k. D depends only on C and the constants r_{i}, r_{i}^{C}, r_{C}.

We prove this theorem in four steps: If k = 0, the Rauch comparison theorem yields the statement. Secondly, we prove the statement for Gaussian coordinates. Here we use step 1. In the third step, we show that the boundary has the same properties as M: | ¯∇^{j}R̄| is bounded for j ≤ k. Last, we derive the statement of the theorem for normal boundary coordinates, using the first three steps.

Step 1: Boundedness of g_{ij}

A.2. Proposition. Suppose we are in the situation of theorem A.1 and k = 0. Suppose x_{i} is a normal coordinate system. Then

C_{1} ≤ |∑ λ_{i} \frac{∂}{∂x_{i}} |_{T M} ≤ C_{2},  if ∑ λ_{i}^{2} = 1.  (A.3)

The constants C_{1,2} depend only on r_{i}, r_{C} and C. Especially, the components g_{ij} and g^{ij} of the metric tensors in normal coordinates are bounded.

Proof. The last statement follows directly from the inequality (A.3). For this inequality, we apply Warner’s generalisation of the Rauch comparison theorem [68, 4.3]. We compare with two complete manifold of constant sectional curvature −C and C, respectively. To compare with normal collar coordinates, choose a hypersurface in this manifold so that all the eigenvalues of its second fundamental form at one (comparison) point are equal to C in the first case and to −C in the second case. Inequality (A.3) is just the statement of the comparison theorem with C_{1,2} depending only on r_{i} or r_{C} and the manifold we compare with (i.e. on C). Here, r_{C} and r_{i} must be sufficiently small (depending only on the manifold we compare with, i.e. on C). The comparison theorem says nothing about the radial tangent vectors ∂_{r} = ∑ x_{i}∂_{i} (in Rauch’s version) and about normal tangent vectors ∂_{n}. But for these vectors Euclidian length and lengt in T M as well as the orthogonal complements coincide by the Gauss lemma A.4. Therefore, the inequality is true in general.

We used here (and at several other places) the Gauss lemma:

A.4. Theorem. Let (M, g) be a Riemannian manifold and exp : B(0, R) → M a Gaussian chart. Pull the metric g back to B(0, R). Then g(∂_{r}, ∂_{r}) = r^{2}, if ∂_{r} = ∑_{i} x_{i}∂_{i} is the radial vector field, and g(∂_{r}, v) = 0 iff v is tangent to the sphere with center 0 and through the base of v. Let K : ∂M ×[0, r_{C}) → M be the Riemannian collar and pull g back to ∂M ×[0, r_{C}). Then g(∂_{t}, ∂_{t}) = 1 and g(∂_{t}, v) = 0 iff v is tangent to the corresponding translate ∂M × {t}.

Proof. Compare Gallot/Hulin/Lafontaine [29, 2.93] — the proof works also for the collar with no more than the obvious changes.

Step 2: Gaussian coordinates

Now we will state a (differential) equation for the metric tensor g_{ij} in terms of the curvature tensor, so that a bound on partial derivatives of the components of the curvature tensor will give corresponding bounds for the metric. Now partial derivatives and the covariant derivative are related by the Christoffel symbols, so we will compute them, too.

Let p ∈ M be an interior point, so that Gaussian coordinates x with radius r_{i} around p exist. Choose an orthonormal base (ONB) {s_{i}} for T_{p}M. Using parallel transport along geodesics emanating from p, we construct a synchronous orthonormal frame {s_{i}(x)} of the tangent space restricted to the chart neighbourhood. Let {θ^{i}} be the frame of 1-forms dual to {s_{i}} (therefore orthonormal). Now consider the connection forms θ_{j}^{i} for this frame defined by

∇s_{j} = ∑_{i} θ_{j}^{i}s_{i}

with associated Christoffel symbols Γ_{jk}^{i} and curvature tensor R_{jkl}^{i} given by

θ_{j}^{i} = ∑_{k} Γ_{jk}^{i}dx_{k}, and

dθ_{j}^{i} − ∑_{k} θ_{k}^{i} ∧ θ_{j}^{k} = ∑_{kl} R_{jkl}^{i}dx_{k} ∧ dx_{l}

Note that this is not the ordinary definition because we use the frame {s_{i}} instead of the frame {∂_{i}}. Additionally, we can express the curvature entirely in terms of s_{i} and θ^{i}:

dθ_{j}^{i} − ∑_{k} θ_{k}^{i} ∧ θ_{j}^{k} = ∑_{kl} K_{jkl}^{i}θ^{k} ∧ θ^{l}.

