Abstract
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves.
We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.
-
[Allard72]
W. K. Allard, "On the first variation of a varifold," Ann. of Math. (2), vol. 95, pp. 417-491, 1972.@ARTICLE{Allard72,
author = {Allard, William K.},
title = {On the first variation of a varifold},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {95},
year = {1972},
pages = {417--491},
issn = {0003-486X},
mrclass = {49F20},
mrnumber = {0307015},
mrreviewer = {M. Klingmann},
doi = {10.2307/1970868},
url = {https://doi.org/10.2307/1970868},
zblnumber = {0252.49028},
} -
[Almgren62] F. J. Almgren Jr., "The homotopy groups of the integral cycle groups," Topology, vol. 1, pp. 257-299, 1962.
@ARTICLE{Almgren62,
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},
url = {https://doi.org/10.1016/0040-9383(62)90016-2},
zblnumber = {0118.18503},
} -
[Almgren65] F. J. Almgren Jr., The theory of varifolds.
@MISC{Almgren65,
author = {Almgren, Jr., Frederick Justin},
title = {The theory of varifolds},
note = {mimeographed notes, Princeton, 1965},
sortyear = {1965},
zblnumber = {},
} -
[BCE88] L. J. Barbosa, M. do Carmo, and J. Eschenburg, "Stability of hypersurfaces of constant mean curvature in Riemannian manifolds," Math. Z., vol. 197, iss. 1, pp. 123-138, 1988.
@ARTICLE{BCE88,
author = {Barbosa, J. Lucas and do Carmo, Manfredo and Eschenburg, Jost},
title = {Stability of hypersurfaces of constant mean curvature in {R}iemannian manifolds},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {197},
year = {1988},
number = {1},
pages = {123--138},
issn = {0025-5874},
mrclass = {53C42},
mrnumber = {0917854},
mrreviewer = {Johan Deprez},
doi = {10.1007/BF01161634},
url = {https://doi.org/10.1007/BF01161634},
zblnumber = {0653.53045},
} -
[Chodosh-Mantoulidis18] O. Chodosh and C. Mantoulidis, "Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates," Ann. of Math. (2), vol. 191, iss. 1, pp. 213-328, 2020.
@ARTICLE{Chodosh-Mantoulidis18,
author = {Chodosh, Otis and Mantoulidis, Christos},
title = {Minimal surfaces and the {A}llen-{C}ahn equation on 3-manifolds: index, multiplicity, and curvature estimates},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {191},
year = {2020},
number = {1},
pages = {213--328},
issn = {0003-486X},
mrclass = {49Q05 (35B25 35J61 35R01 49J35 53C42)},
mrnumber = {4045964},
doi = {10.4007/annals.2020.191.1.4},
url = {https://doi.org/10.4007/annals.2020.191.1.4},
zblnumber = {1431.49045},
} -
[Colding-DeLellis03] T. H. Colding and C. De Lellis, "The min-max construction of minimal surfaces," in Surveys in Differential Geometry, Vol. VIII, Int. Press, Somerville, MA, 2003, vol. 8, pp. 75-107.
@INCOLLECTION{Colding-DeLellis03,
author = {Colding, Tobias H. and De Lellis, Camillo},
title = {The min-max construction of minimal surfaces},
booktitle = {Surveys in Differential Geometry, {V}ol. {VIII}},
venue = {{B}oston, {MA},
2002},
series = {Surv. Differ. Geom.},
volume = {8},
pages = {75--107},
publisher = {Int. Press, Somerville, MA},
year = {2003},
mrclass = {53A10 (49Q05 53C42)},
mrnumber = {2039986},
mrreviewer = {Fei-Tsen Liang},
doi = {10.4310/SDG.2003.v8.n1.a3},
url = {https://doi.org/10.4310/SDG.2003.v8.n1.a3},
zblnumber = {1051.53052},
} -
[Colding-Minicozzi08b] T. H. Colding and W. P. Minicozzi II, "Width and finite extinction time of Ricci flow," Geom. Topol., vol. 12, iss. 5, pp. 2537-2586, 2008.
