Tests for extremality in a problem for quasiconformal mappings
type: Item
instance of: scholarly article
title: Tests for extremality in a problem for quasiconformal mappings
zbMATH Open document ID: 0599.30055
DOI: 10.1007/BF01647625
author: V. G. Sheretov
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: The author considers an extremal problem posed by Belinskij. Consider the class of quasiconformal (qc) mappings from a Riemann surface R to a Riemann surface S which are homotopic to a given quasiconformal mapping f from R to S and which have maximal dilatation Q. Assume the minimal dilatation of mappings homotopic modulo the ideal boundary to f is K(f) and that \(Q>K(f)\). Let \(z_ 1,...,z_ n\) be distinct points of R and let \(F(w_ 1,...,w_ n)\) be a real-valued differentiable function with domain the n-fold product of the surface S and assume F has nonvanishing gradient. One considers the problem (of Belinskij) of finding the maximal value of \(F(f)=F(f(z_ 1),f(z_ 2),...,f(z_ n))\) as f ranges over all qc mappings in the same homotopy class with dilatation Q. The author gives a necessary and sufficient condition for a function \(f_ 0\) to be extremal for this problem. The function \(f_ 0^{-1}\) must satisfy a certain supremum condition relative to quadratic differentials which are holomorphic except for certain simple poles.
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 3966628
zbMATH Keywords: theta series
zbMATH Keywords: infinitesimally trivial curves
zbMATH Keywords: Hamilton sequence
zbMATH Keywords: Beltrami differentials
zbMATH Keywords: modulus problem on Riemann surfaces
zbMATH Keywords: quadratic differentials
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
cites work: Extremal Quasiconformal Mappings with Prescribed Boundary Values
cites work: Q4079891
cites work: Holomorphic convexity of Teichmüller spaces
cites work: Q5332414
cites work: Q5638531
full work available at URL: https://doi.org/10.1007/bf01647625
OpenAlex ID: W2043280261
description: scientific article; zbMATH DE number 3966628