Harmonic properties of the module of a family of curves and invariant metrics in Teichmüller space
type: Item
instance of: scholarly article
title: Harmonic properties of the module of a family of curves and invariant metrics in Teichmüller space
zbMATH Open document ID: 0883.30038
published in: Doklady Mathematics
publication date: 1998-03-19T00:00:00Z
review text: Let \(T(g,n)\) be the Teichmüller space of equivalence classes of finite bordered Riemann surfaces of genus \(g\) and connectivity \(n\) with \(3g+3>n\). On this are defined various metrics, in particular the biholomorphically invariant Carathéodory metric \(c_T\) and the Kobayashi metric \(k_T\). The author obtains a sufficient condition for local coincidence of these metrics utilizing the reviewer's fundamental theorem on the modules of curve families. This is used to define a functional \(m(x)\) on \(T(g,n)\) and the result is as follows (where \(m_0\)) is the module for the model surface). Let \(\varphi\) be the lifting to \(U=\{|z|<1\}\) of the extremal quadratic differential for \(m_0\) and let \(m(x)\) be a pluriharmonic function on \(T(g,n)\). Then the coincidence of the metrics \(c_T\) and \(k_T\) holds in the geodesic Teichmüller disc \(\Phi(t \overline{\varphi}) |\varphi|)\), \(t\in U\). The author's reference to the reviewer's paper [7] is incorrect. The journal is Ann. Math. The data is 1957 not 1975.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 1050640
zbMATH Keywords: Carathéodory metric
zbMATH Keywords: Kobayashi metric
zbMATH Keywords: quadratic differential
author: Alexander Vasil'ev
reviewed by: James A. Jenkins
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1050640