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

Abstract

Consider a metric space (S, d) with an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that if (S, d) is homeomorphically equivalent to the 2-sphere S 2 , then it is conformally equivalent to S 2 . The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by an almost conformal harmonic map. The proof relies on the analysis of the local behavior of harmonic maps between surfaces, and the key step is to show that an almost conformal harmonic map from a compact surface onto a surface with an upper curvature bound is a branched covering.MSC 58E20, 30F10