Distortion of quasiconformal mappings with identity boundary values

TL;DR

This paper investigates the distortion bounds of quasiconformal mappings with fixed boundary values, establishing lower bounds for their maximal dilatation based on the distance ratio metric within various domain types.

Contribution

It extends Teichmüller's classical problem to higher dimensions and diverse domain classes, providing explicit lower bounds for quasiconformal map dilatation.

Findings

Lower bounds for maximal dilatation depend on the distance ratio metric.

Results apply to convex, bounded, and uniformly perfect domains.

The bounds are explicitly characterized in terms of domain geometry.

Abstract

Teichm\"uller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain $D \subset {\mathbb R}^n\,,n\ge 2\,,$ we consider the class of all $K$- quasiconformal maps of $D$ onto itself with identity boundary values and Teichm\"uller's problem in this context. Given a map $f$ of this class and a point $x\in D\,,$ we show that the maximal dilatation of $f$ has a lower bound in terms of the distance of $x$ and $f(x)$ in the distance ratio metric. For instance, convex domains, bounded domains and domains with uniformly perfect boundaries are studied.