On extremal quasiconformal mappings compatible with a Fuchsian group with a dilatation bound
type: Item
instance of: scholarly article
title: On extremal quasiconformal mappings compatible with a Fuchsian group with a dilatation bound
zbMATH Open document ID: 0566.30019
author: Ken-Ichi Sakan
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: Let U be the upper half-plane, \(\Gamma\) a Fuchsian group, \(\sigma\) a \(\Gamma\)-invariant subset of the extended real line \({\hat {\mathbb{R}}}\), which contains the limit set of \(\Gamma\), which, in turn, contains the points 0, 1 and \(\infty\). Let E be a \(\Gamma\) invariant subset of U such that the closure of E/\(\Gamma\) in (U\(\cup ({\hat {\mathbb{R}}}- \sigma))/\Gamma\) is a compact proper subset of \((U\cup ({\hat {\mathbb{R}}}- \sigma))/\Gamma.\) Let b(w) be a nonnegative bounded measurable function on E which is automorphic for \(\Gamma\) and which satisfies \(c_ 1= \sup_{w\in E}b(w)<1.\) Let M(\(\Gamma)\) be the set of Beltrami differentials for \(\Gamma\) with support in U and let \(z=F_{\nu}(w)\) be the unique quasiconformal automorphism of U which has Beltrami coefficient \(\nu\) and which leaves 0, 1 and \(\infty\) fixed. Within M(\(\Gamma)\) is the class \(M_{\mu}=M_{\mu}(\Gamma,\sigma,E,b)\) consisting of those \(\nu\) in M(\(\Gamma)\) for which \(F_{\nu}|_{\sigma}=F_{\mu}|_{\sigma}\) and for which \(| \nu (w)| \leq b(w).\) This article provides interesting sufficient conditions for \(\mu\) to be extremal within the class \(\mu\). The conditions are also necessary with mild assumptions about ess inf b(w). The method makes use of a variational lemma about the existence of trivial curves of Beltrami differentials which vanish on a given set.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 3902987
zbMATH Keywords: extremal quasiconformal mappings
zbMATH Keywords: Teichmüller Beltrami differentials
zbMATH Keywords: quadratic differentials
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228723
OpenAlex ID: W2080412849
cites work: Q5516279
cites work: Q5638531
cites work: A new proof of a fundamental inequality for quasiconformal mappings
cites work: Teichmüller spaces of groups of the second kind
cites work: Reduced Teichmuller Spaces
cites work: Über extremale quasikonforme Abbildungen
cites work: The Existence of Jenkins-Strebel Differentials from Teichmuller Theory
cites work: On the variation of Teichmüller's metric
cites work: On partially Teichmüller Beltrami differentials
cites work: Q5685358
cites work: On Quasiconformal Mappings which Keep the Boundary Points Fixed
cites work: Q4079891
cites work: Q4093707
cites work: Quasiconformal mappings with prescribed boundary values and a dilatation bound
cites work: On extremal quasiconformal mappings compatible with a Fuchsian group
cites work: Ein Konvergenzsatz für Folgen quasikonformer Abbildungen
cites work: On the existence of extremal Teichmueller mappings
cites work: On quasiconformal mappings of open Riemann surfaces
DOI: 10.2748/TMJ/1178228723
description: scientific article; zbMATH DE number 3902987