Analysis and Geometry of Boundary-Manifolds of Bounded Geometry

TL;DR

This paper explores the geometric and analytical properties of boundary-manifolds with bounded geometry, establishing vanishing theorems for L^2-cohomology and proving the Hodge-de Rham theorem in this context.

Contribution

It introduces new vanishing results for L^2-cohomology of boundary-manifolds with bounded geometry and proves the Hodge-de Rham theorem for these manifolds.

Findings

Vanishing of first relative L^2-cohomology under certain curvature conditions

Proof of Hodge-de Rham theorem for boundary-manifolds of bounded geometry

Development of boundary value problem techniques for elliptic operators

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-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.