Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature

Abstract

We prove uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures ${\mathbb{K}}_{g_k}={\mu }_k^1-{\mu }_k^2$, where ${\mu }_k^1,{\mu }_k^2$ are nonnegative Radon measures converging weakly to measures ${\mu }^1,{\mu }^2$ respectively, and ${\mu }^1$ is less than $2\pi$ at each point (no cusps). This is the global version of Yu. G. Reshetnyak's well-known result on uniform convergence of metrics on a domain in $\mathbb{C}$, and answers affirmatively the open question on the metric convergence on a closed surface. We also give an analytic proof of the fact that a (singular) metric $g=e^{2u}{g}_0$ with bounded integral curvature on a closed Riemannian surface $(\Sigma ,g_0)$ can be approximated by smooth metrics in the fixed conformal class $[g_0]$. % in terms of distance functions, curvature measures and conformal factors. Results on a closed surface with varying conformal classes and on complete noncompact surfaces are obtained as well.

To appear in Adv. Math