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

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 ${\mathbb{S}}^2$.

References in corpus (1)

• Quasisymmetric parametrizations of two-dimensional metric spheres