Define functions a_{j}^{i} and b_{j}^{i} via the equations

θ^{i} = ∑_{j} a_{j}^{i}dx_{j}; dx_{i} = ∑_{j} b_{j}^{i}θ^{j}

Then

\begin{aligned}R_{jkl}^{i} & = ∑_{αβ} K_{jαβ}^{i}a_{k}^{α}a_{l}^{β}  and \\ g_{ij} & = ∑_{α} a_{i}^{α}a_{j}^{α};  g^{ij} = ∑_{α} b_{α}^{i}b_{α}^{j} \end{aligned}  (A.5)

Especially, the universal bounds on g_{ij} and g^{ij} from proposition A.2 yield bounds on a_{j}^{i} and b_{j}^{i}. Later, we will also need the covariant differentials of ∂/∂x_{i} =: ∂_{i}:

∇_{∂_{k}}∂_{j} = ∑_{i} Γ̃_{jk}^{i}s_{i}

Since ∂_{j} = ∑_{α} a_{j}^{α}s_{α},

Γ̃_{jk}^{i} = ∂_{k}a_{j}^{i} + ∑_{α} a_{j}^{α}Γ_{α_{k}}^{i}.  (A.6)

Let R be the radial vector field R(x) = ∑_{i} x_{i}∂/∂x_{i}. Then Atiyah/Bott/Patodi [3, a7] derive the following equations

RΓ_{jk}^{i} + Γ_{jk}^{i} = ∑_{l} 2x_{l}R_{jkl}^{i}  ∀i, j, k  (A.7)

(R^{2} + R)a_{l}^{i} = −2 ∑_{jk} R_{jkl}^{i}x_{j}x_{k}  ∀i, l  (A.8)

Set f_{x}(t) = tΓ_{jk}^{i}(tx). Then

f′_{x}(t) = Γ_{jk}^{i}(tx) + t ∑_{l} x_{l}∂_{l}Γ_{jk}^{i}(tx) =^{(A.7)} ∑_{l} 2tx_{l}R_{jkl}^{i}(tx)

Therefore

Γ_{j}^{i}k(x) = f_{x}(1) = ∫_{0}^{1} ∑_{l} 2τx_{l}R_{jkl}^{i}(τx) dτ

D_{x}^{α}Γ_{jk}^{i}(x) = ∫_{0}^{1} τ^{|α|} (D_{x}^{α}(x ↦ ∑_{l} x_{l}R_{jkl}^{i}(x))) (τx) dτ  (A.9)

Set f_{il}(t, x) = a_{l}^{i}(t·x). If′ denotes t-differentiation, tf_{ij}′(t, x) = Ra_{j}^{i}(tx), and t^{2}f′′(t, x)+ tf′(t, x) = R^{2}a(tx). Equation (A.8) implies now

t^{2}f′′_{il} + 2tf′_{il} = −2t^{2} ∑_{kl} R_{jkl}^{i}(tx)x_{j}x_{k}

With w(t, x) = t^{2}f′(t, x) we get w′ = t^{2}f′′ + 2tf′. Since w(0) = 0,

t^{2}f′_{il}(t, x) = −2 ∫_{0}^{t} τ^{2} ∑_{jk} R_{jkl}^{i}(τx)x_{j}x_{k} dτ

⟹^{τ=tu} f′_{il}(t, x) = −2t ∫_{0}^{1} u^{2} ∑_{jkαβ} K_{jαβ}^{i}(tux)a_{k}^{α}(tux)a_{l}^{β}(tux)x_{j}x_{k} du  (A.10)

Now, we are in the position to explain how the bounds on R and its covariant derivatives up to order k imply bounds on g_{ij}, g^{ij} and their partial derivatives up to order k. First, observe that instead of the metric tensor we can consider a_{j}^{i} and b_{j}^{i}. Moreover, the case k = 0 is done by proposition A.2.

A.11. Lemma. Let A, B be matrix functions which are inverse to each other. Therefore,

\frac{∂}{∂x_{i}} B = \frac{∂}{∂x_{i}} A^{−1} = A^{−1}(\frac{∂}{∂x_{i}} A)A^{−1} = B(∂_{i}A)B

Iterated application of this rule and the product rule yields

D_{x}^{α}B = P_{α}(B, D_{x}^{β}A; β ≤ α)

where P_{α} is a well defined non-commutative polynomial. Therefore, bounds for the partial derivatives of A up to order k and on B immediately yield bounds for the partial derivatives of B.

