  New results on the convexity of geodesic-length functions on Teichm\"{u}ller
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.

Differential Geometry