Metrics on a Surface with Bounded Total Curvature

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