Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

TL;DR

This paper explores the connection between hyperbolicity and Cheeger isoperimetric inequalities in Riemannian manifolds and graphs, characterizing when these inequalities hold and applying results to boundary problems.

Contribution

It provides a characterization of hyperbolic manifolds and graphs satisfying isoperimetric inequalities via their Gromov boundary, and analyzes trees without additional assumptions.

Findings

Characterization of hyperbolic manifolds and graphs with isoperimetric inequality

Solution of the Dirichlet problem at infinity for these spaces

Homeomorphism between Martin boundary and Gromov boundary

Abstract

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.