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

Abstract

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 $|A|$. For instance, we prove that if the $L^2$ norm of $|A|$ 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 $|A|$, 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.