Non-convexity of spheres in infinite-dimensional Teichmüller spaces
type: Item
instance of: scholarly article
title: Non-convexity of spheres in infinite-dimensional Teichmüller spaces
zbMATH Open document ID: 0810.30037
author: Zhong Li
published in: Science in China. Series A
publication date: 1994-11-03T00:00:00Z
review text: The Teichmüller space, \(T(G)\), of a Fuchsian group \(G\), is infinite- dimensional if either \(G\) is of the second kind or it is infinitely generated (or both). The Teichmüller metric is always the Kobayashi metric of \(T(G)\). Let \(S_ r\) denote the sphere of points at a fixed distance \(r\) from a given point in \(T(G)\). It is shown that there exist two points on \(S_ r\), and a Teichmüller geodesic joining them, such that the entire geodesic arc lies on that same \(S_ r\). The method is to construct a special holomorphic quadratic differential for \(G\), and thereby certain extremal Beltrami differentials. The same quadratic differential has been used by the author in another paper (``Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces'', Proc. Am. Math. Soc., to appear) to prove stronger results than the above non-convexity.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 682135
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Kobayashi metric
zbMATH Keywords: quadratic differential
zbMATH Keywords: extremal Beltrami differentials
reviewed by: Subhashis Nag
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 682135