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arXiv · 2007.10027

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

Inkang Kim·Xueyuan Wan
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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 $S$.

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Inkang Kim, Xueyuan Wan. 2020-07-20. Harmonic maps between surfaces homotopic to a (branched) covering map. https://arxiv.org/abs/2007.10027

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