Density of minimal hypersurfaces for generic metrics

Abstract

For almost all Riemannian metrics (in the $C^\infty $ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces, thus proving a conjecture of Yau (1982) for generic metrics. \par \par \par

  • [almgren] Go to document F. J. Almgren Jr., "The homotopy groups of the integral cycle groups," Topology, vol. 1, pp. 257-299, 1962.
    @ARTICLE{almgren,
      author = { Almgren, Jr., Frederick Justin},
      title = {The homotopy groups of the integral cycle groups},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {1},
      year = {1962},
      pages = {257--299},
      issn = {0040-9383},
      mrclass = {55.45 (55.42)},
      mrnumber = {0146835},
      mrreviewer = {W. W. Fleming},
      doi = {10.1016/0040-9383(62)90016-2},
      zblnumber = {0118.18503},
      }
  • [almgren-varifolds] F. J. Almgren Jr., The theory of varifolds, 1965.
    @MISC{almgren-varifolds,
      author = {Almgren, Jr., Frederick Justin},
      title = {The theory of varifolds},
      note = {mimeographed notes, Princeton},
      year = {1965},
      zblnumber = {},
      }
  • [besse] Go to document A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987, vol. 10.
    @BOOK{besse,
      author = {Besse, Arthur L.},
      title = {Einstein Manifolds},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {10},
      publisher = {Springer-Verlag, Berlin},
      year = {1987},
      pages = {xii+510},
      isbn = {3-540-15279-2},
      mrclass = {53C25 (53-02 53C21 53C30 53C55 58D17 58E11)},
      mrnumber = {0867684},
      mrreviewer = {S. M. Salamon},
      doi = {10.1007/978-3-540-74311-8},
      zblnumber = {0613.53001},
      }
  • [cristofaro-hutchings-ramos] Go to document D. Cristofaro-Gardiner, M. Hutchings, and V. G. B. Ramos, "The asymptotics of ECH capacities," Invent. Math., vol. 199, iss. 1, pp. 187-214, 2015.
    @ARTICLE{cristofaro-hutchings-ramos,
      author = {Cristofaro-Gardiner, Daniel and Hutchings, Michael and Ramos, Vinicius Gripp Barros},
      title = {The asymptotics of {ECH} capacities},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {199},
      year = {2015},
      number = {1},
      pages = {187--214},
      issn = {0020-9910},
      mrclass = {53D42 (57R17 57R57)},
      mrnumber = {3294959},
      mrreviewer = {David E. Hurtubise},
      doi = {10.1007/s00222-014-0510-7},
      zblnumber = {1315.53091},
      }
  • [gromov0] Go to document M. Gromov, "Dimension, non-linear spectra and width," in Geometric Aspects of Functional Analysis (1986/87), Springer-Verlag, New York, 1988, vol. 1317, pp. 132-184.
    @INCOLLECTION{gromov0,
      author = {Gromov, M.},
      title = {Dimension, non-linear spectra and width},
      booktitle = {Geometric Aspects of Functional Analysis (1986/87)},
      series = {Lecture Notes in Math.},
      volume = {1317},
      pages = {132--184},
      publisher = {Springer-Verlag, New York},
      year = {1988},
      mrclass = {58C40 (47H12 58E05)},
      mrnumber = {0950979},
      mrreviewer = {Friedbert Prüfer},
      doi = {10.1007/BFb0081739},
      zblnumber = {0664.41019},
      }
  • [gromov] Go to document M. Gromov, "Isoperimetry of waists and concentration of maps," Geom. Funct. Anal., vol. 13, iss. 1, pp. 178-215, 2003.
    @ARTICLE{gromov,
      author = {Gromov, M.},
      title = {Isoperimetry of waists and concentration of maps},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {13},
      year = {2003},
      number = {1},
      pages = {178--215},
      issn = {1016-443X},
      mrclass = {53C23},
      mrnumber = {1978494},
      mrreviewer = {Igor Belegradek},
      doi = {10.1007/s000390300004},
      zblnumber = {1044.46057},
      }
  • [guth] Go to document L. Guth, "Minimax problems related to cup powers and Steenrod squares," Geom. Funct. Anal., vol. 18, iss. 6, pp. 1917-1987, 2009.
    @ARTICLE{guth,
      author = {Guth, Larry},
      title = {Minimax problems related to cup powers and {S}teenrod squares},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {18},
      year = {2009},
      number = {6},
