Uniformization of S and Flat Singular Surfaces

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization theorem for the sphere. 1. Uniformization in genus 0 and holomorphic structures on conformal surfaces Let (Σ, g) be a closed, possibly unorientable Riemannian manifold of dimension 2. For background material on surfaces we refer the reader to [1]. The Gaussian curvature κg : Σ → R of the metric g is the sectional curvature function of the tangent planes to Σ. If f ∈ C∞(Σ,R) is any smooth function, an elementary computation shows that the Gaussian curvatures of the conformal metrics g and g′ := e−2fg are related by (1) κg′ = e 2f (κg −∆gf) where ∆g = d ∗d is the Laplacian on functions with respect to the metric g. By the GaussBonnet theorem, ∫