Harmonic maps between surfaces homotopic to a (branched) covering map

TL;DR

This paper investigates harmonic maps between surfaces homotopic to covering or branched covering maps, establishing conditions for uniqueness of critical points and injectivity of Hopf differential, with implications for Teichmüller theory.

Contribution

It proves the uniqueness of harmonic map critical points for covering maps and explores the failure of this uniqueness for non-simple branched coverings, also analyzing Hopf differential injectivity.

Findings

Uniqueness of critical points for harmonic maps homotopic to covering maps.

Failure of uniqueness in non-simple branched coverings.

Injectivity of Hopf differential under certain conditions.

Abstract

In the paper, we consider the harmonic maps between surfaces $\Sigma$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of critical points fails if $u_0$ is a non-simple branched covering, and prove the injectivity of Hopf differential $\Phi:\mc{T}(S)\to \op{QD}(\Sigma,g)$ when $g=[u_0^* h]$ for some hyperbolic metric $h$ on $S$.