Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

Abstract

The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei) of the Allen–Cahn equation on a $3$-manifold. Using these, we are able to show that for generic metrics on a $3$-manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques–Neves in $3$-dimensions regarding min-max constructions of minimal surfaces.

Allen–Cahn min-max constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau’s conjecture on infinitely many minimal surfaces in a $3$-manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a $3$-manifold with a generic metric contains, for every $p = 1, 2, 3, \ldots $, a two-sided embedded minimal surface with Morse index $p$ and area $\sim p^{\frac 13}$, as conjectured by Marques–Neves.

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      title = {On the second inner variation of the {A}llen-{C}ahn functional and its applications},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
      volume = {60},
      year = {2011},
      number = {6},
      pages = {1843--1856},
      issn = {0022-2518},
      mrclass = {49K10 (49J45)},
      mrnumber = {3008253},
      mrreviewer = {Pablo Pedregal},
      doi = {10.1512/iumj.2011.60.4505},
      url = {https://doi.org/10.1512/iumj.2011.60.4505},
      zblnumber = {1273.49017},
      }
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    @ARTICLE{LiZhou,
      author = {Li, Haozhao and Zhou, Xin},
      title = {Existence of minimal surfaces of arbitrarily large {M}orse index},
      journal = {Calc. Var. Partial Differential Equations},
      fjournal = {Calculus of Variations and Partial Differential Equations},
      volume = {55},
      year = {2016},
      number = {3},
      pages = {Art. 64, 12},
      issn = {0944-2669},
      mrclass = {53A10 (49Q05 53C42 58E12)},
      mrnumber = {3509038},
      mrreviewer = {Doan The Hieu},
      doi = {10.1007/s00526-016-1007-6},
      url = {https://doi.org/10.1007/s00526-016-1007-6},
      zblnumber = {1343.53058},
      }
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    @ARTICLE{LMN:Weyl,
      author = {Liokumovich, Yevgeny and Marques, Fernando C. and Neves, André},
      title = {Weyl law for the volume spectrum},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      year = {2018},
      number = {3},
      pages = {933--961},
      issn = {0003-486X},
      mrclass = {53C23 (58E05 58J50)},
      mrnumber = {3779961},
      mrreviewer = {Leonid Friedlander},
      doi = {10.4007/annals.2018.187.3.7},
      url = {https://doi.org/10.4007/annals.2018.187.3.7},
      zblnumber = {1390.53034},
      }
  • [LiuWangWei] Go to document Y. Liu, K. Wang, and J. Wei, "Global minimizers of the Allen-Cahn equation in dimension $n\geq 8$," J. Math. Pures Appl. (9), vol. 108, iss. 6, pp. 818-840, 2017.
    @ARTICLE{LiuWangWei,
      author = {Liu, Yong and Wang, Kelei and Wei, Juncheng},
      title = {Global minimizers of the {A}llen-{C}ahn equation in dimension {$n\geq 8$}},
      journal = {J. Math. Pures Appl. (9)},
      fjournal = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
      volume = {108},
      year = {2017},
      number = {6},
      pages = {818--840},
      issn = {0021-7824},
      mrclass = {35J60 (35B33 35B40 35J20 35J25)},
      mrnumber = {3723158},
      mrreviewer = {Florin Catrina},
      doi = {10.1016/j.matpur.2017.05.006},
      url = {https://doi.org/10.1016/j.matpur.2017.05.006},
      zblnumber = {1380.35071},
      }
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      author = {Mantoulidis, Christos},
      title = {{A}llen-{C}ahn min-max on surfaces},
      arxiv = {1706.05946},
      note ={{\em J. Differential Geom},
      to appear},
      year = {2017},
      }
  • [marques:ICM] F. C. Marques, "Minimal surfaces: variational theory and applications," in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1, 2014, pp. 283-310.
    @INPROCEEDINGS{marques:ICM,
      author = {Marques, Fernando C.},
      title = {Minimal surfaces: variational theory and applications},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians---{S}eoul 2014. {V}ol. 1},
      pages = {283--310},
      publisher = {Kyung Moon Sa, Seoul},
      year = {2014},
      mrclass = {53C42 (49Q05)},
      mrnumber = {3728473},
      zblnumber = {1373.53004},
      }
  • [MarquesNeves:rigidity.min.max] Go to document F. C. Marques and A. Neves, "Rigidity of min-max minimal spheres in three-manifolds," Duke Math. J., vol. 161, iss. 14, pp. 2725-2752, 2012.
    @ARTICLE{MarquesNeves:rigidity.min.max,
      author = {Marques, Fernando C. and Neves, André},
      title = {Rigidity of min-max minimal spheres in three-manifolds},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {161},
      year = {2012},
      number = {14},
      pages = {2725--2752},
      issn = {0012-7094},
      mrclass = {53C24 (53C42)},
