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

2012/07/21 by Theodora Bourni, Bourni, Theodora, Giuseppe Tinaglia +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1207.5129

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

openalex publication_date 2012/07/21 · arxiv created 2014/06/21 · arxiv updated 2014/06/24 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28

Abstract

In this paper we prove several results on the geometry of surfaces immersed in \mathbf R³ with small or bounded L² norm of |A|. For instance, we prove that if the L² 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 L² 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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