Geodesic length functions and the Nielsen problem
type: Item
instance of: scholarly article
title: Geodesic length functions and the Nielsen problem
zbMATH Open document ID: 0616.53039
author: Scott A. Wolpert
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: In view of the author's result establishing the noncompleteness of the Weil-Petersson metric for the Teichmüller space \(T_{g,n}\), the geodesical convexity of \(T_{g,n}\) was an open problem. The present paper is devoted to this question. Roughly speaking, the author shows that the geodesic structure in the Weil-Petersson geometry is quite similar to the one of a complete Riemannian metric of negative curvature. The basic idea in the author's approach is to substitute for completeness the use of the geodesic length functions introduced by Fricke-Klein. The paper contains a comprehensive exposition of Teichmüller theory and the Weil-Petersson geometry by means of harmonic Beltrami differentials. Regarding the geometric properties of \(T_{g,n}\) the author proves that every pair of points is joined by a unique geodesic, the exponential map is a homeomorphism and the Weil-Petersson distance is measured along geodesics. As applications he gives new proofs that \(T_{g,n}\) is Stein and of the Nielsen realization problem.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 3998696
zbMATH Keywords: geodesical convexity
zbMATH Keywords: Weil-Petersson geometry
zbMATH Keywords: geodesic length functions
zbMATH Keywords: Teichmüller theory
zbMATH Keywords: harmonic Beltrami differentials
zbMATH Keywords: exponential map
zbMATH Keywords: Nielsen realization problem
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
Wikidata QID: Q115184815
full work available at URL: https://doi.org/10.4310/jdg/1214440853
OpenAlex ID: W1503851492
DOI: 10.4310/JDG/1214440853
description: scientific article; zbMATH DE number 3998696