The Weil-Petersson Hessian of Length on Teichmuller Space

TL;DR

This paper derives a formula for the Weil-Petersson Hessian of geodesic length on hyperbolic surfaces, proving its convexity and providing estimates that extend to laminations and punctured surfaces, with applications to metric comparisons.

Contribution

It provides a nearly self-contained proof of the Weil-Petersson Hessian formula, extending it to arcs and laminations, and offers new estimates and applications in Teichmüller theory.

Findings

The Hessian formula is the sum of integrals of positive functions over geodesics.

The formula proves convexity of the length functional on Teichmüller space.

Applications include bounds near pinching loci and convexity of length to the half-power.

Abstract

We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert's result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, and recover through a geometric argument Wolpert's recent result on the convexity of length to the half-power.