Geometry and Analysis of Boundary-Manifolds of Bounded Geometry

Thomas Schick^{∗} Dept. of Mathematics — Penn State University

218 McAllister Building — University Park, PA 16802

October 21, 1998

Abstract

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^{2}-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary) then its first relative L^{2}-coho-mology vanishes (this in particular answers a question of Roe).

We prove the Hodge-de Rham-theorem for L^{2}-cohomology of oriented ∂-manifolds of bounded geometry.

The technical basis is the study of (uniformly elliptic) boundary value problems on these manifolds, applied to the Laplacian.

MSC-class: 53C20 (primary); 58G20, 57R19 (secondary)

1 Introduction

The broad theme of this paper is geometry, topology and analysis on non-compact manifolds with boundary, which might be empty. We focus on manifolds of bounded geometry (Definition 3.1). These are (noncompact) Riemannian manifolds with bounds on curvature and second fundamental form and their derivatives. Also, they have positive injectivity radius.

∗e-mail: thomas.schick@math.psu.edu This work was finished during a stay at Penn State funded by the DAAD

Note that, on a non-compact manifold, very different metrics may exist. Therefore, we will concentrate on obstructions for Riemannian metrics in a given bilipschitz class. This is proposed by Roe in [14].

We use the Weizenböck formula (for boundary-manifolds) to derive the following Bochner vanishing Theorem 6.4: if we have a metric with nonnegative Ricci curvature and convex boundary, then the first L^{2}- cohomology is trivial. Then we refine this method so that we can weaken the positivity conditions: they are not required to hold on all of the manifold but only on an eventually large set, defined as follows:

1.1. Definition. Let M be any metric space. A subset X ⊂ M is called eventually large if ∀R > 0 we find x_{R} ∈ X so that B(x_{R}, R) ⊂ X; the R-ball around x_{R} is entirely contained in X.

A corollary is the following generalization of results of Roe [14]:

1.2. Theorem. (compare Theorem 6.7) Let M be a compact manifold, possibly with boundary. Let M̄ be a normal covering of M. Suppose we find a metric g on M̄ which lies in the bilipschitz class of the lifted metric and has bounded geometry. Suppose there exists an eventually large subset X ⊂ M̄ so that on X g has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary). Then b_{(2)}^{1}(M̄, ∂M̄) = 0.

1.3. Corollary. If dim M = 2, or dim M = 4 and M is closed, and if χ(M) < 0 then on the universal covering of M no metric g as described in the theorem exists.

These results answer affirmative [14, Question 6.3] of Roe.

The first (relative) L^{2}-Betti number b_{(2)}^{1} of the theorem above is zero exactly if the space of L^{2}-harmonic 1-forms (with relative boundary conditions) is trivial. It is the von Neumann dimension of this space and a homotopy invariant of M (compare [2]). We prove for an arbitrary orientable manifold M of bounded geometry the L^{2}-Hodge de Rham theorem: the square integrable harmonic forms are isomorphic to simplicially defined L^{2}-cohomology. For this end let ∂M = M_{1} ⨿ M_{2} (M_{1} or M_{2} or both may be empty).

Let K be a smooth triangulation of M. It has to be sufficiently regular, adapted to the bounded geometry condition (compare [3, 2.3]):

1.4. Definition. The triangulation K is called g-bounded if there are ν, K, c > 0 so that (1) the volume of the top-simplexes is bounded from below v,

(2) the diameter of the top-simplexes is bounded from above by K,

(3) |dφ_{p}| ≤ c ∀p where φ_{p} is the barycentric coordinate function for the vertex p of K.

This mimics the injectivity radius and curvature conditions in the definition of bounded geometry.

1.5. 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 (obviously a Hilbert space) (K_{1} is the restriction of K to M_{1}). Since K is g-bounded this is a subcomplex (compare [3]). Its L^{2}-cohomology is

H_{(2)}^{p}(K, K_{1}) := ker d_{p}/\overline{im d_{p−1}}.

The following is our L^{2}-Hodge-de Rham theorem:

1.6. 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 5.5). 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.

If ∂M = ∅ this was done by Dodziuk [3]. We use his result and a doubling trick to prove the result if the metric is a product near the boundary. The last step consists in the comparison of an arbitrary metric with a suitable product metric near the boundary.

Basic technical tools to derive these results are “uniform analysis” like the introduction of uniform Sobolev spaces on manifolds of bounded geometry which have properties similar to the Euclidian case. We introduce uniformly elliptic boundary value problems for which we derive e.g. regularity results as usual. We prove that the Laplacian with absolute/relative boundary conditions belongs to this class. This in turn implies that it is essentially self adjoint. The general theory also gives the following Hodge decomposition (of Sobolev spaces):

1.7. Theorem. (compare 5.12 for the definitions) 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 the Dirichlet/Neumann boundary conditions p⃗ 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.

This paper is based on parts of the author’s dissertation [15]. I thank my advisor, Prof. W. Lück, for his constant support and encouragement.

Organization 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 their main (analytical) properties.

Section 4 deals with elliptic boundary value problems. We define uniform ellipticity as the right condition for manifolds of bounded geometry and derive basic properties.

In Section 5 we proof the Hodge decomposition theorem.

