Convexity of energy function associated to the harmonic maps between surfaces

Abstract

For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichm\"uller space $\mc{T}$ of $\Sigma$ that assigns to a complex structure $t\in \mc{T}$ on $\Sigma$ the energy of the harmonic map $u_t:\Sigma_t:=(\Sigma,t) \to S$ homotopic to $u_0$. We prove that the energy function is convex at its critical points. If $t_0\in\mc{T}$ is a critical point such that $du_{t_0}$ is never zero, then the energy function is strictly convex at this point. As an application, in the case that $u_0$ is a covering map, we prove that there exists a unique critical point $t_0\in \mc{T}$ minimizing the energy function. Moreover, the energy density satisfies $\frac{1}{2}|du|^2(t_0)\equiv 1$ and the Hessian of the energy function is positive definite at this point.