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Curvature estimates for stable free boundary minimal hypersurfaces

Qiang Guang University of California Santa Barbara Martin Man-chun Li Chinese University of Hong Kong Xin Zhou University of California Santa Barbara

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:2001.10003

Journal für die reine und angewandte Mathematik, 2018.5
In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.
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@inproceedings{qiang2018curvature,
  title={Curvature estimates for stable free boundary minimal hypersurfaces},
  author={Qiang Guang, Martin Man-chun Li, and Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103163719746584617},
  booktitle={Journal für die reine und angewandte Mathematik},
  year={2018},
}
Qiang Guang, Martin Man-chun Li, and Xin Zhou. Curvature estimates for stable free boundary minimal hypersurfaces. 2018. In Journal für die reine und angewandte Mathematik. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103163719746584617.
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