Embedded self-similar shrinkers of genus $0$

Abstract

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in $\mathbb{R}^3$ with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

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  • [Angenent] Go to document S. B. Angenent, "Shrinking doughnuts," in Nonlinear Diffusion Equations and Their Equilibrium States, 3, Boston: Birkhäuser, 1992, vol. 7, pp. 21-38.
    @incollection{Angenent, mrkey = {1167827},
      author = {Angenent, Sigurd B.},
      title = {Shrinking doughnuts},
      booktitle = {Nonlinear Diffusion Equations and Their Equilibrium States, 3},
      venue = {{G}regynog, 1989},
      series = {Progr. Nonlinear Differential Equations Appl.},
      volume = {7},
      pages = {21--38},
      publisher = {Birkhäuser},
      address = {Boston},
      year = {1992},
      mrclass = {58E12 (35B99 53A10)},
      mrnumber = {1167827},
      mrreviewer = {Anders Linn{é}r},
      zblnumber = {0762.53028},
      DOI = {10.1007/978-1-4612-0393-3_2},
     }
  • [Brakke] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton, N.J.: Princeton Univ. Press, 1978, vol. 20.
    @book{Brakke, mrkey = {0485012},
      author = {Brakke, Kenneth A.},
      title = {The Motion of a Surface by its Mean Curvature},
      series = {Mathematical Notes},
      volume = {20},
      publisher = {Princeton Univ. Press},
      address = {Princeton, N.J.},
      year = {1978},
      pages = {i+252},
      isbn = {0-691-08204-9},
      mrclass = {49F22 (35K99 49F20 58D25)},
      mrnumber = {0485012},
      mrreviewer = {Jean E. Taylor},
      zblnumber = {0386.53047},
      }
  • [Brendle1] Go to document S. Brendle, "Rotational symmetry of self-similar solutions to the Ricci flow," Invent. Math., vol. 194, iss. 3, pp. 731-764, 2013.
    @article{Brendle1, mrkey = {3127066},
      author = {Brendle, Simon},
      title = {Rotational symmetry of self-similar solutions to the {R}icci flow},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {194},
      year = {2013},
      number = {3},
      pages = {731--764},
      issn = {0020-9910},
      mrclass = {53C44},
      mrnumber = {3127066},
      mrreviewer = {Ye Li},
      doi = {10.1007/s00222-013-0457-0},
      zblnumber = {1284.53044},
      }
  • [Brendle2] Go to document S. Brendle, "Rotational symmetry of Ricci solitons in higher dimensions," J. Differential Geom., vol. 97, iss. 2, pp. 191-214, 2014.
    @article{Brendle2, mrkey = {3231974},
      author = {Brendle, Simon},
      title = {Rotational symmetry of {R}icci solitons in higher dimensions},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {97},
      year = {2014},
      number = {2},
      pages = {191--214},
      issn = {0022-040X},
      mrclass = {53C40 (53C20 53C25)},
      mrnumber = {3231974},
      mrreviewer = {Xiang Gao},
      url = {http://projecteuclid.org/euclid.jdg/1405447804},
      zblnumber = {1304.53042},
      }
  • [Brendle3] Go to document S. Brendle, "Embedded minimal tori in $S^3$ and the Lawson conjecture," Acta Math., vol. 211, iss. 2, pp. 177-190, 2013.
    @article{Brendle3, mrkey = {3143888},
      author = {Brendle, Simon},
      title = {Embedded minimal tori in {$S\sp 3$} and the {L}awson conjecture},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {211},
      year = {2013},
      number = {2},
      pages = {177--190},
      issn = {0001-5962},
      mrclass = {53A10 (49Q05 53C42)},
      mrnumber = {3143888},
      mrreviewer = {Jo{ã}o Lucas Marques Barbosa},
      doi = {10.1007/s11511-013-0101-2},
      zblnumber = {1305.53061},
      }
  • [Brendle4] Go to document S. Brendle, "Two-point functions and their applications in geometry," Bull. Amer. Math. Soc., vol. 51, iss. 4, pp. 581-596, 2014.
    @article{Brendle4, mrkey = {3237760},
      author = {Brendle, Simon},
      title = {Two-point functions and their applications in geometry},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      volume = {51},
      year = {2014},
      number = {4},
      pages = {581--596},
      issn = {0273-0979},
      mrclass = {53C44 (53A10 53C42)},
      mrnumber = {3237760},
      mrreviewer = {James McCoy},
      doi = {10.1090/S0273-0979-2014-01461-2},
      zblnumber = {06377771},
      }