A.12. Corollary. Lemma A.11 applies to the matrices A = (a_{j}^{i}) and B = (b_{j}^{i}). Moreover, by proposition A.2, we have a universal bound for B.

It remains to find bounds for the derivatives of (a_{j}^{i}). To do this, in the next step, we have to relate the covariant derivatives of R with partial derivatives of K_{jkl}^{i}.

A.13. Lemma.

∇_{∂_{i_{1}}} … ∇_{∂_{i_{k}}} R(s_{i}, s_{j}, ∂_{k}, ∂_{l}) =^{α=(i_{1},...,i_{k})} D^{α}K_{jkl}^{i} + P_{α,ijkl}

Here P_{α,ijkl} is a polynomial (only depending on α, i, j, k, l) in partial derivatives up to order |α| − 1 of K_{∗∗∗}^{∗}, Γ_{∗∗}^{∗}, and a_{∗}^{∗} (derivatives of order zero included).

Proof. This follows from the well known formula for covariant differentials in coordinates. Note that for |α| = 1 only Γ_{jk}^{i} shows up, since K_{jkl}^{i} are the components of R entirely in terms of s_{i}. But if we iterate the covariant differentials, we have to take into account that we contracted ∇R with ∂_{i} and not with s_{i}. This yields (via ∇∂_{i}) Γ̃_{jk}^{i} and, since we iterate the covariant differentials, their partial derivatives up to order |α| − 2. Since Γ̃_{jk}^{i} = ∂_{k}a_{i}^{j} + ∑_{α} a_{i}^{α}Γ_{jk}^{α}, the result follows.

Now we argue with a bootstrap argument as follows: By assumption the curvature is bounded. Therefore, since {s_{i}} is orthonormal by construction, K_{jkl}^{i} are M-uniformly bounded. We know already that the same is true for a_{j}^{i}.

Assume by induction that for some r ≥ 0 we have found M-universal bounds on the partial derivatives up to order r of K_{jkl}^{i} and a_{j}^{i} and on the partial derivatives up to order r − 1 of Γ_{jk}^{i}. Suppose we have bounds on |R|, …, |∇^{r+1}R|.

From equation (A.5), relating K_{jkl}^{i} and R_{jkl}^{i}, we get M-universal bounds of the partial derivatives up to order r of R_{jkl}^{i}. Then equation (A.9) yields universal bounds for the partial derivatives of order r of Γ_{jk}^{i}. Then lemma A.13 and the bound on ∇^{r+1}R yields universal bounds on r + 1-order partial derivatives of K_{jkl}^{i} (here we use that the lengths of ∂_{i} are universally bounded (proposition A.2)).

Now remember the following equation (A.10):

f′_{il}(t, x) = −2t ∫_{0}^{1} u^{2} ∑_{jkαβ} (K_{jαβ}^{i}a_{k}^{α}a_{l}^{β}) (tux)x_{j}x_{k} du

Let α be a multi-index with |α| = r + 1. We differentiate the equation with respect to x to get an equation for D_{x}^{α}f(t, x) = t^{|α|}(D^{α}a)(tx). This yields

(D_{x}^{α}f_{il})′(t, x) =

− 2t ∫_{0}^{1} u^{2} ∑_{jkβγ} K_{jβγ}^{i}(tux)((D_{x}^{α}f_{βk})f_{γ,l} + (D^{α}f_{γ,l})f_{βk})(tu, x)x_{j}x_{k} du − 2t ∫_{0}^{1} P_{α} du (A.14)

Here P_{α} is a polynomial in t, u, x, partial derivatives up to order r + 1 of K_{∗∗∗}^{∗} at tux, and partial derivatives up to order r of f_{∗∗}(tu, x) at (tu, x). The induction hypothesis yields for 0 ≤ t ≤ 1 the inequality

|(D_{x}^{α}f)′(t, x)| ≤ C_{1} sup_{0≤τ≤t}{|D_{x}^{α}f(τ, x)|} + C_{2},  (A.15)

where C_{1,2} > 0 can be expressed in the universal bounds of the induction hypothesis. Moreover, note D^{α}f(0, x) = 0 (since |α| ≥ 1).

