Uniformity from Gromov hyperbolicity

Abstract

We show that in a metric space $X$ with annular convexity, uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the Gromov boundary agrees with that on the boundary in $X$. As an application, we show that quasimöbius maps between geodesic spaces with annular convexity preserve uniform domains. These results are quantitative.