On the subinvariance of uniform domains in metric spaces

TL;DR

This paper proves that under certain conditions, the image of a uniform domain remains uniform when mapped by weakly quasisymmetric homeomorphisms between quasiconvex, complete metric spaces, extending known invariance properties.

Contribution

It establishes the subinvariance of uniform domains under weakly quasisymmetric mappings in quasiconvex metric spaces, including cases with locally John domains.

Findings

Uniform domains are preserved under weakly quasisymmetric homeomorphisms.

The result extends to mappings like quasiconformal and quasihyperbolic when domains are locally John.

The paper provides conditions ensuring the subinvariance property in metric space settings.

Abstract

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ 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 $G'$ 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 $G'$ are locally John domains.