Geometric and analytic structures on metric spaces homeomorphic to a manifold

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. We use this to establish (relative) isoperimetric inequalities in metric n-manifolds that are Ahlfors n-regular and linearly locally contractible. As an application, we obtain a short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincaré inequalities in these spaces. In the smooth case the idea for such a proof goes back to Gromov. We also give sufficient conditions for a metric manifold to be rectifiable.

Boundary regularity and stability for spaces with Ricci bounded below

“…Moreover, we can take any s ∈ G p ∩ (0, r) close enough to r, replace B r (p) with B s (p) and repeat the argument above. The estimate (9.6) is obtained arguing as in the proof of [12,Proposition 7.1].…”

“…Our first aim is to exploit the local uniform contractibility of the good Green-spheres to construct a well-behaved retraction from a uniform neighbourhood of a good Green-ball onto the good Green-ball. The idea is classical: see [12,Propostion 7.1] for a recent application in the metric setting. Lemma 9.10.…”