Teichmüller spaces and tame quasiconformal motions
type: Item
instance of: scholarly article
title: Teichmüller spaces and tame quasiconformal motions
zbMATH Open document ID: 1417.32018
author: Yunping Jiang
author: Sudeb Mitra
author: Hiroshige Shiga
author: Zhe Wang
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2019-03-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmj/1546570827
review text: Let $E$ be a subset of the Riemann sphere $\widehat{\mathbb{C}}$ and $X$ a connected Hausdorff space with base point $x_0$; a \textit{motion of $E$ over $X$} is a map $\phi: X \times E \to \widehat{\mathbb{C}}$ such that $\phi$ is the identity map evaluated in $x_0$, and injective for all $x \in X$ (so $X$ is the parameter space of the motion $\phi$). In the paper [Acta Math. 157, 243--286 (1986; Zbl 0619.30026)], \textit{D. P. Sullivan} and \textit{W. P. Thurston} introduced the concept of \textit{quasiconformal motion} and claimed that every quasiconformal motion of a subset of $\widehat{\mathbb{C}}$ over the interval $I = [0,1]$ extends to a quasiconformal motion of $\widehat{\mathbb{C}}$; the first main result of the present paper is a counterexample to that claim. Then the authors introduce the concept of a \textit{tame quasiconformal motion} and show that the claim remains true for tame quasiconformal motions; more generally, they show that every tame quasiconformal motion of a subset of $\widehat{\mathbb{C}}$ over a simpy connected Hausdorff space extends to a quasiconformal motions of $\mathbb{C}$ (in a ``conformally natural way''). ``The fundamental idea is to show that the Teichmüller space of a closed set in the sphere is a univeral parameter space for tame quasiconformal motions of that set over a simply connected Hausdorff space.''
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 37F30
zbMATH DE Number: 7040978
zbMATH Keywords: quasiconformal motions
zbMATH Keywords: tame quasiconformal motions
zbMATH Keywords: holomorphic motions
zbMATH Keywords: continuous motions
zbMATH Keywords: Teichmüller spaces
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
OpenAlex ID: W2907685418
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DOI: 10.2748/TMJ/1546570827
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description: scientific article; zbMATH DE number 7040978