Metrics On A Surface With Bounded Total Curvature

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 $|\nabla u|{L^{2,\infty}(D\frac{1}{2})}$when$|K(g)|{L^1}$issmalland$\frac{\mu(B_r^g(z),g)}{\pi r^2}<\Lambda$forany$r$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 $\mathrm{\parallel }K{\mathrm{\parallel }}_{L^1}$ and give some applications.