Curvature estimates for stable free boundary minimal hypersurfaces

  • 1 Department of Mathematics, University of California Santa Barbara, CA 93106, Santa Barbara, USA
  • 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P. R. China
  • 3 Department of Mathematics, University of California Santa Barbara, CA 93106, Santa Barbara, USA
Qiang GuangORCID iD: https://orcid.org/0000-0003-4772-659X, Martin Man-chun LiORCID iD: https://orcid.org/0000-0002-1877-9409 and Xin Zhou

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.

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