Book contents
- Frontmatter
- Contents
- Preface
- Drilling short geodesics in hyperbolic 3-manifolds
- On topologically tame Kleinian groups with bounded geometry
- An extension of the Masur domain
- Thurston's bending measure conjecture for once punctured torus groups
- Complexity of 3-manifolds
- Moduli of continuity of Cannon–Thurston maps
- Variations of McShane's identity for punctured surface groups
- Train tracks and the Gromov boundary of the complex of curves
- The pants complex has only one end
- The Weil–Petersson geometry of the five-times punctured sphere
- Convexity of geodesic-length functions: a reprise
- A proof of the Ahlfors finiteness theorem
- On the automorphic functions for Fuchsian groups of genus two
- Boundaries for two-parabolic Schottky groups
- Searching for the cusp
- Circle packings on surfaces with projective structures: a survey
- Grafting and components of quasi-fuchsian projective structures
- Computer experiments on the discreteness locus in projective structures
Convexity of geodesic-length functions: a reprise
Published online by Cambridge University Press: 05 November 2011
- Frontmatter
- Contents
- Preface
- Drilling short geodesics in hyperbolic 3-manifolds
- On topologically tame Kleinian groups with bounded geometry
- An extension of the Masur domain
- Thurston's bending measure conjecture for once punctured torus groups
- Complexity of 3-manifolds
- Moduli of continuity of Cannon–Thurston maps
- Variations of McShane's identity for punctured surface groups
- Train tracks and the Gromov boundary of the complex of curves
- The pants complex has only one end
- The Weil–Petersson geometry of the five-times punctured sphere
- Convexity of geodesic-length functions: a reprise
- A proof of the Ahlfors finiteness theorem
- On the automorphic functions for Fuchsian groups of genus two
- Boundaries for two-parabolic Schottky groups
- Searching for the cusp
- Circle packings on surfaces with projective structures: a survey
- Grafting and components of quasi-fuchsian projective structures
- Computer experiments on the discreteness locus in projective structures
Summary
Abstract
New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applications to the behavior of Weil-Petersson geodesics are discussed.
Introduction
In this research brief we describe a new approach to the work [Wol87] (esp. Secs. 3 and 4), as well as new results and applications of the convexity of geodesic-length functions on the Teichmüller space T. Our overall goal is to obtain an improved understanding of the convexity behavior of geodesic-length functions along Weil-Petersson (WP) geodesics. Applications are presented in detail for the CAT(0) geometry of the augmented Teichmüller space. A complete treatment of results is in preparation [Wol04]. Convexity of geodesic-length functions has found application for the convexity of Teichmüller space [Bro02, Bro03, DS03, Ker83, Ker92, McM00, SS01, SS99, Wol87, Yeu03], for the convexity of the WP metric completion [DW03, MW02, Wol03, Yam01], for the study of harmonic maps into Teichmüller space [DKW00, Yam99, Yam01], and for the action of the mapping class group [DW03, MW02]. We consider marked Riemann surfaces R with complete hyperbolic metrics possibly with cusps and consider the lengths of closed geodesics. The length of the unique geodesic in a prescribed free homotopy class provides a function on the Teichmüller space.
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- Spaces of Kleinian Groups , pp. 233 - 246Publisher: Cambridge University PressPrint publication year: 2006
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