  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.