      mrnumber = {2993139},
      mrreviewer = {Jianquan Ge},
      doi = {10.1215/00127094-1813410},
      url = {https://doi.org/10.1215/00127094-1813410},
      zblnumber = {1260.53079},
      }
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    @MISC{MarquesNeves:uper-semi-index,
      author = {Marques, Fernando C. and Neves, André},
      title = {Morse index of multiplicity one min-max minimal hypersurfaces},
      arxiv = {1803.04273},
      year = {2018},
      }
  • [MarquesNeves:Willmore] 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{MarquesNeves:Willmore,
      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},
      url = {https://doi.org/10.4007/annals.2014.179.2.6},
      zblnumber = {1297.49079},
      }
  • [MarquesNeves:multiplicity] 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{MarquesNeves:multiplicity,
      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},
      url = {https://doi.org/10.4310/CJM.2016.v4.n4.a2},
      zblnumber = {1367.49036},
      }
  • [MarquesNeves:spaceOfCycles] Go to document 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{MarquesNeves:spaceOfCycles,
      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 {G}eometry and {M}athematical {P}hysics},
      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},
      doi = {10.4310/SDG.2016.v21.n1.a5},
      url = {https://doi.org/10.4310/SDG.2016.v21.n1.a5},
      }
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    @ARTICLE{MarquesNeves:posRic,
      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},
      mrreviewer = {Martin Man-Chun Li},
      doi = {10.1007/s00222-017-0716-6},
      url = {https://doi.org/10.1007/s00222-017-0716-6},
      zblnumber = {1390.53064},
      }
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    @ARTICLE{MarquesNevesSong,
      author = {Marques, Fernando C. and Neves, André and Song, Antoine},
      title = {Equidistribution of minimal hypersurfaces for generic metrics},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {216},
      year = {2019},
      number = {2},
      pages = {421--443},
      issn = {0020-9910},
      mrclass = {53C42 (49Q05 58D17)},
      mrnumber = {3953507},
      doi = {10.1007/s00222-018-00850-5},
      url = {https://doi.org/10.1007/s00222-018-00850-5},
      zblnumber = {07061102},
      }
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      title = {The gradient theory of phase transitions and the minimal interface criterion},
      journal = {Arch. Rational Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
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      year = {1987},
      number = {2},
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      issn = {0003-9527},
      mrclass = {76T05 (80A15)},
      mrnumber = {0866718},
      mrreviewer = {L. Hsiao},
      doi = {10.1007/BF00251230},
      url = {https://doi.org/10.1007/BF00251230},
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    @INPROCEEDINGS{neves:ICM,
      author = {Neves, André},
      title = {New applications of min-max theory},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians---{S}eoul 2014. {V}ol. {II}},
      pages = {939--957},
      publisher = {Kyung Moon Sa, Seoul},
      year = {2014},
      mrclass = {53C42 (49Q05)},
      mrnumber = {3728646},
      zblnumber = {1373.53084},
      }
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      author = {Pacard, Frank},
      title = {The role of minimal surfaces in the study of the {A}llen-{C}ahn equation},
      booktitle = {Geometric Analysis: Partial Differential Equations and Surfaces},
      series = {Contemp. Math.},
      volume = {570},
      pages = {137--163},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2012},
      mrclass = {35R01 (35B25 35J20 35J61 49Q05)},
      mrnumber = {2963598},
      mrreviewer = {Anna Maria Candela},
      doi = {10.1090/conm/570/11306},
      url = {https://doi.org/10.1090/conm/570/11306},
      zblnumber = {1273.58011},
      }
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      author = {Pacard, Frank and Ritoré,
      Manuel},
      title = {From constant mean curvature hypersurfaces to the gradient theory of phase transitions},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {64},
      year = {2003},
      number = {3},
      pages = {359--423},
      issn = {0022-040X},
      mrclass = {58E12 (35B25 35J20 53A10)},
      mrnumber = {2032110},
      mrreviewer = {Giandomenico Orlandi},
      doi = {10.4310/jdg/1090426999},
      url = {https://doi.org/10.4310/jdg/1090426999},
      zblnumber = {1070.58014},
      }
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      author = {Pacard, Frank and Sun, T.},
      title = {Doubling construction for {CMC} hypersurfaces in {R}iemannian manifolds},
      url={http://www.cmls.polytechnique.fr/perso/pacard/Publications/PR-01.pdf},
