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arXiv · 1203.0427

Distortion of quasiconformal mappings with identity boundary values

Matti Vuorinen·Xiaohui Zhang
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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.

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Matti Vuorinen, Xiaohui Zhang. 2012-03-02. Distortion of quasiconformal mappings with identity boundary values. https://arxiv.org/abs/1203.0427

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