After the technical tools are established, Section 6 examines the relations between L^{2}-cohomology and curvature and proves the Bochner vanishing theorems.

In Section 7 we prove the L^{2}-de Rham theorem.

2 Notation

2.1. Definition. We will often use inequalities to show equivalences of

norms. 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. The same symbol (e.g. c) may be used for different constants.

2.2. Definition. We will come up with equations where the sign of some of the terms does not matter. Then we simply use “±”.

2.3. Convention. A manifold M shall always have dimension m, a bundle E↓M dimension n, if not stated otherwise. 𝕂 stands for ℝ or ℂ.

2.4. Definition. We use multiindices α = (α_{1}, …, α_{m}): tupels of nonnegative integers. Set |α| := max\{α_{1}, …, α_{m}\} and D^{α} := \frac{∂^{α_{1}}}{∂x_{1}^{α_{1}}} … \frac{∂^{αm}}{∂x_{m}^{αm}}.

2.5. Definition. For a metric space M set B(p, r) = {x ∈ M; d(x, p) ≤ r}. If we write B(x′, r) ⊂ ∂M we consider ∂M as a metric space defined by the induced Riemannian metric and construct the ball in this metric.

2.6. Notation. Let M be a manifold of bounded geometry. A constant is called M-universal if it depends only on M, not on local data.

3 Bounded geometry

In this section we introduce the concept of a boundary-manifold of bounded geometry. These manifolds have uniformity properties which e.g. allow the definition of uniform Sobolev spaces. For them, the Sobolev embedding theorem holds; differential operators induce bounded operators H^{s} → H^{s−k}; for s > 1/2 a bounded restriction map H^{s}(M) → H^{s−1/2}(∂M) exists; ….

Manifolds of bounded geometry, but with empty boundary, have been examined and used, among many others, by the following authors: Shubin [19] studies (pseudo)differential operators, Roe [13] derives an index theorem, Dodziuk [3] studies the Laplacian, Cheeger-Gromov-Taylor [1] investigate integral kernels of functions of the Laplacian, Eichhorn deals with various of the above and other aspects [5, 6, 7].

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. Set N_{1/3} := im[0, r_{C}/3) × ∂M and define N_{2/3} similarly.

(IC) Positive injectivity radius of ∂M: 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.

(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 the second fundamental form tensor, ∇ is the Levi-Civita connection of M and ¯∇ the one of ∂M

This is a generalization of corresponding definitions for manifolds without boundary. The embedding of the boundary is described by the second fundamental form; bounds on it will guarantee some homogeneity. Because the injectivity radius does not make sense near the boundary, we replace it by the geodesic collar.

Now we recall Gaussian coordinates and their substitute near the boundary.

3.2. Definition. Choose 0 < r_{i}^{C} < r_{inj}(∂M) and x′ ∈ ∂M. Identify T_{x′}∂M with ℝ^{m−1} via an orthonormal base. 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)

(we compose the exponential maps of ∂M and of M). Here 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).

The main result of [17] is:

3.3. Proposition. A Riemannian manifold M has bounded geometry if and only if (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.

It suffices to check Condition (B1) for an atlas.

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:

3.4. Definition. A bundle E↓M over a manifold of bounded geometry has bounded geometry if it is provided with the following additional structure: a trivialization (called admissible) over each normal chart so that the transition functions for the trivializations, 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}.

3.5. 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, but we omit writing down this chart. Similar remarks apply to sections of bundles of bounded geometry and other objects like differential operators.

3.6. Definition. Let E↓M be a bundle of bounded geometry. Suppose it is equipped with a Hermitian metric. We denote this metric bounded if the conjugate linear isomorphism E → E′ : x ↦ (·, x) is bounded with bounded inverse (in the sense of 3.9).

In an admissible trivialization the Gram’s matrix function b_{ij}(x) = (e_{i}, e_{j})_{x} describes the Hermitian inner product. Let b^{ij}(x) be the inverse. Boundedness translates to the existence of an E-universal C_{K} > 0 so that |∂_{x}^{α}b_{ij}(x)| ≤ C_{K} and |∂_{x}^{α}b^{ij}(x)| ≤ C_{K} ∀ |α| ≤ K.

3.7. Example. (compare [19] and [13])

(1) Every compact Riemannian manifold and every bundle over it are 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 over M is of bounded geometry (if we lift the trivializations).

(3) Compact perturbations (e.g. connected sums) do not change the bounded geometry property.

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

(5) On a manifold of bounded geometry, the trivial as well as the tangent bundle (with trivializations given by normal coordinates) are of bounded geometry.

(6) If M is of bounded geometry then the same is true for ∂M

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

3.8. Definition. Let E be a bundle of bounded geometry over M. Then C^{∞}(E) denotes smooth sections, C_{0}^{∞}(E) smooth sections with compact support. Similarly defined are C^{k} and C_{0}^{k}.

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} through the corresponding admissible trivializations. 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, analysis is only possible if the operator classes and the function spaces match. 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.

3.9. 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 trivializations, A becomes a matrix (∑_{|α|≤k} a_{α}^{ij}(x)D^{α})_{ij}. 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 a differential operator composed with restriction to the boundary.)

3.10. Example. On a manifold of bounded geometry the exterior differentiation d, its formal adjoint δ, the Laplacian Δ and the Hodgeoperator ∗ are bounded.

