Behavior of geodesic-length functions on Teichmueller space

Let $\mathcal T$ be the Teichm\"{u}ller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichm\"{u}ller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}. Provided with the WP metric $\Tbar$ is a complete CAT(0) metric space, \cite{DW2, Wlcomp, Yam2}. An invariant of a marked hyperbolic structure is the length $\ell_{\alpha}$ of the geodesic $\alpha$ in a free homotopy class. A basic feature of Teichm\"{u}ller 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 expansions 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. Bounding configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a discrete group of isometries and bounding sums of the inverse square exponential-distance.