Metrics On A Surface With Bounded Total Curvature

Abstract

Let $g=e^{2u}(d{x}^2+d{y}^2)$ be a conformal metric defined on the unit disk of $\mathbf{C}$. We give an estimate of $\Vert \nabla u{\Vert }_{L^{2,\infty }(D_{\frac{1}{2}})}$ when $\Vert K(g){\Vert }_{L^1}$ 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 $\Vert K{\Vert }_{L^1}$ and give some applications.