On the subinvariance of uniform domains in metric spaces

Abstract

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G^{\prime }\subset Y$ are domains, and that $f:G\to G^{\prime }$ is a homeomorphism. Our main result is the following subinvariance property of the class of uniform domains: Suppose both $f$ and $f^{-1}$ are weakly quasisymmetric mappings and $G^{\prime }$ is a quasiconvex domain. Then the image $f(D)$ of every uniform subdomain $D$ in $G$ under $f$ is uniform. The subinvariance of uniform domains with respect to freely quasiconformal mappings or quasihyperbolic mappings is also studied with the additional condition that both $G$ and $G^{\prime }$ are locally John domains.