@ARTICLE{Colding-Minicozzi08b,
author = {Colding, Tobias H. and Minicozzi, II, William P.},
title = {Width and finite extinction time of {R}icci flow},
journal = {Geom. Topol.},
fjournal = {Geometry \& Topology},
volume = {12},
year = {2008},
number = {5},
pages = {2537--2586},
issn = {1465-3060},
mrclass = {53C44},
mrnumber = {2460871},
mrreviewer = {Andrea Nicole Young},
doi = {10.2140/gt.2008.12.2537},
url = {https://doi.org/10.2140/gt.2008.12.2537},
zblnumber = {1161.53352},
} -
[Colding-Minicozzi11] T. H. Colding and W. P. Minicozzi II, A Course in Minimal Surfaces, American Mathematical Society, Providence, RI, 2011, vol. 121.
@BOOK{Colding-Minicozzi11,
author = {Colding, Tobias Holck and Minicozzi, II, William P.},
title = {A Course in Minimal Surfaces},
series = {Grad. Stud. Math.},
volume = {121},
publisher = {American Mathematical Society, Providence, RI},
year = {2011},
pages = {xii+313},
isbn = {978-0-8218-5323-8},
mrclass = {53A10 (35J93 49Q05)},
mrnumber = {2780140},
mrreviewer = {Andrew Bucki},
doi = {10.1090/gsm/121},
url = {https://doi.org/10.1090/gsm/121},
zblnumber = {1242.53007},
} -
[DeLellis-Ramic16] C. De Lellis and J. Ramic, "Min-max theory for minimal hypersurfaces with boundary," Ann. Inst. Fourier (Grenoble), vol. 68, iss. 5, pp. 1909-1986, 2018.
@ARTICLE{DeLellis-Ramic16,
author = {De Lellis, Camillo and Ramic, Jusuf},
title = {Min-max theory for minimal hypersurfaces with boundary},
journal = {Ann. Inst. Fourier (Grenoble)},
fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
volume = {68},
year = {2018},
number = {5},
pages = {1909--1986},
issn = {0373-0956},
mrclass = {53C42 (49Q05 53A10)},
mrnumber = {3893761},
mrreviewer = {Annalisa Cesaroni},
doi = {10.5802/aif.3200},
url = {https://doi.org/10.5802/aif.3200},
zblnumber = {1408.53079},
} -
[DeLellis-Tasnady13] C. De Lellis and D. Tasnady, "The existence of embedded minimal hypersurfaces," J. Differential Geom., vol. 95, iss. 3, pp. 355-388, 2013.
@ARTICLE{DeLellis-Tasnady13,
author = {De Lellis, Camillo and Tasnady, Dominik},
title = {The existence of embedded minimal hypersurfaces},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {95},
year = {2013},
number = {3},
pages = {355--388},
issn = {0022-040X},
mrclass = {53C42 (53A10)},
mrnumber = {3128988},
mrreviewer = {Fei-Tsen Liang},
doi = {10.4310/jdg/1381931732},
url = {https://doi.org/10.4310/jdg/1381931732},
zblnumber = {1284.53057},
} -
[Frankel66] T. Frankel, "On the fundamental group of a compact minimal submanifold," Ann. of Math. (2), vol. 83, pp. 68-73, 1966.
@ARTICLE{Frankel66,
author = {Frankel, T.},
title = {On the fundamental group of a compact minimal submanifold},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {83},
year = {1966},
pages = {68--73},
issn = {0003-486X},
mrclass = {53.74 (57.00)},
mrnumber = {0187183},
mrreviewer = {R. Hermann},
doi = {10.2307/1970471},
url = {https://doi.org/10.2307/1970471},
zblnumber = {0189.22401},
} -
[Gaspar-Guaraco18] P. Gaspar and M. A. M. Guaraco, "The Allen-Cahn equation on closed manifolds," Calc. Var. Partial Differential Equations, vol. 57, iss. 4, p. 101, 2018.
