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:, Martin Man-chun LiORCID iD: and Xin Zhou


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.

  • [1]

    S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.

    • Crossref
    • Export Citation
  • [2]

    W. K. Allard, On the first variation of a varifold: Boundary behavior, Ann. of Math. (2) 101 (1975), 418–446.

    • Crossref
    • Export Citation
  • [3]

    S. Brendle, A sharp bound for the area of minimal surfaces in the unit ball, Geom. Funct. Anal. 22 (2012), no. 3, 621–626.

    • Crossref
    • Export Citation
  • [4]

    A. Butscher, Deformations of minimal Lagrangian submanifolds with boundary, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1953–1964.

  • [5]

    T. H. Colding and W. P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected, Ann. of Math. (2) 160 (2004), no. 2, 573–615.

    • Crossref
    • Export Citation
  • [6]

    T. H. Colding and W. P. Minicozzi, II, A course in minimal surfaces, Grad. Stud. Math. 121, American Mathematical Society, Providence 2011.

  • [7]

    M. do Carmo and C. K. Peng, Stable complete minimal surfaces in 𝐑 3 {\mathbf{R}}^{3} are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903–906.

    • Crossref
    • Export Citation
  • [8]

    D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211.

    • Crossref
    • Export Citation
  • [9]

    D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics Math., Springer, Berlin 2001.

  • [10]

    M. Grüter and J. Jost, Allard type regularity results for varifolds with free boundaries, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129–169.

  • [11]

    A. Kovalev and J. D. Lotay, Deformations of compact coassociative 4-folds with boundary, J. Geom. Phys. 59 (2009), no. 1, 63–73.

    • Crossref
    • Export Citation
  • [12]

    H. Li and X. Zhou, Existence of minimal surfaces of arbitrarily large Morse index, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Paper No. 64.

  • [13]

    M. Li and X. Zhou, Min-max theory for free boundary minimal hypersurfaces I: Regularity theory, preprint (2016),

  • [14]

    J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes 27, Princeton University Press, Princeton 1981.

  • [15]

    A. Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92.

    • Crossref
    • Export Citation
  • [16]

    R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud. 103, Princeton University Press, Princeton (1983), 111–126.

  • [17]

    R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797.

    • Crossref
    • Export Citation
  • [18]

    R. Schoen, L. Simon and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3–4, 275–288.

    • Crossref
    • Export Citation
  • [19]

    L. Simon, Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral. Natl. Univ. 3, Australian National University, Centre for Mathematical Analysis, Canberra 1983.

  • [20]

    L. Simon, A strict maximum principle for area minimizing hypersurfaces, J. Differential Geom. 26 (1987), no. 2, 327–335.

    • Crossref
    • Export Citation
  • [21]

    A. Volkmann, A monotonicity formula for free boundary surfaces with respect to the unit ball, Comm. Anal. Geom. 24 (2016), no. 1, 195–221.

    • Crossref
    • Export Citation
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