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

Theodora Bourni; Giuseppe Tinaglia

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

Differential Geometry 2014-06-24 v2 Analysis of PDEs

Authors: Theodora Bourni , Giuseppe Tinaglia

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.

Keywords

minimal surfaces surfaces and embeddings del pezzo surfaces

Cite

@article{arxiv.1207.5129,
  title  = {$C^{1,\alpha}$-regularity for surfaces with $H$ in $L^p$},
  author = {Theodora Bourni and Giuseppe Tinaglia},
  journal= {arXiv preprint arXiv:1207.5129},
  year   = {2014}
}

Comments

We wish to express our gratitude to the referee for valuable suggestions. In particular the proof of Corollary 3.2 is now simpler

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

Abstract

Keywords

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