Weyl law for the volume spectrum

Abstract

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega _p(M)\}_{p\in \mathbb {N}}$ satisfies a Weyl law that was conjectured by Gromov.

Note: To view the article, click on the URL link for the DOI number.

  • [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},
      }
  • [federer] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969, vol. 153.
    @BOOK{federer,
      author = {Federer, Herbert},
      title = {Geometric Measure Theory},
      series = {Grundlehren der Math. Wiss.},
      volume = {153},
      publisher = {Springer-Verlag, New York},
      year = {1969},
      pages = {xiv+676},
      mrclass = {28.80 (26.00)},
      mrnumber = {0257325},
      mrreviewer = {J. E. Brothers},
      zblnumber = {0176.00801},
      }
  • [federer-fleming] Go to document H. Federer and W. H. Fleming, "Normal and integral currents," Ann. of Math. (2), vol. 72, pp. 458-520, 1960.
    @ARTICLE{federer-fleming,
      author = {Federer, Herbert and Fleming, Wendell H.},
      title = {Normal and integral currents},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {72},
      year = {1960},
      pages = {458--520},
      issn = {0003-486X},
      mrclass = {28.80 (53.45)},
      mrnumber = {0123260},
      mrreviewer = {L. C. Young},
      doi = {10.2307/1970227},
      zblnumber = {0187.31301},
      }
  • [gal] Go to document P. Glynn-Adey and Y. Liokumovich, "Width, Ricci curvature, and minimal hypersurfaces," J. Differential Geom., vol. 105, iss. 1, pp. 33-54, 2017.
    @ARTICLE{gal,
      author = {Glynn-Adey, Parker and Liokumovich, Yevgeny},
      title = {Width, {R}icci curvature, and minimal hypersurfaces},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {105},
      year = {2017},
      number = {1},
      pages = {33--54},
      issn = {0022-040X},
      mrclass = {53C20 (53C21 53C42)},
      mrnumber = {3592694},
      mrreviewer = {Otis Chodosh},
      doi = {10.4310/jdg/1483655859},
      zblnumber = {1359.53051},
      }
  • [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},
      }
  • [gromov2] Go to document M. Gromov, "Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles," Geom. Funct. Anal., vol. 19, iss. 3, pp. 743-841, 2009.
    @ARTICLE{gromov2,
      author = {Gromov, Mikhail},
      title = {Singularities, expanders and topology of maps. {I}. {H}omology versus volume in the spaces of cycles},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {19},
      year = {2009},
      number = {3},
      pages = {743--841},
      issn = {1016-443X},
      mrclass = {58K15 (05C10 53C23 57M27 57R35 58J50 58K30)},
      mrnumber = {2563769},
      mrreviewer = {Tobias Ekholm},
      doi = {10.1007/s00039-009-0021-7},
      zblnumber = {1195.58010},
      }
  • [gromov3] M. Gromov, Morse spectra, homology measures, spaces of cycles and parametric packing problem, 2015.
    @MISC{gromov3,
      author = {Gromov, Mikhail},
      title = {Morse spectra, homology measures, spaces of cycles and parametric packing problem},
      note = {preprint},
      year = {2015},
      zblnumber = {},
      }
  • [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},
      }
  • [hatcher] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
    @BOOK{hatcher,
      author = {Hatcher, Allen},
      title = {Algebraic Topology},
      publisher = {Cambridge University Press, Cambridge},
      year = {2002},
      pages = {xii+544},
      isbn = {0-521-79160-X; 0-521-79540-0},
      mrclass = {55-01 (55-00)},
      mrnumber = {1867354},
      mrreviewer = {Donald W. Kahn},
      zblnumber = {1044.55001},
      }
  • [hofmann] Go to document S. Hofmann, M. Mitrea, and M. Taylor, "Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains," J. Geom. Anal., vol. 17, iss. 4, pp. 593-647, 2007.
    @ARTICLE{hofmann,
      author = {Hofmann, Steve and Mitrea, Marius and Taylor, Michael},
      title = {Geometric and transformational properties of {L}ipschitz domains, {S}emmes-{K}enig-{T}oro domains, and other classes of finite perimeter domains},
      journal = {J. Geom. Anal.},
      fjournal = {The Journal of Geometric Analysis},
      volume = {17},
      year = {2007},
      number = {4},
      pages = {593--647},
      issn = {1050-6926},
      mrclass = {49Q15 (26B15 26B20 49Q20)},
      mrnumber = {2365661},
      mrreviewer = {Martin Fuchs},
      doi = {10.1007/BF02937431},
      zblnumber = {1142.49021},
      }
  • [irie-marques-neves] Go to document K. Irie, F. C. Marques, and A. Neves, "Density of minimal hypersurfaces for generic metrics," Ann. of Math. (2), vol. 187, pp. 963-972, 2018.
