$C^{1,\alpha }$-regularity for surfaces with $H$ in $L^p$

In this paper we prove several results on the geometry of surfaces immersed in ${\mathbf{R}}^3$ with small or bounded $L^2$ norm of $\mathrm{\mid }A\mathrm{\mid }$. For instance, we prove that if the $L^2$ norm of $\mathrm{\mid }A\mathrm{\mid }$ and the $L^p$ norm of $H$, $p>2$, are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded $L^2$ norm of $\mathrm{\mid }A\mathrm{\mid }$, not necessarily small, then such a disk is graphical away from its boundary, provided that the $L^p$ norm of $H$ is sufficiently small, $p>2$. These results are related to previous work of Schoen-Simon and Colding-Minicozzi.