Geometric and analytic structures on metric spaces homeomorphic to a manifold

TL;DR

This paper explores the geometric and analytic properties of metric spaces homeomorphic to manifolds, establishing the existence of integral currents, rigidity results, isoperimetric inequalities, and rectifiability conditions.

Contribution

It introduces the existence of a non-trivial integral current in metric manifolds, extending classical concepts and proving new rigidity and inequality results.

Findings

Existence of integral currents analogous to fundamental classes.

Riemannian manifolds are Lipschitz-volume rigid among certain metric spaces.

Validation of isoperimetric inequalities and conditions for rectifiability.

Abstract

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. Using our existence result, we establish that Riemannian manifolds are Lipschitz-volume rigid among certain metric manifolds and we show the validity of (relative) isoperimetric inequalities in metric $n$-manifolds that are Ahlfors $n$-regular and linearly locally contractible. The former statement is a generalization of a well-known Lipschitz-volume rigidity result in Riemannian geometry and the latter yields a relatively short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincar\'e inequalities in these spaces. Finally, as a further application, we also give sufficient conditions for a metric manifold to be rectifiable.