The authors show that, in a metric space with annular convexity, the uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the boundary agrees with that on the boundary of .
As an application it is shown that quasi-Möbius maps between geodesic spaces with annular convexity preserve uniform domains.
Theorem 1: Let , , be proper metric spaces, and let be open subsets with . Let be an -quasi-Möbius homeomorphism. If is -uniform and is -quasiconvex and -annular convex, then is -uniform with .