A.16. Lemma. Let h(t) be the unique solution of h′(t) = C_{1}h(t) + C_{2} with h(0) = 0. A Taylor series ansatz easily shows that h is a positive monotonous increasing function, with an explicit bound h(t) ≤ C = C(C_{1}, C_{2}) for 0 ≤ t ≤ 1. Abbreviate g(t) := D_{x}^{α}f(t, x). The inequality (A.15) implies |g(t)| ≤ h(t). Especially

|D^{α}a_{j}^{i}(x)| = |g_{j}^{i}(1)| ≤ h(1) ≤ C

Proof. We only have to show |g(t)| ≤ h(t). This is a more or less standard argument: Let h_{n} be the unique solution of

h′_{n}(t) = C_{1}h_{n}(t) + C_{2} + 1/n.

Then h_{n}(t) →^{n→∞} h(t) uniformly for 0 ≤ t ≤ 1. Therefore, it suffices to show |g(t)| ≤ h_{n}(t). For a contradiction, assume that for some n and t |g(t)| > h_{n}(t). Set t_{0} := inf_{0≤t}{|g(t)| > h_{n}(t)}. Then |g(t_{0})| = h(t_{0}) (since g(0) = 0 = h(0)). Now h_{n} is monotonous and |g(t)| ≤ h_{n}(t) ∀t ≤ t_{0}. Consequently, sup_{0≤t≤t_{0}}|g(t)| = |g(t_{0})|. Then inequality (A.15) shows

|g′(t_{0})| ≤ C_{1}|g(t_{0})| + C_{2} < h′_{n}(t_{0})

Moreover, d/dt|g(t_{0})| ≤ |g′(t_{0})| (if g(t_{0}) = 0 d/dt is understood to be the right derivative): compare Hartmann [35, III.3.2]. It follows |g(t)| < h_{n}(t) for t ∈ [t_{0}, t_{0} + ϵ) and ϵ > 0 sufficiently small. But this contradicts the choice of t_{0}.

Step 3: Curvature of ∂M

Let R̄ be the curvature (0, 4)-tensor on ∂M, R the curvature tensor of M, and l the second fundamental form, considered as (0, 2)-tensor using the inward unit normal vector field ν.

In the following we apply the covariant differential ¯∇ of ∂M to (0, q)-tensors on M. We implicitly assume that we first restrict the tensor to ∂M.

The covariant derivatives ¯∇^{k}R̄ can be computed with the aid of the following rules:

A.17. Lemma. Suppose T is a (0, q)-tensor on M, S a (0, p)-tensor on ∂M, S^{∗_{1}} the (1, p−1)-tensor on ∂M given by g(S^{∗_{1}}(v_{2}, …, v_{p}), v_{1}) = S(v_{1}, …, v_{p}). Let σ be a permutation (operating on a multiple tensor product by permutation of the factors) with σ^{−1}(1) ≤ p. This holds also for the other permutations which occur in the following formulae. c denotes the (unique) contraction of a (0, r)-tensor with a (1, s)-tensor. We adopt the general convention ∇ : C^{∞}(E) → C^{∞}(E ⊗ T^{∗}M). Then the following holds:

(1) ¯∇T = ∇T − ∑_{i} c(T ⊗ l ∘ σ_{i}, ν), where σ_{i} are appropriate permutations

(2) ¯∇((T ⊗ S) ∘ σ) = (( ¯∇T) ⊗ S) ∘ σ + (T ⊗ ¯∇S) ∘ σ

(3) ¯∇c((T ⊗ S) ∘ σ, ν) = c((∇T) ⊗ S ∘ σ′, ν) + c(T ⊗ ¯∇S ∘ σ, ν) + ∑_{i} c(c(T ⊗ S ∘ σ, ν) ⊗ l ∘ σ_{i}, ν) + c(T ⊗ S ∘ σ, l^{∗_{1}}), σ′ and σ_{i} appropriate permutations

(4) ¯∇c(T, ( ¯∇^{k}l)^{∗_{1}}) = c( ¯∇T, ( ¯∇^{k}l)^{∗_{1}}) + c(T, ( ¯∇^{k+1}l)^{∗_{1}})

A.18. Corollary. Iterated application of the rules of lemma A.17 shows that ¯∇^{k}R̄ is a finite sum of tensor products and possibly iterated contractions, composed with permutations, involving

(i) ∇^{j}R  (j ≤ k)  (ii) ¯∇^{j}l  (j < k)
(iii) ( ¯∇^{j}l)^{∗_{1}}  (j < k − 1)  (iv) ν

Especially, if there are bounds for the building blocks (i) and (ii), this yields a concrete bound for ¯∇^{k}R̄ (since tensor products and contractions of bounded tensors remain bounded and permutations are isometric. Note |ν| = 1 and |S^{∗_{1}}| = |S|)

