Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

2016/05/14 by Martínez-Pérez, Álvaro, Rodríguez, José M.
#53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 53C21 #Secondary 58C40

paper · doi:10.48550/arxiv.1605.04394

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.

Related

• A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs
• Isoperimetric inequalities in Riemann surfaces and graphs
• The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces
• The Levy-Gromov Isoperimetric Inequality in Convex Manifolds with Boundary
• The Boundary of a Graph and its Isoperimetric Inequality