DERIVATIVES OF LENGTH FUNCTIONS AND SHEARING COORDINATES ON TEICHMÜLLER SPACES

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\teich(S)$ its Teichm\"uller space. For any isotopy class of closed curves $\g$, we compute the first three derivatives of the length function $\ell_\g:\teich(S)\rightarrow\R_+$ in the shearing coordinates associated to a maximal geodesic lamination $\l$. We show that if $\g$ intersects any leaf of $\l$, then the Hessian of $\ell_\g$ 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\"older distributions and shearing coordinates.