Harmonic branched coverings and uniformization of CAT($k$) spheres

TL;DR

This paper proves that almost conformal harmonic maps from surfaces to Alexandrov-curved surfaces are branched coverings, leading to a uniformization result for CAT(k) spheres that are topologically spheres.

Contribution

It establishes a new link between harmonic maps and branched coverings in the context of Alexandrov spaces, extending uniformization to CAT(k) spheres.

Findings

Harmonic maps are branched coverings under certain conditions

CAT(k) spheres topologically equivalent to the 2-sphere are conformally equivalent to it

Extension of uniformization results to spaces with Alexandrov curvature bounds

Abstract

Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.