Geometric properties of $\phi$-uniform domains

Abstract

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

9 pages, 1 figure