A new method of solving variational problems in the theory of quasiconformal mappings
type: Item
instance of: scholarly article
title: A new method of solving variational problems in the theory of quasiconformal mappings
zbMATH Open document ID: 0664.30018
DOI: 10.1007/BF00969736
published in: Siberian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: An analytic function \(w=F(\zeta_ 1,...,\zeta_ n)\) of n complex variables is given. It is assumed that n distinct points \(z_ 1,z_ 2,...,z_ n\) in \({\mathbb{C}}-\{0,1\}\) are selected and that \(F(z_ 1,...,z_ n)=0\). Let Q be the space of quasiconformal homeomorphisms of the sphere \({\bar {\mathbb{C}}}\) which fix the points 0, 1, and \(\infty\) and Q(k) the subspace of Q consisting of mappings which are \((1+k)/(1-k)\)- quasiconformal. For f in Q, we denote by F(f) the value \(F(f(z_ 1),...,f(z_ n))\). In general, the author considers the problem of finding the supremum of \(| F(f)|\) taken over f in Q(k). Let \[ c_ j=F_{z_ j}(z_ 1,...,z_ n)\quad and\quad \phi_ 0(z)=\frac{1}{z(z-1)}\sum^{n}_{1}\frac{c_ jz_ j(z_ j-1)}{z-z_ j}. \] Theorem. Assume \(\phi_ 0(z)\) has only even order zeroes. Then there exists a constant \(k(F)>0\) depending on F such that for all \(k\leq k(F)\), one has the following inequality: \[ \max_{f\quad in\quad Q(k)}| F(f)| \leq \max_{| t| =k}| F(f^{t\mu})| \equiv d(k), \] where \(\mu =| \phi_ 0(z)| \phi_ 0(z)\) and \(f^{t\mu}\) is a normalized quasiconformal mapping with Beltrami coefficient \(t\mu\). The method of proof is to use the equality of the Kobayashi-Teichmüller metric with the Carathéodory metric on Teichmüller disks of the special form described in the theorem. The conclusion of the theorem holds true on any Teichmüller disk where this equality is in force.
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 4085994
zbMATH Keywords: Beltrami coefficient
zbMATH Keywords: Kobayashi-Teichmüller metric
zbMATH Keywords: Carathéodory metric
zbMATH Keywords: Teichmüller disks
author: Samuel L. Krushkal
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
cites work: On the order of closeness of a spatial quasiconformal mapping to conformal
cites work: On the solution of extremal problems for a class of quasiconforaml mappings
cites work: Tests for extremality in a problem for quasiconformal mappings
cites work: EXTREMAL PROBLEMS ON CLASSES OF QUASI-CONFORMAL HOMEOMORPHISMS
cites work: The Caratheodory metric on abelian Teichmüller disks
cites work: Q5622542
cites work: Approximation of Infinite Dimensional Teichmuller Spaces
cites work: On sections of some holomorphic families of closed Riemann surfaces
cites work: Extremal Quasiconformal Mappings with Prescribed Boundary Values
cites work: On holomorphic cross-sections in Teichmüller spaces
cites work: Q5512337
full work available at URL: https://doi.org/10.1007/bf00969736
OpenAlex ID: W2313528137
description: scientific article; zbMATH DE number 4085994