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

TL;DR

This paper establishes regularity results for surfaces in R^3 with bounded mean curvature in L^p spaces, showing that smallness conditions on curvature norms imply the surface is graphical away from the boundary.

Contribution

It proves new regularity theorems linking small L^2 norms of second fundamental form and L^p norms of mean curvature to surface graphicality.

Findings

Surfaces with small L^2 norm of |A| and L^p norm of H are graphical away from boundary.

Embedded disks with bounded L^2 norm of |A| are graphical if L^p norm of H is sufficiently small.

Results extend previous work by Schoen-Simon and Colding-Minicozzi.

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.