@ARTICLE{Gaspar-Guaraco18,
author = {Gaspar, Pedro and Guaraco, Marco A. M.},
title = {The {A}llen-{C}ahn equation on closed manifolds},
journal = {Calc. Var. Partial Differential Equations},
fjournal = {Calculus of Variations and Partial Differential Equations},
volume = {57},
year = {2018},
number = {4},
pages = {Paper No. 101, 42},
issn = {0944-2669},
mrclass = {53A10 (49J35 49Q05)},
mrnumber = {3814054},
mrreviewer = {Annalisa Cesaroni},
doi = {10.1007/s00526-018-1379-x},
url = {https://doi.org/10.1007/s00526-018-1379-x},
zblnumber = {1396.53064},
} -
[Gilbarg-Trudinger01] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
@BOOK{Gilbarg-Trudinger01,
author = {Gilbarg, David and Trudinger, Neil S.},
title = {Elliptic Partial Differential Equations of Second Order},
series = {Classics Math.},
note = {reprint of the 1998 edition},
publisher = {Springer-Verlag, Berlin},
year = {2001},
pages = {xiv+517},
isbn = {3-540-41160-7},
mrclass = {35-02 (35Jxx)},
mrnumber = {1814364},
zblnumber = {1042.35002},
} -
[Giusti84] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984, vol. 80.
@BOOK{Giusti84,
author = {Giusti, Enrico},
title = {Minimal Surfaces and Functions of Bounded Variation},
series = {Monogr. Math.},
volume = {80},
publisher = {Birkhäuser Verlag, Basel},
year = {1984},
pages = {xii+240},
isbn = {0-8176-3153-4},
mrclass = {58E12 (49F10 53A10)},
mrnumber = {0775682},
mrreviewer = {Helmut Kaul},
doi = {10.1007/978-1-4684-9486-0},
url = {https://doi.org/10.1007/978-1-4684-9486-0},
zblnumber = {0545.49018},
} -
[Gromov88] M. Gromov, "Dimension, non-linear spectra and width," in Geometric Aspects of Functional Analysis (1986/87), Springer, Berlin, 1988, vol. 1317, pp. 132-184.
@INCOLLECTION{Gromov88,
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, Berlin},
year = {1988},
mrclass = {58C40 (47H12 58E05)},
mrnumber = {0950979},
mrreviewer = {Friedbert Prüfer},
doi = {10.1007/BFb0081739},
url = {https://doi.org/10.1007/BFb0081739},
zblnumber = {0664.41019},
} -
[Gromov03] M. Gromov, "Isoperimetry of waists and concentration of maps," Geom. Funct. Anal., vol. 13, iss. 1, pp. 178-215, 2003.
@ARTICLE{Gromov03,
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},
url = {https://doi.org/10.1007/s000390300004},
zblnumber = {1044.46057},
} -
[Guaraco18] M. A. M. Guaraco, "Min-max for phase transitions and the existence of embedded minimal hypersurfaces," J. Differential Geom., vol. 108, iss. 1, pp. 91-133, 2018.
@ARTICLE{Guaraco18,
author = {Guaraco, Marco A. M.},
title = {Min-max for phase transitions and the existence of embedded minimal hypersurfaces},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {108},
year = {2018},
number = {1},
pages = {91--133},
issn = {0022-040X},
mrclass = {49Q05 (53C42 58E12)},
mrnumber = {3743704},
mrreviewer = {Marc Michel Soret},
doi = {10.4310/jdg/1513998031},
url = {https://doi.org/10.4310/jdg/1513998031},
zblnumber = {1387.49060},
} -
[Guth09] L. Guth, "Minimax problems related to cup powers and Steenrod squares," Geom. Funct. Anal., vol. 18, iss. 6, pp. 1917-1987, 2009.
@ARTICLE{Guth09,
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},
url = {https://doi.org/10.1007/s00039-009-0710-2},
zblnumber = {1190.53038},
} -
[Harvey-Lawson75] R. Harvey and B. Lawson, "Extending minimal varieties," Invent. Math., vol. 28, pp. 209-226, 1975.
@ARTICLE{Harvey-Lawson75,
author = {Harvey, R. and Lawson, B.},
title = {Extending minimal varieties},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {28},
year = {1975},
pages = {209--226},
issn = {0020-9910},
mrclass = {49F10},
mrnumber = {0370319},
mrreviewer = {F. J. Almgren, Jr.},
doi = {10.1007/BF01425557},
url = {https://doi.org/10.1007/BF01425557},
zblnumber = {0316.49032},
} -
[Hatcher02] A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.