    @article{irie-marques-neves,
      author={Irie, K. and Marques, F. C. and Neves, A.},
      title={Density of minimal hypersurfaces for generic metrics},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      year={2018},
      pages={963--972},
      doi = {10.4007/annals.2018.187.3.8},
     }
  • [korevaar] Go to document N. Korevaar, "Upper bounds for eigenvalues of conformal metrics," J. Differential Geom., vol. 37, iss. 1, pp. 73-93, 1993.
    @ARTICLE{korevaar,
      author = {Korevaar, Nicholas},
      title = {Upper bounds for eigenvalues of conformal metrics},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {37},
      year = {1993},
      number = {1},
      pages = {73--93},
      issn = {0022-040X},
      mrclass = {58G25 (53A30 58G30)},
      mrnumber = {1198600},
      mrreviewer = {Jack Quine},
      doi = {10.4310/jdg/1214453423},
      zblnumber = {0794.58045},
      }
  • [liokumovich] Go to document Y. Liokumovich, "Families of short cycles on Riemannian surfaces," Duke Math. J., vol. 165, iss. 7, pp. 1363-1379, 2016.
    @ARTICLE{liokumovich,
      author = {Liokumovich, Yevgeny},
      title = {Families of short cycles on {R}iemannian surfaces},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {165},
      year = {2016},
      number = {7},
      pages = {1363--1379},
      issn = {0012-7094},
      mrclass = {53C23 (30F10)},
      mrnumber = {3498868},
      mrreviewer = {Mikhail G. Katz},
      doi = {10.1215/00127094-3450208},
      zblnumber = {1341.53072},
      }
  • [marques-neves] Go to document F. C. Marques and A. Neves, "Min-max theory and the Willmore conjecture," Ann. of Math. (2), vol. 179, iss. 2, pp. 683-782, 2014.
    @ARTICLE{marques-neves,
      author = {Marques, Fernando C. and Neves, André},
      title = {Min-max theory and the {W}illmore conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {179},
      year = {2014},
      number = {2},
      pages = {683--782},
      issn = {0003-486X},
      mrclass = {53C42 (49Q20)},
      mrnumber = {3152944},
      mrreviewer = {Andrea Mondino},
      doi = {10.4007/annals.2014.179.2.6},
      zblnumber = {1297.49079},
      }
  • [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},
      }
  • [marques-neves-cdm] F. C. Marques and A. Neves, "Applications of Almgren-Pitts min-max theory," in Current Developments in Mathematics 2013, Int. Press, Somerville, MA, 2014, pp. 1-71.
    @INCOLLECTION{marques-neves-cdm,
      author = {Marques, Fernando C. and Neves, André},
      title = {Applications of {A}lmgren-{P}itts min-max theory},
      booktitle = {Current Developments in Mathematics 2013},
      pages = {1--71},
      publisher = {Int. Press, Somerville, MA},
      year = {2014},
      mrclass = {58E05 (49K35 53C21)},
      mrnumber = {3307714},
      mrreviewer = {Salvatore A. Marano},
      zblnumber = {1317.49054},
      }
  • [marques-neves-cycles] 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-cycles,
      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},
      }
  • [papasoglu] P. Papasoglu and E. Swenson, A sphere hard to cut, 2015.
    @MISC{papasoglu,
      author = {Papasoglu, P. and Swenson, E.},
      title = {A sphere hard to cut},
      arxiv = {1509.02307},
      year = {2015},
      zblnumber = {},
      }
  • [pitts] 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},
      }
  • [simon] L. Simon, Lectures on Geometric Measure Theory, Australian National University, Centre for Mathematical Analysis, Canberra, 1983, vol. 3.
    @BOOK{simon,
      author = {Simon, Leon},
      title = {Lectures on Geometric Measure Theory},
      series = {Proc. Centre for Mathematical Analysis, Australian National University},
      volume = {3},
      publisher = {Australian National University, Centre for Mathematical Analysis, Canberra},
      year = {1983},
      pages = {vii+272},
      isbn = {0-86784-429-9},
      mrclass = {49-01 (28A75 49F20)},
      mrnumber = {0756417},
      mrreviewer = {J. S. Joel},
      zblnumber = {0546.49019},
      }
  • [weyl] H. Weyl, "Über die Asymptotische Verteilung der Eigenwerte," Gött. Nachr., pp. 110-117, 1911.
    @ARTICLE{weyl,
      author = {Weyl, H.},
      title = {{Ü}ber die Asymptotische Verteilung der Eigenwerte},
      journal={Gött. Nachr.},
      year = {1911},
      pages={110--117},
      zblnumber = {42.0432.03},
      }
  • [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

Yevgeny Liokumovich

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA

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