Proof of the lemma. Formulae (ii) and (iv) are well known. For (i):

¯∇T (v_{1}, …, v_{p}, X) = X.T (v_{1}, …, v_{p}) − T ( ¯∇_{X}v_{1}, …, v_{p}) − ⋯ − T (v_{1}, …, ¯∇_{X}v_{p})

=^{¯∇_{X}Y =∇_{X}Y −l(X,Y)ν} X.T (v_{1}, …, v_{p}) − T (∇_{X}v_{1}, …) − ⋯ − T (v_{1}, …, ∇_{X}v_{p})

+ T (ν, v_{2}, …)l(v_{1}, X) + ⋯ + T (v_{1}, …, v_{p−1}, ν)l(v_{p}, X)

= ∇T (v_{1}, …, v_{p}, X) + ∑_{i} c(T ⊗ l ∘ σ_{i}, ν)(v_{1}, …, v_{p}, X)

For (iii):

¯∇_{X}c(T ⊗ S ∘ σ, ν)(v_{2}, …, v_{p+q}) =^{v_{1}:=ν} (X.T (v_{σ1}, …, v_{σp}))S(v_{σ(p+1)}, …, v_{σ(p+q)})

+ T (v_{σ1}, …, v_{σp})(X.S(v_{σ(p+1)}, …, v_{σ(p+q)})) − ∑_{\begin{matrix}i=1 \\ σ(i)≠1 \end{matrix}}^{p+q} T ⊗ S(v_{σ1}, …, ¯∇_{X}v_{σi}, …, v_{σ(p+q)})

= c(T ⊗ ¯∇_{X}S ∘ σ, ν)(v_{2}, …, v_{p+q})

+ (X.T (v_{σ1}, …, v_{σp}) − ∑_{i=1}^{p} T (v_{σ1}, …, ∇_{X}v_{σi}, …, v_{σp})) S(…)

− ∑_{\begin{matrix}i=1 \\ σ(i)≠1 \end{matrix}}^{p} \underbrace{T (v_{σ1}, …, l(X, v_{σi})ν, …, v_{σp})}_{=T (···)l(X,v_{σi})} S(…) + T (v_{σ1}, …, ∇_{X}ν, …, v_{σp})S(…)

= c(T ⊗ ¯∇_{X}S ∘ σ, ν)(X, …) + c((∇_{X}T) ⊗ S ∘ σ, ν)(…)

− ∑_{i} c(c(T ⊗ S ∘ σ, ν) ⊗ l ∘ σ_{i}, ν)(…) + c(T ⊗ S ∘ σ, ∇_{X}ν)(…)

Now, if Y ∈ C^{∞}(T ∂M) then

0 = X.g(ν, Y) = g(∇_{X}ν, Y) + g(ν, ∇_{X}Y)
⟹ g(∇_{X}ν, Y) = l(X, Y) = l(Y, X)
0 = X.g(ν, ν) = 2g(∇_{X}ν, ν)
Together ∇_{X}ν = l^{∗_{1}}(X) ⟺ ∇ν = l^{∗_{1}}

This finishes the proof.

Step 4: Normal boundary coordinates

First consider ∂M as a Riemannian (m − 1)-dimensional manifold of its own. Let ν be the unit inward normal vector field. Step 3 shows that bounds on the covariant derivatives ∇^{j}R of the curvature of M and ¯∇^{j}l of the second fundamental form of ∂M (0 ≤ j ≤ k) give rise to bounds on the derivatives ¯∇^{j}R̄ (0 ≤ j ≤ k) of the curvature of ∂M. Therefore we can use step two to construct the orthonormal frame {s_{i}}_{1≤i≤m−1} of T ∂M. Extend this to an orthonormal frame of T M|_{∂M} by setting s_{m} = ν. By parallel transport along geodesics with initial speed ν we get a synchronous orthonormal frame of T M on a normal collar neighbourhood. Now we define the dual frame {θ^{i}}, the Christoffel symbols Γ_{jk}^{i} and Γ̃_{jk}^{i}, the curvature coefficients R_{jkl}^{i} and K_{jkl}^{i}, and the base change matrices a_{j}^{i} and b_{j}^{i} in exactly the same way as we did it in step 2. Note that the fundamental relations (A.5) and (A.6) relating R_{jkl}^{i} and K_{jkl}^{i}, Γ̃ and Γ as well as g_{ij} and a_{j}^{i} remain true.