@BOOK{Hatcher02,
author = {Hatcher, Allen},
title = {Algebraic Topology},
publisher = {Cambridge Univ. 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},
} -
[Irie-Marques-Neves18] K. Irie, F. C. Marques, and A. Neves, "Density of minimal hypersurfaces for generic metrics," Ann. of Math. (2), vol. 187, iss. 3, pp. 963-972, 2018.
@ARTICLE{Irie-Marques-Neves18,
author = {Irie, Kei and Marques, Fernando C. and Neves, André},
title = {Density of minimal hypersurfaces for generic metrics},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {187},
year = {2018},
number = {3},
pages = {963--972},
issn = {0003-486X},
mrclass = {53C42 (49Q05)},
mrnumber = {3779962},
mrreviewer = {José Miguel Manzano},
doi = {10.4007/annals.2018.187.3.8},
url = {https://doi.org/10.4007/annals.2018.187.3.8},
zblnumber = {1387.53083},
} -
[Ketover-Marques-Neves16] D. Ketover, F. C. Marques, and A. Neves, "The catenoid estimate and its geometric applications," J. Differential Geom., vol. 115, iss. 1, pp. 1-26, 2020.
@ARTICLE{Ketover-Marques-Neves16,
author = {Ketover, Daniel and Marques, Fernando C. and Neves, André},
title = {The catenoid estimate and its geometric applications},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {115},
year = {2020},
number = {1},
pages = {1--26},
issn = {0022-040X},
mrclass = {53C42 (49J35 49Q05 53A10 58E12)},
mrnumber = {4081930},
doi = {10.4310/jdg/1586224840},
url = {https://doi.org/10.4310/jdg/1586224840},
zblnumber = {1439.53064},
} -
[Li-Zhou16] M. M. Li and X. Zhou, Min-max theory for free boundary minimal hypersurfaces I-regularity theory, 2016.
@MISC{Li-Zhou16,
author = { Li, Martin Man-Chun and Zhou, Xin},
title = {Min-max theory for free boundary minimal hypersurfaces {I}-regularity theory},
note = {{\em J. Differential Geom.},
to appear},
arxiv = {1611.02612},
year = {2016},
zblnumber = {},
} -
[Liokumovich-Marques-Neves18] 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-Neves18,
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},
} -
[Marques-Neves12] 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{Marques-Neves12,
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},
} -
[Marques-Neves14] 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-Neves14,
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},
} -
[Marques-Neves16] 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-Neves16,
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},
} -
[Marques-Neves17] 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-Neves17,
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},
} -
[Marques-Neves18] F. C. Marques and A. Neves, Morse index of multiplicity one min-max minimal hypersurfaces, 2018.
@MISC{Marques-Neves18,
author = {Marques, Fernando C. and Neves, André},
title = {Morse index of multiplicity one min-max minimal hypersurfaces},
arxiv = {1803.04273v1},
year = {2018},
zblnumber = {},
} -
[Marques-Neves-Song17] F. C. Marques, A. Neves, and A. Song, "Equidistribution of minimal hypersurfaces for generic metrics," Invent. Math., vol. 216, iss. 2, pp. 421-443, 2019.
@ARTICLE{Marques-Neves-Song17,
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 49Q20 53A10 58D17 58E12)},
mrnumber = {3953507},
mrreviewer = {S. Timothy Swift},
doi = {10.1007/s00222-018-00850-5},
url = {https://doi.org/10.1007/s00222-018-00850-5},
zblnumber = {1419.53061},
} -
[Pitts81] J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1981, vol. 27.
@BOOK{Pitts81,
author = {Pitts, Jon T.},
title = {Existence and regularity of minimal surfaces on {R}iemannian manifolds},
series = {Mathematical Notes},
volume = {27},
publisher = {Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo},
year = {1981},
pages = {iv+330},
isbn = {0-691-08290-1},
mrclass = {49F22 (53C42)},
mrnumber = {0626027},
mrreviewer = {J. E. Brothers},
zblnumber = {0462.58003},
} -
[Riviere17] T. Rivière, "A viscosity method in the min-max theory of minimal surfaces," Publ. Math. Inst. Hautes Études Sci., vol. 126, pp. 177-246, 2017.
