Compactness analysis for free boundary minimal hypersurfaces

  • Lucas Ambrozio
  • Alessandro CarlottoEmail author
  • Ben Sharp


We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.

Mathematics Subject Classification

Primary 53A10 Secondary 53C42 49Q05 



The authors wish to express their gratitude to André Neves for his interest in this work and for his constant support, and to the anonymous referee for carefully reading the manuscript and providing detailed feedback. A. C. also would like to thank Connor Mooney for several discussions and Francesco Lin for pointing out some relevant references. L. A. was visiting the University of Chicago while this article was written, and he would like to thank the Department of Mathematics for its hospitality. He is supported by the EPSRC on a Programme Grant entitled ‘Singularities of Geometric Partial Differential Equations’ reference number EP/K00865X/1.


  1. 1.
    Aiex, N.S.: Non-compactness of the space of minimal hypersurfaces. Preprint at arXiv:1601.01049
  2. 2.
    Ambrozio, L.: Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds. J. Geom. Anal. 25(2), 1001–1017 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrozio, L., Carlotto, A., Sharp, B.: Compactness of the space of minimal hypersurfaces with bounded volume and \(p\)-th Jacobi eigenvalue. J. Geom. Anal. 26(4), 2591–2601 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrozio, L., Carlotto, A., Sharp, B.: Index estimates for free boundary minimal hypersurfaces. Math. Ann. (to appear) Google Scholar
  5. 5.
    Ambrozio, L., Nunes, I.: A gap theorem for free boundary minimal surfaces in the three-ball. Preprint at arXiv: 1608.05689
  6. 6.
    Azzam, A., Kreyszig, E.: On solutions of elliptic equations satisfying mixed boundary conditions. SIAM J. Math. Anal. 13(2), 254–262 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cairns, S.: Triangulation of the manifold of class one. Bull. Am. Math. Soc. 41(8), 549–552 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Choi, H.I., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81, 387–394 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Courant, R.: The existence of minimal surfaces of given topological structure under prescribed boundary conditions. Acta Math. 72, 51–98 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Courant, R.: Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer. Interscience Publishers Inc., New York (1950)zbMATHGoogle Scholar
  11. 11.
    De Lellis, C., Ramic, J.: Min-max theory for minimal hypersurfaces with boundary. Preprint at arXiv:1611.00926
  12. 12.
    Devyver, B.: Index of the critical catenoid. Preprint at arXiv: 1609.02315
  13. 13.
    Folha, A., Pacard, F., Zolotareva, T.: Free boundary minimal surfaces in the unit 3-ball. Preprint at arXiv:1502.06812
  14. 14.
    Fraser, A., Li, M.: Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary. J. Differ. Geom. 96(2), 183–200 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fraser, A., Schoen, R.: The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math. 226(5), 4011–4030 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fraser, A., Schoen, R.: Minimal surfaces and eigenvalue problems. In: Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations. Contemporary Mathematics, vol. 599, pp. 105–121. American Mathematical Society, Providence, RI (2013)Google Scholar
  17. 17.
    Fraser, A., Schoen, R.: Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203(3), 823–890 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Freidin, B., Gulian, M., McGrath, P.: Free boundary minimal surfaces in the unit ball with low cohomogeneity. Proc. Am. Math. Soc. 145(4), 1671–1683 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)zbMATHGoogle Scholar
  20. 20.
    Grüter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(1), 129–169 (1986)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Guang, Q., Li, M., Zhou, X.: Curvature estimates for stable free boundary minimal hypersurfaces. Preprint at arXiv:1611.02605
  22. 22.
    Hsiang, W.Y.: Minimal cones and the spherical Bernstein problem. I. Ann. Math. 118(1), 61–73 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ketover, D.: Free boundary minimal surfaces of unbounded genus. Preprint at arXiv:1612.08691
  24. 24.
    Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)zbMATHGoogle Scholar
  25. 25.
    Li, M.: A general existence theorem for embedded minimal surfaces with free boundary. Commun. Pure Appl. Math. 68(2), 286–331 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li, M., Zhou, X.: Min–max theory for free boundary minimal hypersurfaces I—regularity theory. Preprint at arXiv:1611.02612
  27. 27.
    Máximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex three-manifolds. J. Differ. Geom. 106(1), 139–186 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    McGrath, P.: A characterization of the critical catenoid. Preprint at arXiv:1603.04114v2
  29. 29.
    Miranda, C.: Sul problema misto per le equazioni lineari ellittiche. Ann. Mat. Pura Appl. 39, 279–303 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34, 741–797 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schoen, R., Simon, L., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sharp, B.: Compactness of minimal hypersurfaces with bounded index. J. Differ. Geom. 106(2), 317–339 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, vol. 3. ANU (1983)Google Scholar
  34. 34.
    Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Smith, G., Zhou, D.: The Morse index of the critical catenoid. Preprint at arXiv:1609.01485
  37. 37.
    Taylor, M.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar
  38. 38.
    Tran, H.: Index characterization for free boundary minimal surfaces. Preprint at arXiv: 1609.01651
  39. 39.
    White, B.: The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Indiana Univ. Math. J. 36(3), 567–602 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    White, B.: Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals. Invent. Math. 88(2), 243–256 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    White, B.: The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40(1), 161–200 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    White, B.: A local regularity theorem for mean curvature flow. Ann. Math. 161, 1487–1519 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    White, B.: Which ambient spaces admit isoperimetric inequalities for submanifolds? J. Differ. Geom. 83, 213–228 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    White, B.: On the bumpy metrics theorem for minimal submanifolds. Am. J. Math. (to appear) Google Scholar
  45. 45.
    Whitehead, J.H.C.: On \(C^1\)-complexes. Ann. Math. (2) 41, 809–824 (1940)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Lucas Ambrozio
    • 1
  • Alessandro Carlotto
    • 2
    Email author
  • Ben Sharp
    • 1
  1. 1.University of WarwickCoventryUK
  2. 2.Department of MathematicsETH ZürichZürichSwitzerland

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