Now we come to the differential equations which relate these quantities. Let x_{1}, …, x_{m} be the normal coordinates and x_{m} the normal direction, i.e. ∂_{m}|_{∂M} = ν. Our approach is quite similar to the one used by Atiyah/Bott/Patodi [3]. By construction {s_{i}} is parallel to ∂_{m}. This translates to

i(∂_{m})θ_{j}^{i} = 0, i.e. Γ_{jm}^{i} = 0 ∀i, j.  (A.19)

(i(∂_{m}) denotes contraction with ∂_{m}). The Lie derivative along ∂_{m} (denoted by ∂_{m}) acts on differential forms via ∂_{m} = i(∂_{m})d + di(∂_{m}). Therefore,

\begin{aligned}∂_{m}θ_{j}^{i} & = i(∂_{m})dθ_{j}^{i} = i(∂_{m})(dθ_{j}^{i} − ∑_{k=1}^{m} θ_{k}^{i} ∧ θ_{j}^{k}) \\ & = (∂_{m})(∑_{k,l} R_{jkl}^{i})dx_{k} ∧ dx_{l} = 2 ∑_{k} R_{jmk}^{i}dx_{k} \end{aligned}

On the other hand

∂_{m}θ_{j}^{i} = ∑_{k} ∂_{m}(Γ_{jk}^{i})dx_{k}

Equating coefficients and applying D_{x}^{α} yields

∂_{m}(D^{α}Γ_{jk}^{i}) = 2D^{α}R_{jmk}^{i} ∀i, j, k  (A.20)

Also, we need an equation for a_{j}^{i}. Here we apply ∂_{m} twice to the dual frame. Since ∂_{m} = s_{m}, i(∂_{m})θ^{i} = δ_{im} (i.e. 1, if i = m, elso zero). Then

∂_{m}θ^{i} = i(∂_{m})dθ^{i} + di(∂_{m})θ^{i} = i(∂_{m})dθ^{i}

The connection is torsion free. This means

\begin{matrix}dθ^{i} & = ∑_{j} θ_{j}^{i} ∧ θ^{j} \\ ⟹ & ∂_{m}(θ^{i}) = ∑_{j}(∂_{m})θ_{j}^{i} ∧ θ^{j} = −θ_{m}^{i} \\ ⟹ & ∂_{m}^{2}(θ^{i}) = −∂_{m}(θ_{m}^{i}) = −2 ∑_{k} R_{mmk}^{i}dx_{k} \end{matrix}

The left hand side can be computed in terms of a_{j}^{i}

∂_{m}(θ^{i}) = ∑_{j} ∂_{m}(a_{j}^{i})dx_{j} ⟹ ∂_{m}^{2}(θ^{i}) − ∑_{j} ∂_{m}^{2}(a_{j}^{i})dx_{j}

Equating coefficients and computing R_{jkl}^{i} in terms of K_{jkl}^{i} yields

∂_{m}^{2}a_{j}^{i} = −2 ∑_{αβ} K_{mαβ}^{i}a_{m}^{α}a_{j}^{β}  ∀i, j  (A.21)

To make use of the differential equation (A.20) and (A.21) we have to supply the initial conditions (x_{m} = 0). For a_{j}^{i} this is very easy: if i, j < m, a_{j}^{i}|_{x_{m}=0} is given by application of step 2 to ∂M. Especially, we get univeral bounds for these functions. And by construction a_{m}^{i} = a_{i}^{m} = δ_{im} for x_{m} = 0 (actually even everywhere on the chart). For the first derivative we have ∂_{m}a_{j}^{i}|_{x_{m}=0} = −Γ_{jm}^{i} (this follows from the two formulae for ∂_{m}θ^{i}).

It remains to compute Γ_{jk}^{i} on ∂M. By definition ¯∇_{∂_{i}}s_{j} = ∇_{∂_{i}}s_{j} −l(∂_{i}, s_{j})ν for i, j < m. If we define l_{j}^{i} := l(∂_{i}, s_{j}), then for j, k < m

Γ_{jk}^{i}|_{x_{m}=0} = \begin{cases}Γ̄_{jk}^{i}|_{x_{m}=0}; & i < m \\ l_{j}^{i}; & i = m \end{cases}

We have already seen that Γ_{jm}^{i} = 0 ∀i, j, since {s_{i}} is synchronous along ∂_{m}. To compute Γ_{mj}^{i} we need