Proof. Boundedness is clear for d. Since the composition of bounded operators is bounded and since δ = ± ∗ d∗ and Δ = dδ + δd, we only have to check it for ∗. But the entries in the matrix locally describing ∗ are polynomials in the metric tensors g_{ij} and g^{ij}. q.e.d.

Sobolev theory for manifolds of bounded geometry

We will define Sobolev spaces using local data. For this, we need particular coverings and partitions of unity, whose existence is established in [17, Proposition 3]:

3.11. Lemma. 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 charts {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.

• 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.

The charts are denoted 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.

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 ∈ ℤ ∀ |α| ≤ K  (in normal coordinates).

3.12. Definition. Let M be a bounded ∂-manifold. Choose data as in Lemma 3.11. 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 f instead of fκ_{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 trivializations) we define H^{s}(E) if E is a bundle of bounded geometry over M.

It is easy to prove that other choices yield equivalent Sobolev norms.

The next proposition lists elementary properties of our Sobolev spaces, which follow from the corresponding properties of ℝ_{+}^{n}.

3.13. 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) Sobolev embedding theorem: we have a bounded embedding

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

(2) 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).

(3) If r, s, t ∈ ℝ with s < t and ϵ > 0 then there is C > 0 such that

|·|_{H^{s}}^{2} ≤ ϵ |·|_{H^{t}}^{2} + C |·|_{H^{r}}^{2} .

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

A : H^{s}(E) → H^{s−k}(F) ∀s.

(5) Let p : C^{∞}(E) → C^{∞}(X) be a 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}.

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 differential operators.

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 map). 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 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 a constant C so that ∥a_{μ}^{−1}(x, ξ)∥ · |ξ|^{μ} ≤ C ∀(x, ξ).

We use the norms induced by admissible trivializations.

Before we define elliptic boundary value problems we make a digression on conventions concerning arbitrary boundary value problems.

Boundary value problems

4.3. 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. We set p⃗ = (p_{0}, …, p_{r}) and P = (A, p⃗). The composition of (A, p⃗) and (B, q⃗) (if this makes sense) is defined as (A ∘ B, (q⃗ ∘ A) ⊕ p⃗).

Next we go on to describe adjoints to boundary value problems. From now on,

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

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

4.4. Definition. We say p⃗ = (p_{0}, …, p_{r}) is normal if in all charts around ∂M the matrix (a_{i,j}(x, ξ))_{ij} of the principal part of p_{α} has rank n_{α} for fixed x and for ξ = (0, …, 0, ξ_{m}); ξ_{m} ≠ 0.

If p⃗ is normal and n_{α} = n ∀α, it is called a Dirichlet system.

Normality says that the different boundary operators p_{i} are independent of each other. Dirichlet systems are maximal normal systems. They have a unique adjoint and are unique up to transformations of lower order.

4.5. Lemma. Suppose p⃗ = (p_{0}, …, p_{r}) is normal. Then a (normal) set of boundary differential operators t⃗ = (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 less than i and A_{i} : E → X_{i} is a linear bundle morphism which is surjective by normality. Set Y_{i} := ker A ⊂ E and t_{i} := pr_{Y_{i}} · ∂_{ν}^{i} (pr_{Y_{i}} being the orthogonal projection E → Y_{i}). q.e.d.

4.6. 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 such that

p⃗ = Ψq⃗; q⃗ = Φp⃗

(this implies Ψ ∘ Φ = 1 = Φ ∘ Ψ) and such that

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

are tangential differential operators of order j − i (i.e. Φ_{i,j} = 0 = Ψ_{i,j} for i > j) and so that Φ_{i,i} and Ψ_{i,i} are linear bundle isomorphisms.

Proof. First observe that it suffices to construct Φ and Ψ locally (glue them together with a partition of unity). Next replace q⃗ by D⃗ : U × 𝕂^{n} → U × (𝕂^{n})^{r+1} with D_{i} = 1 · ∂_{m}^{i} (the general case follows by twice applying this). 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} (p_{0} is invertible by definition of Dirichlet systems). For i > 0 set Ψ_{i,0} = 0. Suppose Ψ_{i,j} is already constructed 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. q.e.d.

4.7. 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}.

Proof. It suffices to consider the local case. Let p⃗ = ΦD⃗ and D⃗ = Ψp⃗ as in Lemma 4.6. Let φ⃗ = (φ_{0}, …, φ_{μ−1}). By Whitney’s [20] we find smooth f with compact support so that D⃗f = α. Then

p⃗f = ΦD⃗f = Φ(α⃗) = ΦΨ(φ⃗) = φ⃗. q.e.d.

