Derivatives of length functions and shearing coordinates on teichm{ü}ller spaces

Abstract

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $\gamma$, we compute the first three derivatives of the length function $\ell \_\gamma :\mathcal{T}(S)\to \mathbf{R}\_+$ 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{ö}lder distributions and shearing coordinates.

References in corpus (1)

• Convexity of length functions and Thurston's shear coordinates