∇_{∂_{j}}∂_{m} = ∇_{∂_{m}}∂_{j} = ∑_{i} l_{j}^{i}s_{i}

The first equation is due to the fact that [∂_{m}, ∂_{i}] = 0. The second one is stated in Warner [68] and is a consequence of ∂_{m} ⊥ ∂_{i} for i < m all over the collar (the latter follows from the Gauss lemma A.4). It follows for i < m

Γ_{mj}^{i} = \begin{cases}l_{j}^{i}; & j < m \\ 0; & j = m \end{cases}

The arguments of step two show that the initial functions and their partial derivatives up to order k in directions x_{1}, …, x_{m−1} are uniformly bounded, if bounds exist on the covariant derivatives up to order k of the second fundamental form and of R (hence by step 3 also on R̄).

We proceed with a bootstrap argument similar to the argumentation in step 2: First, K_{jkl}^{i} and (since the length of ∂_{i} is bounded by proposition A.2) R_{jkl}^{i} are M-uniformly bounded functions. From equation (A.20) we find M-universal bounds for the Γ_{jk}^{i}. With these, the bound on ∇R translates to M-universal bounds on ∂_{α}K_{jkl}^{i}. From (A.21) it follows that ∂_{α}a_{j}^{i} are M-universally bounded. This, together with the relation (A.5) between R_{jkl}^{i} and K_{jkl}^{i} yields bounds on ∂_{α}R_{jkl}^{i}. Now we start again with (A.20) to get bounds on the first order derivatives of Γ_{jk}^{i}. Inductively, this concludes the proof of our main theorem A.1.

Changes of normal coordinates

In this section, we describe properties of change-of-variable maps for normal charts on manifolds of bounded geometry. We prove the following theorem:

A.22. Theorem. Suppose, M is a bounded Riemannian ∂-manifold, i.e. C > 0 is a bound for partial derivatives up to order k + 1 of the metric tensors g^{ij} and g_{ij} in normal coordinates. Then, an M-universal D > 0 exists so that, if κ_{1} : U_{1} ⊂ ℝ^{m} → M and κ_{2} : U_{2} ⊂ ℝ^{m} → M are arbitrary normal charts, the following holds for f := κ_{1}^{−1} ∘ κ_{2} : U_{0} ⊂ ℝ^{m} → ℝ^{m} (U_{0} the natural domain of definition of the composition):

|D^{α}f| ≤ D ∀|α| ≤ k.

If we have an M-universal bound C on the curvature tensor R, then

|Df| ≤ D with Df the differential of f

The maps κ_{i} are given as solutions of ordinary differential equations (o.d.e.) (namely, the equation for geodesics). Therefore, first we state some results for arbitrary differential equations.

Boundedness properties for the flow of a differential equation

A.23. Proposition. Let x′ = F (t, x) be a system of o.d.e. (t ∈ ℝ, x ∈ ℝ^{n}), F ∈ C^{∞}(ℝ × ℝ^{n}, ℝ^{n}). Suppose φ(t, x) is the flow of this equation. Then

|\frac{∂φ}{∂x_{k}} (t, x_{0})| ≤ exp(∥D_{x}F ∥t) ∀t ≥ 0.  (A.24)

For higher derivatives, we find a universal expression Expr_{α}, only depending on α and involving iterated exponentials, so that

|D_{x}^{α}(x_{0}, t)| ≤ Expr_{α}(∥D_{x}^{β}F ∥, t_{0} | β ≤ α)

Proof. The theory of o.d.e. tells us (Hartmann [35, V.3.1.]) that we have the linear differential equation

α′(t) = \frac{∂F}{∂x} (t, φ(t, x)) · α(t); α(0) = e_{k} = (0, …, 1, …, 0)

for ∂_{k}φ(t, x). For linear differential equations, we have inequalities (Hartmann [35, IV.4.2]) which directly imply (A.24). Higher derivatives also fulfill a linear differential equation:

(D_{x}^{α}φ)′(x, t) = (D_{x}F)(t, φ(x, t)) · D_{x}^{α}φ(x, t) + P_{α}(D_{x}^{γ}φ, (D_{x}^{β}F)(t, φ(x, t)); γ < α, β ≤ α)

where P_{α} is a polynomial matrix which depends only on α. By induction and using the inequalities for solutions of such linear differential equations, the proposition follows.

A.25. Corollary. Reduction of order shows that a corresponding statement holds for o.d.e. of order k: x^{(k)} = F (t, x, x′, …, x^{(k−1)}).

