Geometric properties of $\varphi$-uniform domains

TL;DR

This paper studies the geometric properties of $$-uniform domains in Euclidean spaces, demonstrating their invariance under quasiconformal maps and providing conditions for uniformity and counterexamples.

Contribution

It establishes the preservation of $$-uniform domains under quasiconformal mappings and introduces new geometric criteria for uniformity.

Findings

$$-uniform domains are preserved under quasiconformal maps

A geometric condition ensures $$-uniformity implies uniformity

Constructs a planar $$-uniform domain with a non-$$-uniform complement

Abstract

We consider proper subdomains $G$ of $\mathbb{R}^n$ and their images $G'=f(G)$ under quasiconformal mappings $f$ of $\mathbb{R}^n$. We compare the distance ratio metrics of $G$ and $G'$; as an application we show that $\varphi$-uniform domains are preserved under quasiconformal mappings of $\mathbb{R}^n$. A sufficient condition for $\varphi$-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If $G\subset\mathbb{R}^n$ is $\phi$-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar $\phi$-uniform domain whose complement is not $\psi$-uniform for any $\psi$.