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Inventiones mathematicae

, Volume 218, Issue 2, pp 441–490 | Cite as

Min–max theory for constant mean curvature hypersurfaces

  • Xin ZhouEmail author
  • Jonathan J. Zhu
Article

Abstract

In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature c. Moreover, if c is nonzero then our min–max solution always has multiplicity one.

Notes

Acknowledgements

Both authors are grateful to Prof. Shing-Tung Yau for suggesting this problem and for his generous support. X. Zhou would also like to thank Prof. Richard Schoen and Prof. Neshan Wickramasekera for valuable comments. J. Zhu would also like to thank Prof. William Minicozzi for his invaluable guidance and encouragement. X. Zhou is partially supported by NSF grant DMS-1704393 and DMS-1811293. J. Zhu is partially supported by NSF grant DMS-1607871. Finally, both authors would like to thank the anonymous referees for their comments.

References

  1. 1.
    Agol, I., Marques, F.C., Neves, A.: Min–max theory and the energy of links. J. Am. Math. Soc. 29(2), 561–578 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almgren Jr., F.J.: The Theory of Varifolds, Mimeographed Notes. Princeton University, Princeton (1965)Google Scholar
  3. 3.
    Almgren Jr., F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4(165), viii+199 (1976)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Arnold’s problems. Springer-Verlag, Berlin; PHASIS, Moscow. Translated and revised edition of the 2000 Russian original. Philippov, A. Yakivchik and M. Peters, With a preface by V (2004)Google Scholar
  5. 5.
    Almgren, J., Justin, F.: The homotopy groups of the integral cycle groups. Topology 1, 257–299 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barbosa, J.L., do Carmoand, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197(1), 123–138 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bellettini, C., Wickramasekera, N.: Stable CMC integral varifolds of codimension 1: regularity and compactness. arXiv preprint arXiv:1802.00377 (2018)
  8. 8.
    Bérard, P., Meyer, D.: Inégalités isopérimétriques et applications. Ann. Sci. École Norm. Sup. (4) 15(3), 513–541 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chambers, G.R., Liokumovich, Y.: Existence of minimal hypersurfaces in complete manifolds of finite volume. arXiv:1609.04058 (2016)
  10. 10.
    Chruściel, P.T., Galloway, G.J., Pollack, D.: Mathematical general relativity: a sampler. Bull. Am. Math. Soc. (N.S.) 47(4), 567–638 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Colding, T.H., De Lellis, C.: The min–max construction of minimal surfaces. Surv Differ Geom 8:75–107 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Colding, T.H., Minicozzi, W.P.: II. A Course in Minimal Surfaces Volume 121 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)Google Scholar
  13. 13.
    De Lellis, C., Ramic, J.: Min–max theory for minimal hypersurfaces with boundary. Ann. Inst. Fourier (Grenoble) 68(5), 1909–1986 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    De Lellis, C., Tasnady, D.: The existence of embedded minimal hypersurfaces. J. Differ. Geom. 95(3), 355–388 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duzaar, F., Steffen, K.: Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds. Indiana Univ. Math. J. 45(4), 1045–1093 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ginzburg, V.L.: On closed trajectories of a charge in a magnetic field. An application of symplectic geometry. In: Contact and symplectic geometry (Cambridge 1994). Publications of the Newton Institute, vol. 8, pp. 131–148. Cambridge University Press, Cambridge, UK (1996)Google Scholar
  17. 17.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Guaraco, M.A.M.: Min–max for phase transitions and the existence of embedded minimal hypersurfaces. J. Differ. Geom. 108(1), 91–133 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Harvey, R., Lawson, B.: Extending minimal varieties. Invent. Math. 28, 209–226 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Heinz, E.: über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung. Math. Ann. 127, 258–287 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Commun. Pure Appl. Math. 23, 97–114 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hoffman, D., Meeks, W.H.: III. The strong halfspace theorem for minimal surfaces. Invent. Math. 101(2), 373–377 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. (2) 131(2), 239–330 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ketover, D.: Equivariant min–max theory. arXiv:1612.08692 (2016)
  26. 26.
    Ketover, D., Zhou, X.: Entropy of closed surfaces and min–max theory. J. Differ. Geom. 110(1), 31–71 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, Volume 4 of Basler Lehrbücher [Basel Textbooks]. Birkhäuser Verlag, Basel (1992)CrossRefGoogle Scholar
  28. 28.
    Li, M., Zhou, X.: Min–max theory for free boundary minimal hypersurfaces I-regularity theory. J. Differ. Geom. arXiv:1611.02612 (2016)
  29. 29.