@ARTICLE{Riviere17,
author = {Rivière, Tristan},
title = {A viscosity method in the min-max theory of minimal surfaces},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
volume = {126},
year = {2017},
pages = {177--246},
issn = {0073-8301},
mrclass = {49Q05 (53C42 58E12)},
mrnumber = {3735867},
mrreviewer = {Panayotis Vyridis},
doi = {10.1007/s10240-017-0094-z},
url = {https://doi.org/10.1007/s10240-017-0094-z},
zblnumber = {1387.53084},
} -
[Schoen-Simon81] R. Schoen and L. Simon, "Regularity of stable minimal hypersurfaces," Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981.
@ARTICLE{Schoen-Simon81,
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},
url = {https://doi.org/10.1002/cpa.3160340603},
zblnumber = {0497.49034},
} -
[Schoen-Simon-Yau75] R. Schoen, L. Simon, and S. T. Yau, "Curvature estimates for minimal hypersurfaces," Acta Math., vol. 134, iss. 3-4, pp. 275-288, 1975.
@ARTICLE{Schoen-Simon-Yau75,
author = {Schoen, Richard and Simon, Leon and Yau, S. T.},
title = {Curvature estimates for minimal hypersurfaces},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {134},
year = {1975},
number = {3-4},
pages = {275--288},
issn = {0001-5962},
mrclass = {53C40 (49F10)},
mrnumber = {0423263},
mrreviewer = {Robert Gulliver},
doi = {10.1007/BF02392104},
url = {https://doi.org/10.1007/BF02392104},
zblnumber = {0323.53039},
} -
[Sharp17] B. Sharp, "Compactness of minimal hypersurfaces with bounded index," J. Differential Geom., vol. 106, iss. 2, pp. 317-339, 2017.
@ARTICLE{Sharp17,
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},
mrreviewer = {Luciano Mari},
doi = {10.4310/jdg/1497405628},
url = {https://doi.org/10.4310/jdg/1497405628},
zblnumber = {1390.53065},
} -
[Simon83] L. Simon, Lectures on Geometric Measure Theory, Australian National Univ. , Centre for Mathematical Analysis, Canberra, 1983, vol. 3.
@BOOK{Simon83,
author = {Simon, Leon},
title = {Lectures on Geometric Measure Theory},
series = {Proc. Centre Math. Anal. Aust. Nat. Univ.} , volume = {3},
publisher = {Australian National Univ. , 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},
} -
[Simon87] L. Simon, "A strict maximum principle for area minimizing hypersurfaces," J. Differential Geom., vol. 26, iss. 2, pp. 327-335, 1987.
@ARTICLE{Simon87,
author = {Simon, Leon},
title = {A strict maximum principle for area minimizing hypersurfaces},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {26},
year = {1987},
number = {2},
pages = {327--335},
issn = {0022-040X},
mrclass = {49F10 (35J60 49F20 53C42)},
mrnumber = {0906394},
mrreviewer = {Michael Grüter},
doi = {10.4310/jdg/1214441373},
url = {https://doi.org/10.4310/jdg/1214441373},
zblnumber = {0625.53052},
} -
[Simons] J. Simons, "Minimal varieties in riemannian manifolds," Ann. of Math. (2), vol. 88, pp. 62-105, 1968.
@ARTICLE{Simons,
author = {Simons, James},
title = {Minimal varieties in riemannian manifolds},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {88},
year = {1968},
pages = {62--105},
issn = {0003-486X},
mrclass = {53.04 (35.00)},
mrnumber = {0233295},
mrreviewer = {W. F. Pohl},
doi = {10.2307/1970556},
url = {https://doi.org/10.2307/1970556},
zblnumber = {0181.49702},
} -
[Smith82] F. R. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary Riemannian metric, 1982.
@MISC{Smith82,
author = {Smith, Francis R.},
title = {On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary {R}iemannian metric},
note = {Ph.D. thesis, Australian National Univ., supervisor: Leon Simon},
year = {1982},
zblnumber = {},
} -
[Song18] A. Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, 2018.
