Rough isometry between Gromov hyperbolic spaces and uniformization

TL;DR

This paper investigates the relationship between Gromov hyperbolic spaces and uniform domains, showing that rough isometries preserve the uniformization property under certain conditions.

Contribution

It establishes a dichotomy for the uniformization of roughly isometric Gromov hyperbolic spaces, extending the understanding of their geometric structure.

Findings

Uniformization preserves the uniform domain property under small parameters.

Rough isometries imply either both spaces are uniform domains after uniformization or neither.

The result links hyperbolic geometry with the theory of uniform domains.

Abstract

In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and $\varepsilon>0$, either the uniformization of both spaces with parameter $\varepsilon$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from the work of Bonk, Heinonen and Koskela, where it is shown that the uniformization, with parameter $\varepsilon>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\varepsilon$ is small enough.