Derivatives of length functions and shearing coordinates on teichm{\"u}ller spaces

TL;DR

This paper computes derivatives of length functions in Teichmüller space using shearing coordinates, showing positive definiteness of the Hessian under certain conditions and extending results to measured laminations.

Contribution

It introduces a method to compute higher derivatives of length functions in Teichmüller space using shearing coordinates and Bonahon's theory.

Findings

Hessian of length functions is positive-definite when intersecting all lamination leaves.

Provides formulas for first three derivatives of length functions.

Extends results to measured laminations.

Abstract

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{\"u}ller space. For any isotopy class of closed curves $\gamma$, we compute the first three derivatives of the length function $\ell\_\gamma:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maximal geodesic lamination $\lambda$. We show that if $\gamma$ intersects every leaf of $\lambda$, then the Hessian of $\ell\_\gamma$ is positive-definite. We extend this result to length functions of measured laminations. We also provide a method to compute higher derivatives of length functions of geodesics. We use Bonahon's theory of transverse H{\"o}lder distributions and shearing coordinates.