Rough isometry between Gromov hyperbolic spaces and uniformization

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.