Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces

TL;DR

This paper introduces a new Gromov-Hausdorff distance with boundary to analyze convergence of noncomplete metric spaces and demonstrates its applications in the stability of Gromov hyperbolic and uniform spaces.

Contribution

It defines a novel boundary-aware Gromov-Hausdorff distance and applies it to establish convergence and stability results for Gromov hyperbolic and uniform spaces.

Findings

Established completeness of bounded A-uniform spaces under the new metric

Proved stability of Gromov hyperbolicity under convergence

Demonstrated applications to boundary and uniformization properties

Abstract

In this paper we introduce a notion of the Gromov-Hausdorff distance with boundary, denoted by $d_{GHB}$, to construct a framework of convergence of noncomplete metric spaces. We show that a class of bounded $A$-uniform spaces with diameter bounded from below is a complete metric space with respect to $d_{GHB}$. As an application we show the stability of Gromov hyperbolicity, roughly starlike property, uniformization, quasihyperbolization, and boundary of Gromov hyperbolic spaces under appropriate notions of convergence and assumptions.