Equidistribution of minimal hypersurfaces for generic metrics

  • Fernando C. Marques
  • André NevesEmail author
  • Antoine Song


For almost all Riemannian metrics (in the \(C^\infty \) Baire sense) on a closed manifold \(M^{n+1}\), \(3\le (n+1)\le 7\), we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of the main result of Irie et al. (Ann Math 187(3):963–972, 2018), that established density of minimal hypersurfaces for generic metrics. As in Irie et al. (2018), the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich et al. (Ann Math 187(3):933–961, 2018).



  1. 1.
    Almgren, F.: The Theory of Varifolds, Mimeographed Notes. Princeton University, Princeton (1965)Google Scholar
  2. 2.
    Bowen, R.: The equidistribution of closed geodesics. Am. J. Math. 94, 413–423 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eskin, A., Oh, H.: Ergodic theoretic proof of equidistribution of Hecke points. Ergod. Theory Dyn. Syst. 26(1), 163–167 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gromov, M.: Dimension, nonlinear spectra and width. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric aspects of functional analysis, (1986/87). Lecture Notes in Mathematics, vol. 1317, pp. 132–184. Springer, Berlin (1988)CrossRefGoogle Scholar
  5. 5.
    Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guth, L.: Minimax problems related to cup powers and Steenrod squares. Geom. Funct. Anal. 18, 1917–1987 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Irie, K.: Dense existence of periodic Reeb orbits and ECH spectral invariants. J. Mod. Dyn. 9, 357–363 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Irie, K., Marques, F.C., Neves, A.: Density of minimal hypersurfaces for generic metrics. Ann. Math. 187(3), 963–972 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liokumovich, Y., Marques, F.C., Neves, A.: Weyl law for the volume spectrum. Ann. Math. 187(3), 933–961 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Margulis, G.: On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: periodic orbits of hyperbolic flows (translated from the Russian by Valentina Vladimirovna Szulikowska. Springer Monographs in Mathematics. Springer, Berlin) (2004)Google Scholar
  11. 11.
    Marques, F.C., Neves, A.: Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math. 4(4), 463–511 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marques, F.C., Neves, A.: Topology of the Space of Cycles and Existence of Minimal Varieties, Surv. Differ. Geom., 21, pp. 165–177. International Press, Somerville, MA (2016)Google Scholar
  13. 13.
    Marques, F.C., Neves, A.: Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. Invent. Math. 209(2), 577–616 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McMullen, C., Mohammadi, A., Amir, Oh, H.: Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209(2), 425–461 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mozes, S., Shah, N.: On the space of ergodic invariant measures of unipotent flows. Ergod. Theory Dyn. Syst. 15(1), 149–159 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pitts, J.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Mathematical Notes 27. Princeton University Press, Princeton (1981)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ratner, M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, 235–280 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34, 741–797 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shah, N.: Closures of totally geodesic immersions in manifolds of constant negative curvature. In: Ghys, E., Haefliger, A., Verjovsky, A. (eds.) Group Theory from a Geometrical Viewpoint, pp. 718–732. World Scientific, River Edge, New Jersey (1990)Google Scholar
  20. 20.
    Sharp, B.: Compactness of minimal hypersurfaces with bounded index. J. Differ. Geom. 106(2), 317–339 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Song, A.: Existence of infinitely many minimal hypersurfaces in closed manifolds. arXiv:1806.08816 [math.DG] (2018)
  23. 23.
    White, B.: The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40, 161–200 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    White, B.: On the bumpy metrics theorem for minimal submanifolds. Am. J. Math. 139(4), 1149–1155 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yau, S.-T.: Problem section. In: Seminar on Differential Geometry, pp. 669–706, Ann. of Math. Stud., 102. Princeton Univ Press, Princeton, NJ (1982)Google Scholar
  26. 26.
    Zelditch, S.: Trace formula for compact \(\Gamma \setminus PSL2({\mathbb{R}})\) and the equidistribution theory of closed geodesics. Duke Math. J. 59(1), 27–81 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Fernando C. Marques
    • 1
  • André Neves
    • 2
    • 3
    Email author
  • Antoine Song
    • 1
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Imperial College LondonLondonUK

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