Abstract
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.
Funding source: Research Grants Council, University Grants Committee
Award Identifier / Grant number: CUHK 24305115
Funding source: Chinese University of Hong Kong
Award Identifier / Grant number: 4053118
Award Identifier / Grant number: 3132705
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1406337
Funding statement: The work of Martin Man-chun Li described in this paper is substantially supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 24305115) and partially supported by direct grants from CUHK (Project Code 4053118 and 3132705). Xin Zhou is partially supported by NSF grant DMS-1406337.
Acknowledgements
The authors would like to thank Professor Richard Schoen for his continuous encouragement. They also want to thank Professor Shing Tung Yau, Professor Tobias Colding and Professor Bill Minicozzi for their interest in this work. The authors are grateful for the anonymous referee for valuable comments.
References
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