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 an arbitrary \(\epsilon>0\) (not necessarily small), 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 [BHK], 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.