Sphericalization with its applications in Gromov hyperbolic spaces

TL;DR

This paper explores the use of sphericalization in Gromov hyperbolic spaces, revealing new insights into boundary properties, domain characterization, and boundary equivalences in hyperbolic metric spaces.

Contribution

It provides novel results on boundary doubling properties, domain characterization via sphericalization, and boundary topological equivalences in Gromov hyperbolic spaces.

Findings

Doubling properties of boundary metrics coincide

Unbounded Gromov hyperbolic domains characterized by sphericalization

Topological equivalence of boundary domains in hyperbolic spaces

Abstract

In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that the doubling property regarding two classes of metrics on the Gromov boundary of hyperbolic spaces are coincided. Next, we obtain a characterization of unbounded Gromov hyperbolic domains via metric spaces sphericalization. Finally, we investigate the topological equivalence of Gromov hyperbolic $\varphi$-uniform domains between the Gromov boundary and the inner metric boundary.