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C^(1,α)-regularity for surfaces with H in L^p

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arxiv 1207.5129 v2 pith:F5Z2SKUT submitted 2012-07-21 math.DG math.AP

C^(1,α)-regularity for surfaces with H in L^p

classification math.DG math.AP
keywords normsmallproveawayboundaryboundeddiskgraphical
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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.

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