Unique extremality
type: Item
instance of: scholarly article
title: Unique extremality
zbMATH Open document ID: 0929.30017
DOI: 10.1007/BF02788704
published in: Journal d'Analyse Mathématique
publication date: 2000-02-02T00:00:00Z
review text: Let \(R\) be a Riemann surface and \(T(R)\) be the Teichmüller space of \(R\), viewed as the Teichmüller equivalence classes of Beltrami coefficients. The following theorems are proved in this paper: A Beltrami coefficient \(\mu\) is uniquely extremal in its Teichmüller class if, and only if, it is uniquely extremal in its infinitesimal Teichmüller class. Let \(\varphi\) be a nonzero holomorphic quadratic differential on \(R\) and let \(f\) be a quasiconformal mapping of \(R\) with Beltrami coefficient \(\mu=k|\varphi|/\varphi\), \(0<k<1\). Then \(f\) is uniquely extremal if and only if there exists a sequence \(\varphi_n\) of holomorphic quadratic differentials of finite norm such that (1) \(\varphi_n(z)\) converges to \(\varphi(z)\) uniformly on compact subsets of \(R\), and (2) \(k\|\varphi_n\|-\text{Re}\int_R\varphi_n\mu\to 0\). As an application of these theorems, an example of a uniquely extremal Beltrami coefficient with non-constant modulus is given. This solves a well-known problem which was open for a long period.
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 1221961
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Beltrami coefficients
zbMATH Keywords: quadratic differential
author: Vladimir Božin
author: Nikola Lakic
author: Vladimir Markovic
author: Miodrag Mateljević
reviewed by: Zhong Li
MaRDI profile type: Publication
Wikidata QID: Q56565757
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full work available at URL: https://doi.org/10.1007/bf02788704
OpenAlex ID: W3190906677
description: scientific article; zbMATH DE number 1221961