      pages = {1917--1987},
      issn = {1016-443X},
      mrclass = {53C23},
      mrnumber = {2491695},
      mrreviewer = {John F. Oprea},
      doi = {10.1007/s00039-009-0710-2},
      zblnumber = {1190.53038},
      }
  • [irie] Go to document K. Irie, "Dense existence of periodic Reeb orbits and ECH spectral invariants," J. Mod. Dyn., vol. 9, pp. 357-363, 2015.
    @ARTICLE{irie,
      author = {Irie, Kei},
      title = {Dense existence of periodic {R}eeb orbits and {ECH} spectral invariants},
      journal = {J. Mod. Dyn.},
      fjournal = {Journal of Modern Dynamics},
      volume = {9},
      year = {2015},
      pages = {357--363},
      issn = {1930-5311},
      mrclass = {37J45 (53D42)},
      mrnumber = {3436746},
      mrreviewer = {Sheila Sandon},
      doi = {10.3934/jmd.2015.9.357},
      zblnumber = {1353.37125},
      }
  • [lawson] Go to document B. H. Lawson Jr., "Complete minimal surfaces in $S^{3}$," Ann. of Math. (2), vol. 92, pp. 335-374, 1970.
    @ARTICLE{lawson,
      author = {Lawson, Jr., H. Blaine},
      title = {Complete minimal surfaces in {$S\sp{3}$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {92},
      year = {1970},
      pages = {335--374},
      issn = {0003-486X},
      mrclass = {53.04},
      mrnumber = {0270280},
      mrreviewer = {C. S. Weaver},
      doi = {10.2307/1970625},
      zblnumber = {0205.52001},
      }
  • [liokumovich-marques-neves] Go to document Y. Liokumovich, F. C. Marques, and A. Neves, "Weyl law for the volume spectrum," Ann. of Math. (2), vol. 187, iss. 3, pp. 933-961, 2018.
    @ARTICLE{liokumovich-marques-neves,
      author = {Liokumovich, Y. and Marques, F. C. and Neves, A.},
      title = {{W}eyl law for the volume spectrum},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      number = {3},
      year = {2018},
      pages = {933--961},
      doi = {10.4007/annals.2018.187.3.7},
      zblnumber = {},
      }
  • [marques-neves-index] Go to document F. C. Marques and A. Neves, "Morse index and multiplicity of min-max minimal hypersurfaces," Camb. J. Math., vol. 4, iss. 4, pp. 463-511, 2016.
    @ARTICLE{marques-neves-index,
      author = {Marques, Fernando C. and Neves, André},
      title = {Morse index and multiplicity of min-max minimal hypersurfaces},
      journal = {Camb. J. Math.},
      fjournal = {Cambridge Journal of Mathematics},
      volume = {4},
      year = {2016},
      number = {4},
      pages = {463--511},
      issn = {2168-0930},
      mrclass = {49J35 (58E12)},
      mrnumber = {3572636},
      mrreviewer = {Giandomenico Orlandi},
      doi = {10.4310/CJM.2016.v4.n4.a2},
      zblnumber = {1367.49036},
      }
  • [marques-neves-topology] F. C. Marques and A. Neves, "Topology of the space of cycles and existence of minimal varieties," in Surveys in Differential Geometry 2016. Advances in Geometry and Mathematical Physics, Int. Press, Somerville, MA, 2016, vol. 21, pp. 165-177.
    @INCOLLECTION{marques-neves-topology,
      author = {Marques, Fernando C. and Neves, André},
      title = {Topology of the space of cycles and existence of minimal varieties},
      booktitle = {Surveys in Differential Geometry 2016. {A}dvances in Geometry and Mathematical Physics},
      series = {Surv. Differ. Geom.},
      volume = {21},
      pages = {165--177},
      publisher = {Int. Press, Somerville, MA},
      year = {2016},
      mrclass = {58E05 (49Q05 53C42)},
      mrnumber = {3525097},
      mrreviewer = {Giandomenico Orlandi},
      zblnumber = {1361.53048},
      }
  • [marques-neves-infinitely] Go to document F. C. Marques and A. Neves, "Existence of infinitely many minimal hypersurfaces in positive Ricci curvature," Invent. Math., vol. 209, iss. 2, pp. 577-616, 2017.
    @ARTICLE{marques-neves-infinitely,
      author = {Marques, Fernando C. and Neves, André},
      title = {Existence of infinitely many minimal hypersurfaces in positive {R}icci curvature},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {209},
      year = {2017},
      number = {2},
      pages = {577--616},
      issn = {0020-9910},
