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Uniformity from Gromov hyperbolicity. (English) Zbl 1189.30055
The authors show that, in a metric space $X$ 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 $X$. As an application it is shown that quasi-Möbius maps between geodesic spaces with annular convexity preserve uniform domains. Theorem 1: Let $(X_i,d_i)$, $i= 1, 2$, be proper metric spaces, and let $\Omega_i\subset X_i$ be open subsets with $\partial\Omega_1\ne\emptyset$. Let $h:\Omega_1\to \Omega_2$ be an $\eta$-quasi-Möbius homeomorphism. If $\Omega_1$ is $c_1$-uniform and $(X_2,d_2)$ is $c_2$-quasiconvex and $c_2$-annular convex, then $\Omega_2$ is $c$-uniform with $c= c(\eta,c_1, c_2)$.

30C65Quasiconformal mappings in ${\Bbb R}^n$ and other generalizations
53C23Global geometric and topological methods; differential geometric analysis on metric spaces
Full Text: Euclid
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