@MISC{Song18,
author = {Song, Antoine},
title = {Existence of infinitely many minimal hypersurfaces in closed manifolds},
arxiv = {1806.08816v1},
year = {2018},
zblnumber = {},
} -
[White87] B. White, "Curvature estimates and compactness theorems in $3$-manifolds for surfaces that are stationary for parametric elliptic functionals," Invent. Math., vol. 88, iss. 2, pp. 243-256, 1987.
@ARTICLE{White87,
author = {White, B.},
title = {Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {88},
year = {1987},
number = {2},
pages = {243--256},
issn = {0020-9910},
mrclass = {58E12 (49F10 53C20 58D10)},
mrnumber = {0880951},
mrreviewer = {Helmut Kaul},
doi = {10.1007/BF01388908},
url = {https://doi.org/10.1007/BF01388908},
zblnumber = {0615.53044},
} -
[White91] B. White, "The space of minimal submanifolds for varying Riemannian metrics," Indiana Univ. Math. J., vol. 40, iss. 1, pp. 161-200, 1991.
@ARTICLE{White91,
author = {White, Brian},
title = {The space of minimal submanifolds for varying {R}iemannian metrics},
journal = {Indiana Univ. Math. J.},
fjournal = {Indiana Univ. Mathematics Journal},
volume = {40},
year = {1991},
number = {1},
pages = {161--200},
issn = {0022-2518},
mrclass = {58D10 (53C42)},
mrnumber = {1101226},
mrreviewer = {Jo\~{a}o Lucas Marques Barbosa},
doi = {10.1512/iumj.1991.40.40008},
url = {https://doi.org/10.1512/iumj.1991.40.40008},
zblnumber = {0742.58009},
} -
[White10] B. White, "The maximum principle for minimal varieties of arbitrary codimension," Comm. Anal. Geom., vol. 18, iss. 3, pp. 421-432, 2010.
@ARTICLE{White10,
author = {White, Brian},
title = {The maximum principle for minimal varieties of arbitrary codimension},
journal = {Comm. Anal. Geom.},
fjournal = {Communications in Analysis and Geometry},
volume = {18},
year = {2010},
number = {3},
pages = {421--432},
issn = {1019-8385},
mrclass = {53C42 (49Q20)},
mrnumber = {2747434},
mrreviewer = {Fei-Tsen Liang},
doi = {10.4310/CAG.2010.v18.n3.a1},
url = {https://doi.org/10.4310/CAG.2010.v18.n3.a1},
zblnumber = {1226.53061},
} -
[White17] B. White, "On the bumpy metrics theorem for minimal submanifolds," Amer. J. Math., vol. 139, iss. 4, pp. 1149-1155, 2017.
@ARTICLE{White17,
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},
mrreviewer = {Otis Chodosh},
doi = {10.1353/ajm.2017.0029},
url = {https://doi.org/10.1353/ajm.2017.0029},
zblnumber = {1379.53084},
} -
[White18] B. White, Generic transversality of minimal submanifolds and generic regularity of two-dimensional area-minimizing integral currents, 2019.
@MISC{White18,
author = {White, Brian},
title = {Generic transversality of minimal submanifolds and generic regularity of two-dimensional area-minimizing integral currents},
arxiv = {1901.05148},
year = {2019},
zblnumber = {},
} -
[Yau82] S. T. Yau, "Problem section," in Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J., 1982, vol. 102, pp. 669-706.
@INCOLLECTION{Yau82,
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},
} -
[Zhou10] X. Zhou, "On the existence of min-max minimal torus," J. Geom. Anal., vol. 20, iss. 4, pp. 1026-1055, 2010.
@ARTICLE{Zhou10,
author = {Zhou, Xin},
title = {On the existence of min-max minimal torus},
journal = {J. Geom. Anal.},
fjournal = {Journal of Geometric Analysis},
volume = {20},
year = {2010},
number = {4},
pages = {1026--1055},
issn = {1050-6926},
mrclass = {58E12 (58E20)},
mrnumber = {2683775},
mrreviewer = {Andreas Gastel},
doi = {10.1007/s12220-010-9137-0},
url = {https://doi.org/10.1007/s12220-010-9137-0},
zblnumber = {1203.58001},
} -
[Zhou15] X. Zhou, "Min-max minimal hypersurface in $(M^{n+1},g)$ with $Ric>0$ and $2 \leq n\break\leq 6$," J. Differential Geom., vol. 100, iss. 1, pp. 129-160, 2015.
