On Teichmüller's proof of Teichmüller's theorem
type: Item
instance of: scholarly article
title: On Teichmüller's proof of Teichmüller's theorem
zbMATH Open document ID: 0606.30024
DOI: 10.1007/BF02796573
author: Lipman Bers
published in: Journal d'Analyse Mathématique
publication date: 1986-01-01T00:00:00Z
review text: The author gives further explanations of his proof of Teichmüller's theorem, first presented in his expository paper in Princeton, Math. Ser. 24, 84-119 (1960; Zbl 0100.289). A key element in the proof was the assertion that parameters determining a marked Fuchsian group depend continuously on the parameters determining the Teichmüller Beltrami coefficient. From Bers' exposition it appeared to some readers that the theory of quasiconformal mappings with bounded measurable Beltrami coefficients was indispensable to the proof of Teichmüller's theorem. Not being able to comprehend Teichmüller's arguments, some of them erroneously concluded that Teichmüller had given an incomplete proof. In this paper, Bers shows that the continuity can be proved in an elementary way which does not involve the complete theory of quasiconformal mappings and, in fact, is a way very close to the one employed by Teichmüller. The route taken is to use Grötzsch's inequality bounding the distortion of the modulus of a rectangle or an annulus in terms of the average dilatation of a quasiconformal mapping. Then, in order to get the continuous dependence of the parameters determining a marked Fuchsian group, one only needs to show the continuous dependence of the values of a quasiconformal mapping restricted to the real axis. The values on the real axis can be seen as depending on the moduli of rectangles with vertices on the real axis. Then one shows the convergence of the moduli of rectangles by showing that the local dilatations are bounded functions and showing that they converge uniformly on compact subsets.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 3979466
zbMATH Keywords: Grötzsch's inequality
zbMATH Keywords: Teichmüller Beltrami coefficient
zbMATH Keywords: Teichmüller's theorem
zbMATH Keywords: Fuchsian group
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
cites work: Q5512337
cites work: On a Theorem of Mori and the Definition of Quasiconformality
cites work: Q3282113
cites work: Fricke spaces
cites work: On the Solutions of Quasi-Linear Elliptic Partial Differential Equations
full work available at URL: https://doi.org/10.1007/bf02796573
OpenAlex ID: W4246990914
description: scientific article; zbMATH DE number 3979466