4.8. Definition. Suppose (A, p⃗) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁_{i} C^{∞}(X_{i}) is a normal boundary value problem. Choose t⃗: C^{∞}(E) → ⨁_{i} C^{∞}(Y_{i}) so that p⃗ ⊕ t⃗ is Dirichlet. A similar system

(B, s⃗ ⊕ q⃗) : C^{∞}(F) → C^{∞}(E) ⊕ ⨁ C^{∞}(X_{i}) ⨁ C^{∞}(Y_{i})

is called the formal adjoint (with respect to given metrics on E, F, X_{i}, Y_{i} and on M) if ∀f ∈ C_{0}^{∞}(E), g ∈ C_{0}^{∞}(F)

(Af, g) − (f, Bg) = (p⃗f, s⃗g) − (t⃗f, q⃗g).  (4.9)

Here (p⃗f, s⃗g) := ∑_{i} ∫_{∂M}(p_{i}f(x), s_{i}g(x))_{X_{i}|_{x}}dx, etc. In particular B is the formal adjoint of A. We call (B, q⃗) the formal adjoint system to (A, p⃗) (with respect to theGreenian formula (4.9)).

If the manifold and the bundles are of bounded geometry and all operators are bounded then by continuity the Greenian formula (4.9) extends to f ∈ H^{μ}(E), g ∈ H^{μ}(F).

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

Proof. Uniqueness: Only considering f with supp f ∩ ∂M = ∅ we see that B must be the unique formal adjoint of A. Let (B, s⃗, q⃗) and (B, s⃗_{1}, q⃗_{1}) be two adjoint systems. By Lemma 4.7 and the Greenian formula:

\begin{matrix}(φ, s⃗g) − (ψ, q⃗g) = (φ, s⃗_{1}g) − (ψ, q⃗_{1}g) ∀φ, ψ ∈ C_{0}^{∞} \\ ⟹ s⃗ = s⃗_{1} and q⃗ = q⃗_{1}. \end{matrix}

Existence: First we construct the adjoint locally. On a chart, integration by parts as in [11, p. 218] yields

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

for g with support in a chart neighborhood. (D⃗ is defined in the proof of 4.6). Now, by 4.6 D⃗ = Φ(p⃗ ⊕ t⃗) with a tangential 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).

Let s⃗ and q⃗ be the corresponding components of Φ^{∗}N⃗ which by uniqueness are globally defined. The remaining conditions on them are checked as in [11, p. 218]. q.e.d.

Now we will define uniformly elliptic boundary value problems. The idea is 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.

We use the existence of fundamental solutions for elliptic boundary value problems and pose uniformity conditions on these. In Proposition 5.8 we will give a condition for uniform ellipticity which can be checked directly on the coefficients of the operators.

4.11. 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. We call (A, p⃗) uniformly elliptic if it is an elliptic boundary value problem in the sense of Hörmander [9, 10.6.2], if A is uniformly elliptic and if the following holds: Since (A, p⃗) is elliptic, ∀x ∈ ∂M there is 0 < ϵ_{x} < r_{C} and a bounded fundamental solution [9, 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}) 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} so that its norm is bounded uniformly in x.

4.12. Remark. In [15, 4.10, 4.11] it is shown that for a uniformly elliptic boundary value problem not only the induced map H^{μ} → L^{2} has uniform fundamental solutions but every map H^{μ+k} → H^{k}.

4.13. Example. On a covering manifold M̃↓M the lift of any elliptic differential boundary value problem is uniformly elliptic.

We will now derive elliptic regularity in our context.

4.14. Theorem. Let P = (A, p⃗) : C^{∞}(E) → C^{∞}(F) ⊕ ⨁_{i} C^{∞}(X_{i}) be uniformly elliptic of order μ (situation of 4.11). 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})) 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.

Proof. By [9, p. 270] the hypotheses implies 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 3.11 and so small, that we have local fundamental solutions R_{i}. Let M_{f} be the covering dimension. 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} (Remark 4.12)

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

≤ C′(|Pu|_{s−μ}^{2} + ∑_{i} ∑_{j} |P_{i}φ_{j}u|_{s−μ}^{2}) (C′ = C2^{M_{f} +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 M_{f} + 1 non-trivial summands, and this, together with |a + b|^{2} ≤ 2(|a|^{2} + |b|^{2}) yields the factor 2^{M_{f}}. As the derivatives of the coefficients of P_{i} are bounded independently of i we get:)

≤ \underbrace{C′(M_{f} + 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})

≤^{3.13(3)} 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}}).

The a priori estimate follows. q.e.d.

The following proposition is proved in [15, Chapter 4].

4.15. Proposition. The composition and the adjoint of uniformly elliptic boundary value problems is uniformly elliptic, as well.

Now we will derive regularity theorems for adjoint boundary value problems. We start with a theorem of Hörmander [9, Theorem 10.4.3 and p. 270] which describes the local situation:

4.16. 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 4.11. 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.

We will need the following lemma to apply Theorem 4.16.

4.17. Lemma. Let (A, p⃗) be an elliptic boundary value problem as in 4.11. Suppose f ∈ L^{2}(F), g ∈ L^{2}(E) so that

(Au, f) = (u, g) ∀u ∈ C_{0}^{∞}(E) with p⃗u = 0.  (4.18)

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 (possible by Lemma 4.7 since p⃗ is normal by ellipticity). Set

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

Then T_{i} ∈ H_{loc}^{−∞}(X_{i}). In particular,

(Au, f) + ∑_{i=0}^{r} T_{i}(p_{i}u) = (u, g)  (4.19)

and, by Theorem 4.16, f ∈ H_{loc}^{μ} and A^{∗}f = g.

Proof. It follows directly from (4.18) 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 neighborhood with bundle trivialization) the functional φ·T_{i} is H^{N}-bounded for some N ∈ ℕ.