      }
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      author = {Pacard, Frank and Wei, Juncheng},
      title = {Stable solutions of the {A}llen-{C}ahn equation in dimension 8 and minimal cones},
      journal = {J. Funct. Anal.},
      fjournal = {Journal of Functional Analysis},
      volume = {264},
      year = {2013},
      number = {5},
      pages = {1131--1167},
      issn = {0022-1236},
      mrclass = {35J91 (35B09)},
      mrnumber = {3010017},
      doi = {10.1016/j.jfa.2012.03.010},
      url = {https://doi.org/10.1016/j.jfa.2012.03.010},
      zblnumber = {1281.35046},
      }
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      author = {del Pino, Manuel and Kowalczyk, Micha\l and Wei, Juncheng},
      title = {On {D}e {G}iorgi's conjecture in dimension {$N\geq 9$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {174},
      year = {2011},
      number = {3},
      pages = {1485--1569},
      issn = {0003-486X},
      mrclass = {35J91 (35B05 35B06 35B25 35J20)},
      mrnumber = {2846486},
      mrreviewer = {Rolando Magnanini},
      doi = {10.4007/annals.2011.174.3.3},
      url = {https://doi.org/10.4007/annals.2011.174.3.3},
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      }
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    @ARTICLE{delPinoKowalczykWei,
      author = {del Pino, Manuel and Kowalczyk, Michal and Wei, Juncheng},
      title = {Entire solutions of the {A}llen-{C}ahn equation and complete embedded minimal surfaces of finite total curvature in {$\Bbb R^3$}},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {93},
      year = {2013},
      number = {1},
      pages = {67--131},
      issn = {0022-040X},
      mrclass = {35J91 (35B08 53A10)},
      mrnumber = {3019512},
      mrreviewer = {Sanjiban Santra},
      doi = {10.4310/jdg/1357141507},
      url = {https://doi.org/10.4310/jdg/1357141507},
      zblnumber = {1275.53015},
      }
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    @ARTICLE{delPinoKowalczykWeiYang:interface,
      author = {del Pino, Manuel and Kowalczyk, Michal and Wei, Juncheng and Yang, Jun},
      title = {Interface foliation near minimal submanifolds in {R}iemannian manifolds with positive {R}icci curvature},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {20},
      year = {2010},
      number = {4},
      pages = {918--957},
      issn = {1016-443X},
      mrclass = {53C21 (35Q79 58E50)},
      mrnumber = {2729281},
      mrreviewer = {Enrico Valdinoci},
      doi = {10.1007/s00039-010-0083-6},
      url = {https://doi.org/10.1007/s00039-010-0083-6},
      zblnumber = {1213.35219},
      }
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      series = {Math. Notes},
      volume = {27},
      publisher = {Princeton Univ. 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},
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      title = {On the stability of minimal surfaces},
      journal = {Dokl. Akad. Nauk SSSR},
      fjournal = {Doklady Akademii Nauk SSSR},
      volume = {260},
      year = {1981},
      number = {2},
      pages = {293--295},
      issn = {0002-3264},
      mrclass = {49F10},
      mrnumber = {0630142},
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      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {169},
      year = {2009},
      number = {1},
      pages = {41--78},
      issn = {0003-486X},
      mrclass = {58E12 (35J60 58E50)},
      mrnumber = {2480601},
      mrreviewer = {Vicen\c{t}iu D. R\u{a}dulescu},
      doi = {10.4007/annals.2009.169.41},
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      fjournal = {Indiana University Mathematics Journal},
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      issn = {0022-2518},
      mrclass = {53C42},
      mrnumber = {2779062},
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      doi = {10.1512/iumj.2010.59.4013},
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      author = {Schoen, Richard},
      title = {Estimates for stable minimal surfaces in three-dimensional manifolds},
      booktitle = {Seminar on Minimal Submanifolds},
      series = {Ann. of Math. Stud.},
      volume = {103},
      pages = {111--126},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {1983},
      mrclass = {53C42 (53A10 58E12)},
      mrnumber = {0795231},
      mrreviewer = {Richard H. Escobales, Jr.},
      zblnumber = {0532.53042},
      doi = {10.1515/9781400881437-006},
      url = {https://doi.org/10.1515/9781400881437-006},
      }
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      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},
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Authors

Otis Chodosh

Princeton University, Princeton, NJ and Institute for Advanced Study, Princeton, NJ

Christos Mantoulidis

Massachusetts Institute of Technology, Cambridge, MA