Thomas Schick: Geometry and Analysis of Boundary-Manifolds of Bounded Geometry
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-cohomology vanishes (this
in particular answers a question of Roe).
We prove the Hodge-de Rham-theorem for L^2-cohomology of oriented
boundary-manifolds of bounded geometry.
The technical basis is the study of (uniformly elliptic) boundary
value problems on these manifolds, applied to the Laplacian.
Thomas
Schick