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

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.

arXiv comment42pages, comments welcome

References in corpus (3)

• Convergence of Manifolds and Metric Spaces with Boundary
• Sequences of Open Riemannian Manifolds with Boundary
• Uniformizing Gromov hyperbolic spaces with Busemann functions