Changes of normal coordinates

A.26. Corollary. Let p ∈ U ⊂ ℝ^{m}, g_{ij} a Riemannian metric on U and exp_{p} : B(r, 0) → U the exponential map at p (we identify T_{p}U with ℝ^{m} via an orthonormal frame). If r is sufficiently small, then exp_{p} is a diffeomorphism onto some open subset of U, and the derivatives up to order k of exp_{p} are bounded in universal expressions containing g_{ij}, their derivatives up to order k + 1, and r.

Proof. We have exp_{p}(x) = φ(x, p, 1), where φ is the flow of the differential equation for geodesics (φ(x, q, 0) = q, φ′(x, q, 0) = x). To this differential equation: x′′ = F (x), proposition A.23 applies, and F (x) = − ∑_{ij} Γ_{ij}(x)x′_{i}x′_{j} is given by g_{ij} and its first order derivatives.

A.27. Corollary. Suppose p ∈ U′ ⊂ ℝ^{m−1}, κ : U′ × [0, r_{C}) → V × [0, r_{C}) is a normal boundary chart on the Riemannian manifold V × [0, r_{C}) with metric g_{ij} (g_{in} = δ_{in} = g_{ni}). Then κ and its derivatives up to order k are bounded by a universal expression in g_{ij} and their derivatives up to order k + 1.

Proof. κ(q, s) = φ_{1}(s · ∂_{n}, φ_{2}(q, p, 1), 0, 1), where φ_{1} is the flow of the differential equation for the geodesics in V × [0, r_{C}) (φ_{1}(v, p′, s′, 0) = (p′, s′), φ′_{1}(v, (p′, s′), 0) = v), and φ_{2} is the flow of the differential equation for geodesics on V. κ is therefore the composition of two flows to which proposition A.23 applies exactly as in the previous corollary.

Proof of theorem A.22. By theorem A.1 we have universal bounds for g_{ij} and their derivatives up to order k + 1 in normal coordinates. To apply the preceding two corollaries we use one intermediate chart:

κ_{1}^{−1} ∘ κ_{2} = (κ_{1}^{−1} ∘ κ_{0}) ∘ (κ_{0}^{−1} ∘ κ_{2}).

This covers the following cases: κ_{2} Gaussian coordinates and κ_{1} arbitrary normal coordinates (then κ_{0} shall be a Gaussian chart around points in the common range of κ_{1} and κ_{2}), or κ_{1} and κ_{2} are both normal boundary coordinates (then their common range is U × [0, r_{C}) —in terms of the geodesic collar— with U ⊂ ∂M, and κ_{0} shall be a normal boundary chart centered at points of U. It is immediate that the preceding corollaries apply to κ_{0}^{−1} ∘ κ_{2} and to κ_{1}^{−1} ∘ κ_{0} and yield the desired bounds. The next proposition covers the remaining case κ_{1} a Gaussian chart and κ_{2} a normal collar chart and concludes the proof.

A.28. Proposition. In the situation of the theorem, let f = κ_{1}^{−1} ∘ κ_{2} : U → V with κ_{2} a normal boundary chart, κ_{1} a Gaussian chart. Let g = f^{−1}. We have found C > 0 independend of g so that |D^{α}g| ≤ C ∀|α| ≤ k. There is C′ independent of f so that |D^{α}f| ≤ C′ ∀|α| ≤ k.

Proof. For the differentials Df = (Dg)^{−1} holds. By lemma A.11 a bound on Df together with the bounds for the derivatives of Dg (given by C) yields a bound C′ for the higher derivates of f. Moreover, C′ only depends on k, C and the bound on Df.

Now Df = Dx_{1} · Dx_{2}^{−1}. Take v ∈ T U with |v|_{ℝ^{m}} = 1 ⟹^{A.2} |Dx_{2}^{−1}v|_{T M} ≤ C_{2}. Moreover

(again by lemma A.2)

C_{1} ≤ \frac{|Dx_{2}^{−1}v|_{M}}{|Dx_{1}Dx_{2}^{−1}v|_{ℝ^{m}}} ⟹ |Dx_{1}Dx_{2}^{−1}v|_{ℝ^{m}} ≤ \frac{|Dx_{2}^{−1}v|_{M}}{C_{1}} ≤ \frac{C_{2}}{C_{1}}

i.e. we have the (by lemma A.2 M-universal) bound C_{2}/C_{1} for ∥Df∥. The last computation concludes the proof of the additional statement of the theorem.

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