Metrics On A Surface With Bounded Total Curvature

TL;DR

This paper provides estimates for the gradient of conformal metrics with small total curvature and bounded area ratios, and applies these to analyze Gromov-Hausdorff convergence of such metric sequences.

Contribution

It introduces new gradient estimates for conformal metrics with bounded total curvature and explores their implications for metric convergence.

Findings

Gradient estimates under small total curvature

Convergence results for metric sequences with bounded curvature

Applications to geometric analysis of conformal surfaces

Abstract

Let $g=e^{2u}(dx^2+dy^2)$ be a conformal metric defined on the unit disk of $\mathbf{C}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty}(D_\frac{1}{2})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{\mu(B_r^g(z),g)}{\pi r^2}<\Lambda$ for any $r$ and $z\in D_\frac{3}{4}$. Then we will use this estimate to study the Gromov-Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.