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Published online by Cambridge University Press: 05 November 2011
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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