A quasiconformal dynamic property of the disk
type: Item
instance of: scholarly article
title: A quasiconformal dynamic property of the disk
zbMATH Open document ID: 0911.30015
DOI: 10.1007/BF02843154
author: Samuel L. Krushkal
author: Reiner Kühnau
published in: Journal d'Analyse Mathématique
publication date: 1999-05-04T00:00:00Z
review text: Let \(D\) be a hyperbolic simply-connected domain in \(\widehat{\mathbb{C}}\), \(z_0\in D\) and \(0<r<1\). The \(r\)-circle of \(D\) with centre \(z_0\) is \(\gamma(z_0,r)= f^{-1} \{z:| z|= \frac 1r\}\) where \(f\) is a conformal map of \(D\) onto the disc \(\Delta= \{z:| z|>1\), \(z\in \widehat{\mathbb{C}}\}\) such that \(f(z_0)= \infty\). \(D\) is said to have the \(r^2\)-property if for every \(r\), \(0<r<1\) and every \(z_0\in D\) and every conformal map \(g\) of \(D\) into \(\widehat{\mathbb{C}}\) there is an \(r^2\)-quasiconformal map \(h:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}\) whose restriction to the interior of \(\gamma(z_0,r)\) is \(g\). \textit{S. L. Krushkal} [Sib. Math. J. 30, 730-744 (1989; Zbl 0701.30019)] proved that any disc has the \(r^2\)-property. The authors give two proofs of the converse result that among the hyperbolic simply-connected domains only discs have the \(r^2\)-property.
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 1146574
zbMATH Keywords: quasiconformal extensions
reviewed by: I. N. Baker
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02843154
OpenAlex ID: W2005174137
description: scientific article; zbMATH DE number 1146574