A fundamental variational lemma for extremal quasiconformal mappings compatible with a Fuchsian group
type: Item
instance of: scholarly article
title: A fundamental variational lemma for extremal quasiconformal mappings compatible with a Fuchsian group
zbMATH Open document ID: 0614.30022
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1987-01-01T00:00:00Z
review text: In this paper the author carries over the fundamental variational proved by \textit{R. Fehlmann} in Comment. Math. Helv. 61, 565-580 (1986) to the setting of Fuchsian groups \(\Gamma\). The lemma enables one to replace an infinitesimally trivial Beltrami differential by an equivalent Beltrami differential with much smaller norm. This replacement is accomplished with the side condition that the Beltrami coefficients be identically equal to zero on a specified closed \(\Gamma\)-invariant subset E of the unit disk. The set E is assumed to be measurable and to have a boundary of measure zero. It is also assumed to satisfy a relative compactness condition on the quotient surface equal to the unit disk factored by the Fuchsian group. The purpose of this lemma is to enable one to extend the results of \textit{R. Fehlmann} [Ann. Acad. Sci. Fenn., Ser. A I 7, 337-347 (1982; Zbl 0487.30008)], \textit{F. P. Gardiner} [Mich. Math. J. 29, 237-242 (1982; Zbl 0476.30015)] and \textit{K. Sakan} [J. Math. Kyoto Univ. 26, 31- 37 (1986; Zbl 0604.30023)] to Riemann surfaces and to infinite dimensional Teichmüller spaces.
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 3993936
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: Beltrami differential
author: Ken-Ichi Sakan
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3993936