The Teichmüller distance on the space of spherical CR structures
type: Item
instance of: scholarly article
title: The Teichmüller distance on the space of spherical CR structures
zbMATH Open document ID: 1109.32031
author: Wei Wang
published in: Science in China. Series A
publication date: 2007-02-16T00:00:00Z
review text: It is well known that quasiconformal mappings between Riemann surfaces can be used to define a distance, the Teichmüller distance, on the Teichmüller space of Riemann surfaces with a fixed genus. Such a construction was generalized by H.~Izeki to define a distance on the space of conformally flat structures. The complex counterpart of a conformally flat manifold is the spherical CR manifold. Denote by \(\text{Diff\,}_0^r(M)\) the group of \(C^n\) diffeomorphisms of the manifold \(M\) isotropic to the identity mapping. Define the space \[ \mathcal T(M) = \{\text{spherical CR structures on } M\}/\text{Diff\,}_0^\infty(M). \] Denote by \([J]\) the element of \(\mathcal T(M)\) represented by a spherical CR structure \(J\). The topology of the space \(\mathcal T(M)\) is simply given by that of the space \(\mathcal T^\infty(M)\) constructed from developing maps. This is the counterpart of the Teichmüller space in the spherical CR case. There is a natural definition of quasiconformal mappings between spherical CR manifolds. It is shown that the distortion of a quasiconformal mapping \(f\) between two spherical CR manifolds \((M, J)\) and \((N, J')\), denoted by \(K(f;(M, J),(N,J'))\), is well defined. The Teichmüller pseudodistance \(d\) on the space \(\mathcal T(M)\) defined by \[ d([J], [J'])=\inf_f\log K (f, (M, J), (M, J)'),\tag{1} \] for \([J], [J']\in\mathcal T(M)\), is well defined, i.e. the right-hand side of (1) is independent of the choices of spherical CR structures \(J\in [J]\) and \(J'\in [J']\), where the infimum is taken over all orientation preserving quasiconformal mappings \(f\) from \((M, J)\) to \((M, J')\), which are homotopic to the identity as homeomorphisms. Let \(M\) be a \(C^\infty\) orientable, connected, compact manifold without boundary. Then the Teichmüller pseudodistance on the space \(\mathcal T(M)\) is shown to be a complete distance by establishing a compactness result. If a spherical CR structure \(J'\) is a smooth deformation of the othe…
Mathematics Subject Classification ID: 32V99
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 5127958
zbMATH Keywords: Teichmüller pseudodistance
zbMATH Keywords: spherical CR structures
zbMATH Keywords: quasiconformal mappings between spherical CR manifolds
zbMATH Keywords: developing mappings
zbMATH Keywords: CR diffeomorphisms
zbMATH Keywords: Heisenberg group
zbMATH Keywords: John domains
zbMATH Keywords: compactness.
MaRDI profile type: Publication
full work available at URL: s11425-006-2052-y
OpenAlex ID: W1628330835
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DOI: 10.1007/S11425-006-2052-Y
description: scientific article; zbMATH DE number 5127958