Extend p⃗ to a Dirichlet system p⃗ ⊕ s⃗. By 4.6 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). Via a chart we can work in Euclidian space. We consider α^{i} as a tuple of functions on ℝ^{m−1}.

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}}) = (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}. Now (4.19) follows from (4.18) and from p⃗(u) = p⃗(∑_{i} u_{i}). q.e.d.

The regularity result allows the following decomposition of L^{2}.

4.20. Corollary. Make the same assumptions as in Theorem 4.16. 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}^{∞}.

Setting D_{A}^{∞} := {g ∈ C_{0}^{∞}; p⃗g = 0} we get an orthogonal decomposition

L^{2}(F) = \underbrace{\{f ∈ C^{∞} ∩ L^{2}; Bf = 0, q⃗f = 0\}}_{=:ker B} ⊕\overline{A(D_{A}^{∞})}.  (4.21)

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\} ⊕ \overline{A\{g ∈ C_{0}^{∞}(E); p^{∞}g = 0\}}. (4.22)

Proof. The Greenian formula implies that ker B and A(D_{A}^{∞}) in (4.21) are orthogonal to each other. Suppose that f ∈ L^{2}(F) is orthogonal to A(D_{A}^{∞}). Then Lemma 4.17 and Theorem 4.16 show that Bf = 0 and f ∈ H_{loc}^{∞} = C^{∞}. It remains to show that q⃗f = 0: Choose an arbitrary smooth boundary value t⃗g with compact support. By Lemma 4.7 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

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

For the bounded geometry formula (4.22) observe first ker B ⊂ H^{∞}(E) because of the a priori estimates of Theorem 4.14. 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}. This follows from the following approximation result, proved in [15, 3.42]:

If E↓M and X_{i}↓∂M are bundles of bounded geometry and

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

is an infinite Dirichlet system 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 ∈ ℕ and K a fixed compact set), we find u ∈ C_{0}^{∞}(E) with

p_{i}u = u_{i} ∀i ∈ ℕ  so that |u|_{H^{k}(E)} < ϵ.  q.e.d.

On a manifold of bounded geometry, the above can be strengthened to a condition for (essential) self adjointness:

4.23. 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).

Set 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 4.17 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 4.7 we find 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 4.14:

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

This concludes the proof. q.e.d.

We will see in the next section how Theorem 4.23 shows that the Laplacian with Dirichlet/Neumann boundary conditions is self adjoint.

Next, we have to recall further aspects of Hilbert space theory.

4.24. Lemma. Let A : D → H be a positive (≥ c > 0) 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 isometry.

4.25. Corollary. Suppose A : D ⊂ L^{2} → L^{2} is a strictly 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 and A : D → L^{2} is an isometry if we equip D ⊂ H^{μ} with | · |_{A}.

Proof. By Lemma 4.24, (D, | · |_{A}) is complete. Because A is of order μ,

|ω|_{A} = |Aω|_{L^{2}} ≤^{∥A∥} |ω|_{H^{μ}} ∀ω ∈ D. Since (D, | · |_{H^{μ}}) is complete, too,

the norms are equivalent by the open mapping theorem. q.e.d.

4.26. Theorem. Let (A, p⃗) be a uniformly elliptic formally self adjoint boundary value problem of order μ. Choose ϵ > 0 so that (ϵ + A, p⃗) is again uniformly elliptic (this is possible by [15, 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 ∈ ℕ.

Suppose A is a non-negative 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 Proposition 4.15 and 4.23. 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 4.14 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 4.25 applies and yields the isomorphism. By Corollary 4.20 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. q.e.d.

5 The Hodge decomposition

We start with a discussion of the proper boundary conditions for the Laplacian.

Let M be a complete oriented Riemannian manifold with ∂M = M_{1} ⨿ M_{2} (possibly empty). Let d be the differential on forms, δ = ± ∗ d its formal adjoint and Δ = dδ + δd the Laplacian.

5.1. Notation. For a form ω, ω|_{M_{1}} means the pullback to the submanifold M_{1}.

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, …

Define now Dirichlet/Neumann boundary conditions (k ≥ 0)

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

5.3. 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)}.

The following statements are more or less well known. Proofs may be found in [15, Chapter 5].

5.4. 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 4.8 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 as follows (∗ denotes the Hodge-∗-operator of M; ˆ∗ the one of ∂M):

\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}

5.5. 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)}.

If M is complete then these forms are closed and coclosed, i.e. H^{p}(M, M_{1}) = {ω ∈ C^{∞} ∩ L^{2}(Λ^{p}M); dω = 0 = δω, ω|_{M_{1}} = 0 = ∗ω|_{M_{2}}}.

As application of Corollary 4.20 we obtain:

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

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

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

Hodge decomposition for manifolds of bounded geometry

We want to show that on a manifold of bounded geometry the Dirichlet/Neumann boundary value problem for the Laplacian (Δ, p^{1}) is uniformly elliptic.

To do this we investigate how to check for uniform ellipticity directly from the coefficients of the boundary value problem, without explicitly constructing the fundamental solutions.

