Harmonic branched coverings and uniformization of CAT(k) spheres

2020/11/25 by Christine Breiner, Breiner, Christine, Chikako Mese +1
Mathematics · #30F10 #58E20 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2011.12802

openalex publication_date 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 S², then it is conformally equivalent to \mathbb S².

Citations

• Quasisymmetric parametrizations of two-dimensional metric spheres
• On the solutions of quasi-linear elliptic partial differential equations

Related

• Uniformization of embedded surfaces
• Metric curvature of infinite branched covers
• Harmonic maps between surfaces homotopic to a (branched) covering map
• Uniformization of S² and flat singular surfaces
• Polyhedral approximation and uniformization for non-length surfaces