Localized deformation of Riemann surfaces
type: Item
instance of: scholarly article
title: Localized deformation of Riemann surfaces
zbMATH Open document ID: 1071.30042
DOI: 10.5565/PUBLMAT_49105_12
author: Peter Haïssinsky
published in: Publicacions Matemàtiques
publication date: 2005-07-05T00:00:00Z
full work available at URL: https://eudml.org/doc/41563
review text: Let \(Y\) be a compact bordered Riemann surface inside an analytically finite Riemann surface \(X\). Each complex deformation of \(Y\) induces naturally a complex deformation of \(X\). In this way, one has a natural holomorphic map \(\phi\) from the deformation space of \(Y\), say \(D(Y)\), to the Teichmüller space of \(X\). In this article, the author proves that, if \(X\) is analytically finite hyperbolic Riemann surface, then: (i) the image \(\phi(D(Y))\) is an open set in the Teichmüller space of \(X\); (ii) \(\phi(D(Y))\) is relatively compact if and only if \(Y\) can be homotoped to either a point or to a punctured disc in \(X\); (iii) \(\phi\) is surjective if and only if \(X-Y\) is disjoint union of discs and/or punctured discs. In particular, the author observes that the deformations of an analytically finite hyperbolic Riemann surface induced by the Schiffer variations remain inside a compact part of Teichmüller space of the surface.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 2186478
zbMATH Keywords: Teichmüller theory
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: Riemann surface
zbMATH Keywords: simple closed curves
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
OpenAlex ID: W2087067154
description: scientific article; zbMATH DE number 2186478