@ARTICLE{Zhou15,
author = {Zhou, Xin},
title = {Min-max minimal hypersurface in {$(M^{n+1},
g)$} with {$Ric>0$} and {$2 \leq n\break\leq 6$}},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {100},
year = {2015},
number = {1},
pages = {129--160},
issn = {0022-040X},
mrclass = {53C42},
mrnumber = {3326576},
mrreviewer = {Xiaoli Chao},
doi = {10.4310/jdg/1427202766},
url = {https://doi.org/10.4310/jdg/1427202766},
zblnumber = {1331.53092},
} -
[Zhou17] X. Zhou, "Min-max hypersurface in manifold of positive Ricci curvature," J. Differential Geom., vol. 105, iss. 2, pp. 291-343, 2017.
@ARTICLE{Zhou17,
author = {Zhou, Xin},
title = {Min-max hypersurface in manifold of positive {R}icci curvature},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {105},
year = {2017},
number = {2},
pages = {291--343},
issn = {0022-040X},
mrclass = {49Q20 (53C21 58E12)},
mrnumber = {3606731},
mrreviewer = {M\'{a}rcio Fabiano da Silva},
doi = {10.4310/jdg/1486522816},
url = {https://doi.org/10.4310/jdg/1486522816},
zblnumber = {1367.53054},
} -
[Zhou17b] X. Zhou, "On the existence of min-max minimal surface of genus $g\geq 2$," Commun. Contemp. Math., vol. 19, iss. 4, p. 1750041, 2017.
@ARTICLE{Zhou17b,
author = {Zhou, Xin},
title = {On the existence of min-max minimal surface of genus {$g\geq 2$}},
journal = {Commun. Contemp. Math.},
fjournal = {Communications in Contemporary Mathematics},
volume = {19},
year = {2017},
number = {4},
pages = {1750041, 36},
issn = {0219-1997},
mrclass = {58E20 (49J35 58E12)},
mrnumber = {3665358},
mrreviewer = {Tian Xu},
doi = {10.1142/S0219199717500419},
url = {https://doi.org/10.1142/S0219199717500419},
zblnumber = {1369.49059},
} -
[Zhou-Zhu17] X. Zhou and J. J. Zhu, "Min-max theory for constant mean curvature hypersurfaces," Invent. Math., vol. 218, iss. 2, pp. 441-490, 2019.
@ARTICLE{Zhou-Zhu17,
author = {Zhou, Xin and Zhu, J. J.},
title = {Min-max theory for constant mean curvature hypersurfaces},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {218},
year = {2019},
number = {2},
pages = {441--490},
issn = {0020-9910},
mrclass = {53C42 (28A75 49J35 49Q15 53A15)},
mrnumber = {4011704},
mrreviewer = {John McCuan},
doi = {10.1007/s00222-019-00886-1},
url = {https://doi.org/10.1007/s00222-019-00886-1},
zblnumber = {1432.53086},
} -
[Zhou-Zhu18] X. Zhou and J. J. Zhu, "Existence of hypersurfaces with prescribed mean curvature I — generic min-max," Camb. J. Math., vol. 8, iss. 2, pp. 311-362, 2020.
@ARTICLE{Zhou-Zhu18,
author = {Zhou, Xin and Zhu, J. J.},
title = {Existence of hypersurfaces with prescribed mean curvature {I} --- generic min-max},
journal = {Camb. J. Math.},
fjournal = {Cambridge Journal of Mathematics},
volume = {8},
year = {2020},
number = {2},
pages = {311--362},
issn = {2168-0930},
mrclass = {58E12 (49Q05 53C42)},
mrnumber = {4091027},
doi = {10.4310/CJM.2020.v8.n2.a2},
url = {https://doi.org/10.4310/CJM.2020.v8.n2.a2},
zblnumber = {07192738},
}
Full Article