5.7. Definition. Let P = (A, p⃗) be a boundary value problem as in Definition 4.3. For each point b ∈ ∂M we get normal collar coordinates and admissible bundle trivializations. In these coordinates the problem is described by matrices (∑_{α} A_{α}^{(ij)}(x)D^{α})_{ij} and (∑_{α} p_{α,k}^{(ij)}D^{α})_{ij} (k = 0, …, μ − 1) with corresponding symbols A(x, ξ), … (omiting an additional index b). In the following we inspect the exposition in Hörmander’s [9, 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 [9, 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}).

The smallest possible constant depends continuously on the matrices. Let C_{x}^{b} < ∞ denote the best possible constant valid ∀ |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 matrices. For λ ≥ λ_{x}^{b} := max{λ_{x,ξ′}; |ξ′| = 1} set C_{x,λ}^{b} := max_{|ξ′|=1}{C_{x,ξ′}^{λ}}.

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}.

5.8. Proposition. If λ, r, C > 0 exist so that (notation as above)

(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 such that in normal coordinates around b a fundamental solution R_{b} on B(b, ρ) in the sense of Definition 4.11 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 [9, 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. q.e.d.

5.9. 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 Schwarz [18]). The principal symbol of 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}. 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 5.8. What is the local expression for Δ and p^{1}? We will not write down the complicated formulas (see Schwarz [18]) 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 5.8 is fulfilled. Concerning (C0) and (C1) we have the map

\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) and 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 on this set bounds for λ_{x}^{b} and for C_{x}^{b} and C_{x,λ}^{b} exist 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 relabeling) (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 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}). In particular, they come from an elliptic boundary value problem, i.e. they lie in E. q.e.d.

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

5.10. Lemma. The operator Δ is non-negative on its domain 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}.  (5.11)

Proof. Simply integrate by parts. By continuity, this 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} to L^{2}. q.e.d.

Theorem 4.14 implis H^{p}(M, M_{1}) ⊂ H^{∞}(Λ^{p}T^{∗}M). Applying 4.26 to our situation, we get:

5.12. Theorem. Let M be an orientable manifold of bounded geometry, possibly 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}}.

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

6 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 Weizenböck formula Δ = ∇^{∗}∇ − R^{W} with its integrated consequence (for ∂-manifolds)

\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} (6.1)

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}} (proved in Schwarz [18, 2.1.5 and 2.1.7]). Here 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}}, (6.2)

where ν is the unit inward normal field, and i_{X} denotes contraction of a form with the vector field X.

We derive algebraic properties of the operators S and R^{W}, especially sufficient conditions for negativity. With this at hand, it is an easy task to get vanishing results for square integrable harmonic forms.

6.3. Proposition. Let M be a Riemannian manifold, R_{p}^{W} the Weizenböck tensor and S_{p} the boundary tensor 6.2.

(1) R_{1}^{W} = − Ric (Ric the Ricci operator). If the curvature operator is ≥ 0 then R_{p}^{W} ≤ 0 ∀p.

(2) S_{p}|_{x} ≤ 0 if the sum of the m−p smallest eigenvalues of l|_{x} is nonnegative (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.

Note: Negativity of the second fundamental form measures convexity of the boundary inside the manifold.

Proof of Proposition 6.3. (1) is classical (compare [10, II.8.3 and II.8.6]). For (2) we explicitly perform the computations: We consider the second fundamental form as a symmetric fiber-wise 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}. 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 formulas of Schwarz [18, (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 if and only if 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. q.e.d.

6.4. 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 S are both negative semidefinite on k-forms, then H^{k}(M, ∂M) vanishes. If S_{m−k} ≤ 0 instead of S_{k}, then H^{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 H^{1}(M, ∂M) vanishes.

Proof. First note that the second statement is a direct consequence of the first by Proposition 6.3. The first statement is proved similar to the way Dodziuk treats manifolds without boundary in [4], using equation (6.1) (details are given in [15, Theorem 7.7]). q.e.d.

One can apply this to infinite coverings of a compact manifolds. Using the L^{2}-Hodge de Rham Theorem 1.6, 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}) vanish.

6.5. Example. Take ℝP^{3}#ℝP^{3}#ℝP^{3} or (ℝP^{3}#ℝP^{3}#ℝP^{3})−D^{3}. An explicit calculation of Lott and Lück [12, 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. (This does not follow from the classical Bochner theorem.)

In the case of normal coverings of compact manifolds (and for other manifolds of bounded geometry with sufficient symmetry), we can give much weaker condition for the vanishing of L^{2}-cohomology groups. On a non-compact 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 (rough) 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 [14]. Recall the following:

6.6. 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.

We will strengthen the L^{2}-Bochner theorem by imposing positivity conditions on the Weitzenböck endomorphism only on (the small) eventually large sets (Definition 1.1). Observe that in a Riemannian manifold with infinite diameter, the complement of any compact set is eventually large.

6.7. Theorem. Let (M, g_{1}) be an oriented manifold of bounded geometry (possibly with boundary), so that R_{p}^{W} ≤ 0 on an eventually large set and additionally S_{p} ≤ 0 on an eventually large set. Let g be a metric on M which is bilipschitz to g_{1} and so that (M, g) admits a cocompact group Γ of isometries.

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

6.8. 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 is the existence of an enormous subset where the curvature assumption is not required to hold. This is due to the uniformity of the metric and does not hold in general for manifolds of bounded geometry as shows [15, Example 7.2].

