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

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 $\mathrm{\ell }\mathrm{\_}\gamma :\mathcal{T}(S)\to \mathbf{R}\mathrm{\_}+$ 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 $\mathrm{\ell }\mathrm{\_}\gamma$ 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.