Quadrilaterals and extremal quasiconformal extensions
type: Item
instance of: scholarly article
title: Quadrilaterals and extremal quasiconformal extensions
zbMATH Open document ID: 0839.30017
DOI: 10.1007/BF02566018
author: J. Milne Anderson
author: Aimo Hinkkanen
published in: Commentarii Mathematici Helvetici
publication date: 1996-06-23T00:00:00Z
full work available at URL: https://eudml.org/doc/140379
review text: If \(f\), a sense-preserving homeomorphism of the boundary \(\partial D\) of the unit disc \(D\), admits a (continuous) \(K\)-quasiconformal extension to \(D\), for the module \(M(Q)\) of any quadrangle \(Q\) with domain \(D\) and the corresponding module \(M(f(Q))\) we have \(M(f(Q))/M(Q)\leq K\) (only a one-sided inequality need to be stated because of the conjugate problem). Let \(K_0= K_0(f)\) be the least upper bound of such ratio for all such \(Q\). Let \(K_1= K_1(f)= \text{g.l.b.} K\) for all quasiconformal extensions. Evidently \(K_0\leq K_1\) on the other hand consideration of the Beurling-Ahlfors necessary and sufficient conditions for the existence of quasiconformal extensions shows that when \(K_0\) is finite there will be a quasiconformal extension with maximal dilation bounded by some function of \(K_0\). It has been conjectured by Yang that always \(K_0= K_1\). This paper shows that this conjecture is false. The basic idea of the proof is simple. The parallelogram \(V\) with vertices \(0, 1, \alpha+ 1+ i\beta\), \(\alpha+i\beta\), \(\alpha> 0\), \(\beta> 0\), is mapped by the affine mapping \(F_k\) taking \(x+ iy\) to \(Kx+ iy\) \((K> 1)\) on the parallelogram \(F_k(V)\) and this mapping is uniquely extremal quasiconformal for its boundary values with maximal dilation \(K\). If \(\Phi_j\), \(j= 1, 2\), map \(V\), \(F_k(V)\) conformally on the unit disc with homeomorphic extension to the boundaries, \(\widetilde F_k= \Phi_2 F_k \Phi^{- 1}_1\) maps \(D\) quasiconformally onto itself with \(\partial D\) mapped homeomorphically and \(K_1(\widetilde F_k)=K\). It is then shown that \(K_1(\widetilde F_k)< K\). If we had \(K_0(\widetilde F_k)= K\) there would be basically two possibilities. Either \(K_0(\widetilde F_k)\) would be a maximum attained for some quadrangle \(Q\) and its image or it would not be and there would exist quadrangles \(Q_n\) with \(\lim_{n\to \infty} m(\widetilde F_n(Q_n))/m(Q_n)= K_0(\widetilde F_k)\). The authors consider the equivale…
reviewed by: James A. Jenkins
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 838581
zbMATH Keywords: quasiconformal extensions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 838581