      mrclass = {53C42 (49Q05 53C21 58E12)},
      mrnumber = {3674223},
      doi = {10.1007/s00222-017-0716-6},
      zblnumber = {06786951},
      }
  • [pitts] Go to document J. T. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Princeton University Press, Princeton, N.J., 1981, vol. 27.
    @BOOK{pitts,
      author = {Pitts, Jon T.},
      title = {Existence and Regularity of Minimal Surfaces on {R}iemannian Manifolds},
      series = {Math. Notes},
      volume = {27},
      publisher = {Princeton University Press, Princeton, N.J.},
      year = {1981},
      pages = {iv+330},
      isbn = {0-691-08290-1},
      mrclass = {49F22 (53C42)},
      mrnumber = {0626027},
      mrreviewer = {J. E. Brothers},
      zblnumber = {0462.58003},
      doi = {10.1515/9781400856459},
      }
  • [schoen-simon] Go to document R. Schoen and L. Simon, "Regularity of stable minimal hypersurfaces," Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981.
    @ARTICLE{schoen-simon,
      author = {Schoen, Richard and Simon, Leon},
      title = {Regularity of stable minimal hypersurfaces},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {34},
      year = {1981},
      number = {6},
      pages = {741--797},
      issn = {0010-3640},
      mrclass = {49F22 (53C42 58E15)},
      mrnumber = {0634285},
      mrreviewer = {F. J. Almgren, Jr.},
      doi = {10.1002/cpa.3160340603},
      zblnumber = {0497.49034},
      }
  • [sharp] Go to document B. Sharp, "Compactness of minimal hypersurfaces with bounded index," J. Differential Geom., vol. 106, iss. 2, pp. 317-339, 2017.
    @ARTICLE{sharp,
      author = {Sharp, Ben},
      title = {Compactness of minimal hypersurfaces with bounded index},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {106},
      year = {2017},
      number = {2},
      pages = {317--339},
      issn = {0022-040X},
      mrclass = {53C42 (53C21)},
      mrnumber = {3662994},
      doi = {10.4310/jdg/1497405628},
      zblnumber = {},
      }
  • [white2] Go to document B. White, "The space of minimal submanifolds for varying Riemannian metrics," Indiana Univ. Math. J., vol. 40, iss. 1, pp. 161-200, 1991.
    @ARTICLE{white2,
      author = {White, Brian},
      title = {The space of minimal submanifolds for varying {R}iemannian metrics},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
      volume = {40},
      year = {1991},
      number = {1},
      pages = {161--200},
      issn = {0022-2518},
      mrclass = {58D10 (53C42)},
      mrnumber = {1101226},
      mrreviewer = {João Lucas Marques Barbosa},
      doi = {10.1512/iumj.1991.40.40008},
      zblnumber = {0742.58009},
      }
  • [white] Go to document B. White, "On the bumpy metrics theorem for minimal submanifolds," Amer. J. Math., vol. 139, iss. 4, pp. 1149-1155, 2017.
    @ARTICLE{white,
      author = {White, Brian},
      title = {On the bumpy metrics theorem for minimal submanifolds},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {139},
      year = {2017},
      number = {4},
      pages = {1149--1155},
      issn = {0002-9327},
      mrclass = {53C42},
      mrnumber = {3689325},
      doi = {10.1353/ajm.2017.0029},
      }
  • [yau1] S. T. Yau, "Problem section," in Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J., 1982, vol. 102, pp. 669-706.
    @INCOLLECTION{yau1,
      author = {Yau, Shing Tung},
      title = {Problem section},
      booktitle = {Seminar on {D}ifferential {G}eometry},
      series = {Ann. of Math. Stud.},
      volume = {102},
      pages = {669--706},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1982},
      mrclass = {53Cxx (58-02)},
      mrnumber = {0645762},
      mrreviewer = {Yu. Burago},
      zblnumber = {0479.53001},
      }

Authors

Kei Irie

Research Institute for Mathematical Sciences Kyoto University, Kyoto Japan and Simons Center for Geometry and Physics State, University of New York Stony Brook, NY

Fernando C. Marques

Department of Mathematics, Princeton University, Princeton, NJ

André Neves

Department of Mathematics, University of Chicago, Chicago, IL and Imperial College London, London, UK