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Curvature estimates for stable free boundary minimal hypersurfaces

Curvature estimates for stable free boundary minimal hypersurfaces AbstractIn 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

Curvature estimates for stable free boundary minimal hypersurfaces

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
DOI
10.1515/crelle-2018-0008
Publisher site
See Article on Publisher Site

Abstract

AbstractIn 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.

Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: Feb 1, 2020

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