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 $\epsilon >0$, either the uniformization of both spaces with parameter $\epsilon$ 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 $\epsilon >0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\epsilon$ is small enough.

17 pages

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