For the proof of Theorem 6.7 we need the following proposition which is essentially due to Roe [14, 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).

6.9. Proposition. Suppose M is a ∂-manifold of bounded geometry. If R_{p}^{W} and S_{p} are non-positive on an eventually large subset X of M, then ∀ϵ > 0 an eventually large subset Y ⊂ M exists so that

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

Proof. Set K := M − X. Suppose s ∈ L^{2} and set s_{t} := e^{−tΔ}s. Then s_{t} →^{t→∞} P s in L^{2} by the spectral theorem and s_{t} ∈ D_{Δ}. Even better: s_{t} ∈ D_{Δ^{k}}, the domain of the k-th power of (1 + Δ), ∀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 ( 5.12)). Let ψ : M → [0, 1] be a smooth function with support contained in X so that ψ = 1 on an eventually large set Y ⊂ X and |∇ψ| < ϵ (take f.i. an approximation of the Lipschitz function x ↦ max{1, ϵ dist(x, M − X)}). Observe that we find M-universal constants C_{k,l} > 0 so that for t > 1

\begin{matrix}|Δ^{l}s_{t}|_{H^{k}} =^{5.12} |(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}  (6.10)

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

∫_{Y} |∇s_{t}|^{2} ≤ ∫_{M} ψ(∇s_{t}, ∇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 (6.1). Note (ψs_{t})|_{∂M} = 0. Moreover, R^{W} ≤ 0 and S ≤ 0 on supp ψ. 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 (6.10), then Cauchy-Schwartz and (Δu, u) = (du, du) + (δu, δu) for u ∈ D_{Δ} by (5.11))

≤ϵ |s|_{L^{2}}^{2} + |Δs_{t}|_{L^{2}} |s_{t}|_{L^{2}} + ϵ · (∥d∥ + ∥δ∥)C_{1,0} |s|_{L^{2}}^{2} (by (6.10))
≤^{C}ϵ |s|_{L^{2}}^{2} + t^{−1} |s|_{L^{2}}^{2}  (again by (6.10)).

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 (6.10) for k, l ≤ 1. From now on, we proceed exactly as Roe does in [14, 1.11] to conclude

|s_{t}(y)| ≤ ϵ_{1}(ϵ) |s|_{L^{2}} ∀y ∈ Y_{ϵ}  (6.11)

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

• the estimates (6.10)

• 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 → ∞.

Such a function can be constructed as follows:

Bounded geometry implies that we can construct V_{0}(r) for r sufficiently small [17, Lemma 5]. 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). Mark successive points x_{0} = x, x_{1}, …, x_{[r/d]} on the path with d(x_{i}, x_{i+1}) = d.

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.

Having constructed all ingredients to use Roe’s method, it remains to note that (1 + Δ)^{k}e^{−tΔ}s →^{t→∞} (1 + Δ)^{k}P s = P s in L^{2}-norm ∀k. Theorem 5.12 shows that s_{t} →^{t→∞} P s in the Sobolev space H^{k} ∀k. The Sobolev embedding theorem and (6.11) imply

|P s(y)| ≤ ϵ_{1}(ϵ) |s|_{L^{2}} ∀y ∈ Y_{ϵ}. q.e.d.

We will also need the following lemma in the proof of Theorem 6.7:

6.12. Lemma. Let M be a metric space with a cocompact action by isometries Γ. If X ⊂ M is eventually large and K ⊂ M is compact then there is γ ∈ Γ with γ(K) ⊂ X.

Proof. Choose x ∈ K. Then {B(x, n)}_{n∈ℕ} is an open covering of M. Since the action of Γ is by homeomorphism, the image of this covering under the projection M → M/Γ gives an open covering of the compact space M/Γ. Therefore we find R > 0 so that every point in M has a translate under Γ which lies in B(x, R). Let D be the diameter of K. D < ∞ because K is compact. Since X is eventually large, there is x_{D+R+1} with B(x_{D+R+1}, D + R + 1) ⊂ X. We just observed that γ ∈ Γ exists with γ(x) ⊂ B(x_{D+R+1}, R) (γ is an isometry!). Then

γ(K) ⊂ γ(B(x, D)) ⊂ B(x_{D+R+1}, D + R + 1) ⊂ X. q.e.d.

Proof of Theorem 6.7. Let D > 0 be a constant so that

\begin{aligned}& 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})} . \end{aligned}

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. The proof of Lemma 7.5 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)})}.

By 6.9 we find an eventually large set Y ⊂ M so that

|P_{1}f(x)| ≤ δ |f|_{L^{2}} ∀x ∈ Y ∀f ∈ L^{2}.

Since the isometry group Γ acts cocompactly, we find by Lemma 6.12 γ ∈ Γ so that γ^{−1}(K) ⊂ Y. 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 ⊂ Y. Then (in terms of the metric g),

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}.

This is a contradiction, therefore no non-trivial relative L^{2}-harmonic forms exist. For the absolute case, simply replace S by S∗ from equation (6.1). q.e.d.

