Necessary and sufficient conditions for extremality in certain classes of quasiconformal mappings
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for extremality in certain classes of quasiconformal mappings
zbMATH Open document ID: 0604.30023
published in: Journal of Mathematics of Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The author gives necessary and sufficient conditions for quasiconformal mappings to be extremal in certain classes. To define the class one starts with a Fuchsian group \(\Gamma\) acting on the upper half plane U. \(L_{\infty}(\Gamma)\) is the Banach space of Beltrami differentials \(\nu\) satisfying \(\nu\) (\(\gamma\) (w))\({\bar \gamma}\)'(w)/\(\gamma\) '(w)\(=\nu (w)\) for every \(\gamma\) in \(\Gamma\). For \(\nu\) in \(L_{\infty}(\Gamma)\) with \(\| \nu \|_{\infty}<1\), \(F_{\nu}(w)\) is the uniquely determined quasiconformal self-mapping of \(\bar U\) with complex dilatation \(\nu\) (w) and normalized at 0,1,\(\infty\). \(\sigma\) is a T-invariant closed subset of the real line \({\hat {\mathbb{R}}}\). E is a T-invariat measurable subset of U such that \(D=U\setminus E\) has positive measure. b(w) is a nonnegative bounded measurable function on E, automorphic for \(\Gamma\), satisfying \(\| b\|_{\infty}=es\sup b(w)<1\). \(h: {\hat {\mathbb{R}}}\to {\hat {\mathbb{R}}}\) is the boundary mapping included by some \(F_{\nu}.\) The class \(Q=Q(\Gamma,h,\sigma,E,b)\) consists of those \(F_{\nu}\) with \(\| \nu \|_{\infty}<1\), \(\nu\) in \(L_{\infty}(\Gamma)\), which satisfy \(F_{\nu}|_{\sigma}=h| \sigma\) and \(| \nu (w)| \leq b(w)\) a.e. in E. This paper gives necessary and sufficient conditions for a mapping to be extremal in Q provided that the space of integrable holomorphic quadraic differentials, real off of \(\sigma\), is finite dimensional or provided that E/\(\Gamma\) is relatively compact in \(\{\) \(U\cup ({\hat {\mathbb{R}}}-\sigma)\}/\Gamma\).
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 3975308
zbMATH Keywords: extremal quasiconformal mappings
zbMATH Keywords: curves of trivial Beltrami coefficients
zbMATH Keywords: dilatation bound
zbMATH Keywords: Fuchsian group
zbMATH Keywords: quadraic differentials
author: Ken-Ichi Sakan
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520962
OpenAlex ID: W1820980076
DOI: 10.1215/KJM/1250520962
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description: scientific article; zbMATH DE number 3975308