Analysis and Geometry of Boundary-Manifolds of Bounded Geometry

Details Analysis and Geometry of Boundary-Manifolds of Bounded Geometry Analysis and Geometry of Boundary-Manifolds of Bounded Geometry

1998-10-17

arxiv.org/abs/math/9810107v1

Mathematics

Geometric Topology

AMS-Latex2e, 41 pages

Scientific paper

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.

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