6.13. Corollary. Let M be a compact manifold (possibly with boundary). Let M̃ be an orientable infinite normal covering of M. In the following, we are considering only metrics of 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 and tr l ≥ 0 on any eventually large sets. In particular this is the case if χ(M) < 0 and either dim M = 2 or dim M = 4 and ∂M = ∅.

• 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 on eventually large sets.

Proof. These are direct consequences of the algebraic results in Proposition 6.3 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 [16, Proposition 6.4]). Moreover, if M is even-dimensional then because of Poincare duality χ(M) = χ(M, ∂M). The statement concerning closed 4-manifold also follows from Poincare duality which implies b_{(2)}^{1}(M̃) = b_{(2)}^{3}(M̃). q.e.d.

6.14. Remark. John Roe [14] proves a theorem which implies the same for coverings of manifolds without boundary and amenable covering group. Roe does not use eventually large, but more exhaustive sets — in particular complements of compact sets. His question whether the theorem holds for arbitrary coverings, is answered affirmatively by our result.

7 Proof of the L^{2}-de Rham theorem

The proof of Theorem 1.6 is done in three steps.

Step 1: ∂M = ∅

This is a classical result of Dodziuk [3], compare also [2].

Step 2: product metrics

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 “quadruple”:

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, which 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. Dodziuk’s theorem says that A is an isomorphism

A : H^{∗}(W) → H_{(2)}^{∗}(K̄).

Restriction to V_{4}-invariant subspaces yields that 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. It remains to identify these eigenspaces.

7.1. 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 (4.14 and 5.9) ω̄ ∈ H^{∞} and Δω̄ = 0 smoothly. That weakly Δω̄ = 0 is shown using integration by parts.

q.e.d.

7.2. 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. q.e.d.

Step 3: general case

We have the following strategy: If the metric g is not a product near the boundary, the doubling trick does not work since the metric would no longer be smooth. However, we now construct a metric ḡ which is a product near the boundary and which allows an isomorphism of the spaces of harmonic forms commuting with the de Rham map. Therefore, the positive result for product metrics implies the general result.

7.3. Proposition. 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 1 in a neighborhood of 0 and 0 outside another neighborhood 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 and their topology do not depend on the metric: H^{s}(E, g) = H^{s}(E, ḡ).

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 [8, 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})

= 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 says g(v_{t}, Z) = 0 ⟺ g′(v_{t}, Z) = 0 and moreover g(v_{t}, v_{t}) = 1 = g′(v_{t}, v_{t}). This is proved in [8, 2.93] for radial geodesics, and the prove with not more than the obvious changes applies also to our product situation. Apparently ∇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 constant 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).

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 unchanged!) 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. Now the product rule for derivatives implies 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 matrix (ḡ_{ij})_{ij}. By assumption, we find corresponding M-universal bounds for (g_{ij}(x′, t))_{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) ≥ c. This implies that we get an M-universal bound for |(ḡ^{ij})|; and (M, ḡ) has bounded geometry. For the definition of the Sobolev spaces H^{s}(E), we have seen now that we can choose identical data for g and for ḡ. q.e.d.

To compare the spaces of harmonic forms (and the de Rham map) for two different metrics we use the following functional analytical lemma:

7.4. 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} : X = U + A → U onto U along A, when restricted to V is an isomorphism V → U of topological vector spaces with inverse p_{V}.

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.

7.5. 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 7.4. 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}}.

This is true because it is exactly the kernel of d, which does not depend on the metric. The same is true for \overline{dC_{0}^{∞}(Λ^{p−1}(T^{∗}M); ·|_{M_{1}} = 0)} = \overline{dΩ_{d}^{p−1}}, the image of d. q.e.d.

7.6. Lemma. Suppose we are in the situation of Lemma 7.5 and, additionally, (M, g_{1}) and (M, g_{2}) are manifolds of bounded geometry such that all the Sobolev spaces 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 5.12). 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. q.e.d.

7.7. Lemma. Suppose x ∈ H^{∞} ∩ \overline{dΩ_{∞}^{p−1}} and x|_{M_{1}} = 0 and dx = 0. Then [Ax] = 0 ∈ H_{(2)}^{p}(M, M_{1}), where A is the de Rham map.

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 [2, 3] remains valid). 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 5.12, 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 = lim_{n} ddy_{n} = 0. Since (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

⟹ x = (1 + Δ)^{k}y = y + ∑_{i=1}^{k} \binom{k}{i} ((dδ)^{i}y + \underbrace{(δd)^{i}y}_{=0}) = y + dz

with z = δ ∑_{i=1}^{k} \binom{k}{i}(dδ)^{i−1}y. Note that z ∈ H^{∞} since y ∈ H^{∞}. H^{2k}- boundedness of the boundary differential operators implies z|_{M_{1}} = 0 as y_{n} fulfill bd(∞). This implies A(dz) = d(Az) = 0 ∈ H_{(2)}^{p}(M, M_{1})

⟹ 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}). q.e.d.

The metric ḡ constructed in Lemma 7.3 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, and therefore the Hodge-De Rham theorem is proved in general. 7.8. Remark. Dodziuk [3] uses L^{2}-de Rham cohomology to prove his theorem. This can also be introduced for ∂-manifolds of bounded geometry. There are several L^{2}-de Rham complexes whose cohomology is isomorphic to the simplicial cohomology defined above, as shown in [15, Chapter 6].

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