    Liokumovich, Y., Marques, F.C., Neves, A.: Weyl law for the volume spectrum. Ann. Math. (2) 187(3), 933–961 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lobaton, E.J., Salamon, T.R.: Computation of constant mean curvature surfaces: application to the gas-liquid interface of a pressurized fluid on a superhydrophobic surface. J. Colloid Interface Sci. 314, 184–198 (2007)CrossRefGoogle Scholar
  31. 31.
    López, R.: Wetting phenomena and constant mean curvature surfaces with boundary. Rev. Math. Phys. 17(7), 769–792 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mahmoudi, F., Mazzeo, R., Pacard, F.: Constant mean curvature hypersurfaces condensing on a submanifold. Geom. Funct. Anal. 16(4), 924–958 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2), 683–782 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Marques, F.C., Neves, A.: Morse index and multiplicity of min–max minimal hypersurfaces. Camb. J. Math. 4(4), 463–511 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Marques, F.C., Neves, A.: Existence of infinitely many minimal hypersurfaces in positive ricci curvature. Invent. Math. (2017).  https://doi.org/10.1007/s00222-017-0716-6 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Meeks III, W., Simon, L., Yau, S.T.: Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. Math. (2) 116(3), 621–659 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Meeks III, W.H., Mira, P., Perez, J., Ros, A.: Constant mean curvature spheres in homogeneous three-spheres. arXiv preprint arXiv:1308.2612 (2013)
  38. 38.
    Meeks III, W.H., Mira, P., Perez, J., Ros, A.: Constant mean curvature spheres in homogeneous three-manifolds. arXiv preprint arXiv:1706.09394 (2017)
  39. 39.
    Montezuma, R.: Min–max minimal hypersurfaces in non-compact manifolds. J. Differ. Geom. 103(3), 475–519 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Morgan, F.: Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. Am. Math. Soc. 355(12), 5041–5052 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nardulli, S.: The isoperimetric profile of a smooth Riemannian manifold for small volumes. Ann. Glob. Anal. Geom. 36(2), 111–132 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Novikov, S.P.: The Hamiltonian formalism and a multivalued analogue of Morse theory. Uspekhi Mat. Nauk 37(5(227)), 3–49 (1982)MathSciNetGoogle Scholar
  43. 43.
    Pacard, F.: Constant mean curvature hypersurfaces in Riemannian manifolds. Riv. Mat. Univ. Parma (7) 4, 141–162 (2005)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Pitts, J.T.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Volume 27 of Mathematical Notes. Princeton University Press, Princeton (1981)Google Scholar
  45. 45.
    Qing, J., Tian, G.: On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds. J. Am. Math. Soc. 20(4), 1091–1110 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ros, A.: The isoperimetric problem. Global Theory of Minimal Surfaces. In: Clay Mathematics Proceedings, vol. 2, pp. 175–209. American Mathematical Society, Providence, RI (2005)Google Scholar
  47. 47.
    Rosenberg, H., Schneider, M.: Embedded constant-curvature curves on convex surfaces. Pacific J. Math. 253(1), 213–218 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Rosenberg, H., Smith, G.: Degree theory of immersed hypersurfaces. arXiv:1010.1879v3 (2016)
  49. 49.
    Schneider, M.: Closed magnetic geodesics on \(S^2\). J. Differ. Geom. 87(2), 343–388 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34(6), 741–797 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Schoen, R., Simon, L., Yau, S.-T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134(3–4), 275–288 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, vol. 3. Centre for Mathematical Analysis, Australian National University, Canberra (1983)Google Scholar
  53. 53.
    Smith, F.R: On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary Riemannian metric. Ph.D. thesis, Ph.D. thesis, Supervisor: Leon Simon, University of Melbourne (1982)Google Scholar
  54. 54.
    Song, A.: Local min–max surfaces and strongly irreducible minimal Heegaard splittings. arXiv:1706.01037 (2017)
  55. 55.
    Struwe, M.: Large \(H\)-surfaces via the mountain-pass-lemma. Math. Ann. 270(3), 441–459 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160(1–2), 19–64 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Tamanini, I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wente, H.C.: Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121(1), 193–243 (1986)CrossRefzbMATHGoogle Scholar
  59. 59.
    White, B.: The maximum principle for minimal varieties of arbitrary codimension. Commun. Anal. Geom. 18(3), 421–432 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Wickramasekera, N.: A general regularity theory for stable codimension 1 integral varifolds. Ann. Math. (2) 179(3), 843–1007 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Ye, R.: Foliation by constant mean curvature spheres. Pacific J. Math. 147(2), 381–396 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Zhou, X.: Min-max hypersurface in manifold of positive Ricci curvature. J. Differ. Geom. 105(2), 291–343 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California Santa BarbaraSanta BarbaraUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

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