Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space
type: Item
instance of: scholarly article
title: Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space
zbMATH Open document ID: 0873.32019
author: Luen-Fai Tam
author: Tom Yau-Heng Wan
published in: Journal of Differential Geometry
publication date: 1997-09-28T00:00:00Z
review text: The universal Teichmüller space, \(T\), is an infinite dimensional complex-analytic manifold modelled on a nonseparable Banach space. \(T\) can be thought of as the space of (three-point-normalized) quasi-symmetric homeomorphisms of the unit circle \(S^1\). A homeomorphism, \(h: S^1 \rightarrow S^1\), is quasi-symmetric if and only if it allows a quasiconformal (q.c.) extension \(H: D \rightarrow D\), where \(D\) is the unit disc -- the boundary of \(D\) being \(S^1\). For this basic background, see, for instance, \textit{L. Ahlfors}, ``Lectures on quasiconformal mappings'', Van Nostrand (1966; Zbl 0138.06002), or, \textit{S. Nag} ``The Complex analytic theory of Teichmüller spaces'', Wiley-Interscience (1988; Zbl 0667.30040). Consider the set of those \(h\) in \(T\) which allow a quasiconformal and diffeomorphic extension, \(H\), such that this extension \(H\) is furthermore required to be a harmonic self-mapping of the unit disc (the disc being equipped with the Poincaré hyperbolic metric). Denote this subset of the universal Teichmüller space by \(T'\). For every \(h \in T'\), it is known that this harmonic and quasiconformal extension, \(H = H_{h}\), is a uniquely determined diffeomorphism of \(D\) on itself. Consequently, one may associate to every \(h\) in \(T'\) the standard holomorphic ``Hopf differential'' on \(D\) arising from this harmonic mapping \(H_{h}\). That Hopf differential is a holomorphic function \(\phi\) on \(D\) which is a ``bounded quadratic differential'' (with respect to the Bers-Nehari norm): namely, \(sup_{D}|(1-|z|^{2})^{2}{\phi}(z) |\) is finite. Denote this Banach space of bounded holomorphic quadratic differentials on the disc by \(B(D)\). The authors explain the inverse relationship: they show how to find the quasi-symmetric homeomorphism \(h\) whose associated Hopf differential is any given function \(\phi \in B(D)\). Thus one has a bijective correspondence, say \({\mathcal {B}}\), between the Banach space \(B(D)\) and the subset \(…
reviewed by: Subhashis Nag
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 894386
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Hopf quadratic differential.
zbMATH Keywords: harmonic diffeomorphisms
Wikidata QID: Q115177872
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214457235
OpenAlex ID: W2058951856
DOI: 10.4310/JDG/1214457235
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description: scientific article; zbMATH DE number 894386