Convexity of energy function associated to the harmonic maps between surfaces

TL;DR

This paper proves convexity properties of the energy function associated with harmonic maps between surfaces, establishing uniqueness and strict convexity at certain critical points, with applications to covering maps.

Contribution

It demonstrates convexity and strict convexity of the energy function on Teichmüller space and proves uniqueness of energy-minimizing harmonic maps for covering maps.

Findings

Energy function is convex at critical points.

Strict convexity at points where the differential is non-zero.

Unique energy-minimizing harmonic map for covering maps.

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.