Teichm\"uller's problem for Gromov hyperbolic domains

TL;DR

This paper investigates how far a point in a Gromov hyperbolic domain can be moved by a K-quasiconformal automorphism fixing the boundary, using two metrics to estimate the displacement.

Contribution

It extends Teichmüller's problem to Gromov hyperbolic domains and provides bounds on point displacement using the distance ratio and quasihyperbolic metrics.

Findings

Derived bounds for point displacement in Gromov hyperbolic domains.

Extended results to $ ext{ extmu}$-uniform and inner uniform domains.

Analyzed Teichmüller's problem with identity boundary conditions at infinity.

Abstract

Let $\mathcal{T}_K(D)$ be the class of $K$-quasiconformal automorphisms of a domain $D\subsetneq \mathbb{R}^n$ with identity boundary values. Teichm\"uller'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 Teichm\"{u}ller's problem for Gromov hyperbolic domains in $\mathbb{R}^n$ with identity values at the boundary of infinity. As applications, we obtain results on Teichm\"{u}ller's problem for $\psi$-uniform domains and inner uniform domains in $\mathbb{R}^n$.