Geometric implications of fast volume growth and capacity estimates

TL;DR

This paper explores how volume growth and capacity estimates influence the geometric structure of metric measure spaces, leading to stability results for Harnack inequalities and diffusions on manifolds with ends.

Contribution

It establishes connectivity of annuli under certain conditions and applies this to prove stability of the elliptic Harnack inequality under perturbations.

Findings

Connectivity of annuli in volume doubling spaces with Poincaré inequality.

Stability of elliptic Harnack inequality under radial perturbations.

Applicability to diffusions on manifolds with ends.

Abstract

We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincar\'e inequality, a capacity estimate and a fast volume growth condition. This type of connectivity was introduced by Grigor'yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial type weights.