Kobayashi's and Teichmüller's metrics and Bers complex manifold structure on circle diffeomorphisms
type: Item
instance of: scholarly article
title: Kobayashi's and Teichmüller's metrics and Bers complex manifold structure on circle diffeomorphisms
zbMATH Open document ID: 1443.37032
author: Yunping Jiang
publication date: 2020-04-15T00:00:00Z
review text: For \(0<\alpha \leq 1\), denote by \(\mathcal{C}^{1+\alpha}\) the set of orientation-preserving diffeomorphisms of the unit circle \(S^1\) whose derivative is \(\alpha\)-Hölder continuous. Then the corresponding Teichmüller space \(\mathcal{T} \mathcal{C}^{1+\alpha}\) can be identified with the subset of \(\mathcal{C}^{1+\alpha}\) which fixes \(1,-1,i\). Moreover, denote by \(\mathcal{T} \mathcal{C}^{1+H}\) the union \(\bigcup_{0<\alpha \leq 1} \mathcal{T} \mathcal{C}^{1+\alpha}\). After an extensive introduction to the required material, the first main result of the paper is Theorem 4.2, which states that \(\mathcal{T} \mathcal{C}^{1+H}\) can be equipped with a Bers complex manifold structure. This can be viewed as a smooth version of the analogous result for universal Teichmüller spaces. Moreover, this is the largest subspace in the space of all \(C^1\) circle diffeomorphisms on which this structure can be given. The author then proves in Theorem 5.1 that the Kobayashi and Teichmüller metrics coincide, again making a parallel with well-known results in quasiconformal Teichmüller theory. The proofs of the results use well-known techniques in Teichmüller theory, such as Schwarzian derivatives, quasiconformal extensions of quasisymmetric maps, and computations involving Beltrami differentials.
reviewed by: Alastair Fletcher
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 37F34
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 7189515
zbMATH Keywords: Bers complex manifold structure
zbMATH Keywords: circle diffeomorphism
zbMATH Keywords: modulus of continuity
zbMATH Keywords: quasisymmetric circle homeomorphism
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Kobayashi's metric
zbMATH Keywords: Teichmüller's metric
MaRDI profile type: Publication
full work available at URL: s10114-020-7502-x
OpenAlex ID: W3009999704
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DOI: 10.1007/S10114-020-7502-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 7189515