Behavior of geodesic-length functions on Teichmüller space

Abstract

Let T be the Teichmuller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichmuller space of marked Riemann surfaces with nodes, (Abi77, Ber74). Pro- vided with the WP metric T is a complete CAT(0) metric space, (DW03, Wol03, Yam04). An invariant of a marked hyperbolic struc- ture is the lengthof the geodesic in a free homotopy class. A basic feature of Teichmuller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop the understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include ex- pansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bound- ing configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a dis- crete group of isometries and bounding sums of the inverse square exponential-distance.