Teichmüller's problem for Gromov hyperbolic domains

Abstract

Let ${\mathcal{T}}_K(D)$ be the class of $K$-quasiconformal automorphisms of a domain $D⊊{\mathbb{R}}^n$ with identity boundary values. Teichmüller's problem is to determine how far a given point $x\in D$ can be mapped under a mapping $f\in {\mathcal{T}}_K(D)$. We estimate this distance between $x$ and $f(x)$ from the above by using two different metrics, the distance ratio metric and the quasihyperbolic metric. We study Teichmüller's problem for Gromov hyperbolic domains in ${\mathbb{R}}^n$ with identity values at the boundary of infinity. As applications, we obtain results on Teichmüller's problem for $\psi$-uniform domains and inner uniform domains in ${\mathbb{R}}^n$.

arXiv comment22 pages