Metrics On A Surface With Bounded Total Curvature

Yuxiang Li, Jianxin Sun, Hongyan Tang

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 $\Vert K(g){\Vert }_{L^1}$…