Uniform Convergence of Metrics on Alexandrov Surfaces with Bounded Integral Curvature

TL;DR

This paper proves that metrics with bounded integral curvature on closed surfaces converge uniformly under certain conditions, extending known results and providing new approximation methods within the Alexandrov geometry framework.

Contribution

It establishes the global uniform convergence of metrics with bounded integral curvature on closed surfaces and offers an analytic approximation approach by smooth metrics within the same conformal class.

Findings

Uniform convergence of metrics with bounded integral curvature on closed surfaces.

Approximation of singular metrics by smooth metrics in the same conformal class.

Extension of Reshetnyak's local convergence result to global surface setting.

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 $\mathbb{K}_{g_k}=\mu^1_k-\mu^2_k$, where $\mu^1_k,\mu^2_k$ 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.