On the space of schlicht projective structures on compact Riemann surfaces with boundary
type: Item
instance of: scholarly article
title: On the space of schlicht projective structures on compact Riemann surfaces with boundary
zbMATH Open document ID: 0862.30044
author: Toshiyuki Sugawa
published in: Journal of Mathematics of Kyoto University
publication date: 1997-05-25T00:00:00Z
review text: Let \(\Gamma\) be a Fuchsian group; to \(\Gamma\) we associate the totality of univalent holomorphic functions \(f\) for which \(f\circ\gamma=\chi(\gamma)\circ f\) for \(\gamma\in\Gamma\) and where \(\chi(\gamma)\) is a Möbius transformation. We denote by \(S(\Gamma)\) the Schwarzian derivatives of all of these functions; \(S(\Gamma)\) can be considered as a subset of the space of quadratic differentials of bounded norm. Likewise one can represent the Teichmüller space \(T(\Gamma)\) of \(\Gamma\) as a subset of the same space, and it is, by the Bers' construction, a subset of \(S(\Gamma)\). The purpose of this paper is to show that \(T(\Gamma)=\text{Int}(S(\Gamma))\) if \(\Gamma\) is purely hyperbolic of the second kind. The analogous result for groups of the first kind is due to H. Shiga. It is worth noting in this connection that, in general, \(S(\Gamma)\) is larger than the closure of \(T(\Gamma)\), even if \(\Gamma\) is restricted to the class of purely hyperbolic groups of the second kind. For a finitely generated Fuchsian group \(\Gamma\) it was proved by Zhuravlev that \(T(\Gamma)\) is the connected component of \(0\) in \(\text{Int}(S(\Gamma))\). The proof given here follows a method developed by F. Gehring but it requires a very delicate analysis for its execution.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 871817
zbMATH Keywords: Teichmüller theory
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: quadratic differentials
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518656
OpenAlex ID: W1567814855
DOI: 10.1215/KJM/1250518656
description: scientific article; zbMATH DE number 871817