  • [Cheng] Go to document S. Y. Cheng, "Eigenfunctions and nodal sets," Comment. Math. Helv., vol. 51, iss. 1, pp. 43-55, 1976.
    @article{Cheng, mrkey = {0397805},
      author = {Cheng, Shiu Yuen},
      title = {Eigenfunctions and nodal sets},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {51},
      year = {1976},
      number = {1},
      pages = {43--55},
      issn = {0010-2571},
      mrclass = {58G99 (35P15)},
      mrnumber = {0397805},
      mrreviewer = {Shukichi Tanno},
      doi = {10.1007/BF02568142},
      zblnumber = {0334.35022},
      }
  • [Colding-Ilmanen-Minicozzi-White] Go to document T. H. Colding, T. Ilmanen, W. P. Minicozzi II, and B. White, "The round sphere minimizes entropy among closed self-shrinkers," J. Differential Geom., vol. 95, iss. 1, pp. 53-69, 2013.
    @article{Colding-Ilmanen-Minicozzi-White, mrkey = {3128979},
      author = {Colding, Tobias H. and Ilmanen, Tom and Minicozzi II, William P. and White, Brian},
      title = {The round sphere minimizes entropy among closed self-shrinkers},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {95},
      year = {2013},
      number = {1},
      pages = {53--69},
      issn = {0022-040X},
      mrclass = {53C44 (53A07 53C21)},
      mrnumber = {3128979},
      mrreviewer = {Alina Stancu},
      url = {http://projecteuclid.org/euclid.jdg/1375124609},
      zblnumber = {1278.53069},
      }
  • [Colding-Minicozzi] Go to document T. H. Colding and W. P. Minicozzi II, "Generic mean curvature flow I: generic singularities," Ann. of Math., vol. 175, iss. 2, pp. 755-833, 2012.
    @article{Colding-Minicozzi, mrkey = {2993752},
      author = {Colding, Tobias H. and Minicozzi II, William P.},
      title = {Generic mean curvature flow {I}: generic singularities},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {175},
      year = {2012},
      number = {2},
      pages = {755--833},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53C44 (35K55 35K99 53A10 53C42 58E30 58K60)},
      mrnumber = {2993752},
      mrreviewer = {Paul Bryan},
      doi = {10.4007/annals.2012.175.2.7},
      zblnumber = {1239.53084},
      }
  • [Colding-Minicozzi-Pedersen] Go to document T. H. Colding, W. P. Minicozzi II, and E. K. Pedersen, "Mean curvature flow," Bull. Amer. Math. Soc., vol. 52, iss. 2, pp. 297-333, 2015.
    @article{Colding-Minicozzi-Pedersen, mrkey = {3312634},
      author = {Colding, Tobias H. and Minicozzi II, William P. and Pedersen, Erik Kj{æ}r},
      title = {Mean curvature flow},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {American Mathematical Society. Bulletin. New Series},
      volume = {52},
      year = {2015},
      number = {2},
      pages = {297--333},
      issn = {0273-0979},
      mrclass = {53C44},
      mrnumber = {3312634},
      doi = {10.1090/S0273-0979-2015-01468-0},
      zblnumber = {06431061},
     }
  • [Drugan] Go to document G. Drugan, "An immersed $S^2$ self-shrinker," Trans. Amer. Math. Soc., vol. 367, iss. 5, pp. 3139-3159, 2015.
    @article{Drugan, mrkey = {3314804},
      author = {Drugan, Gregory},
      title = {An immersed {$S\sp 2$} self-shrinker},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {367},
      year = {2015},
      number = {5},
      pages = {3139--3159},
      issn = {0002-9947},
      mrclass = {53C44 (53C42)},
      mrnumber = {3314804},
      mrreviewer = {Jun Sun},
      doi = {10.1090/S0002-9947-2014-06051-0},
      zblnumber = {06429009},
     }
  • [Ecker] Go to document K. Ecker, Regularity Theory for Mean Curvature Flow, Boston: Birkhäuser, 2004, vol. 57.
    @book{Ecker, mrkey = {2024995},
      author = {Ecker, Klaus},
      title = {Regularity Theory for Mean Curvature Flow},
      series = {Progr. Nonlinear Differential Equations Appl.},
      volume = {57},
      publisher = {Birkhäuser},
      address = {Boston},
      year = {2004},
      pages = {xiv+165},
      isbn = {0-8176-3243-3},
      mrclass = {53C44 (35-02 35B65 35K55)},
      mrnumber = {2024995},
      mrreviewer = {Xi Ping Zhu},
      doi = {10.1007/978-0-8176-8210-1},
      zblnumber = {1058.53054},
      }
  • [Ecker-Huisken] Go to document K. Ecker and G. Huisken, "Mean curvature evolution of entire graphs," Ann. of Math., vol. 130, iss. 3, pp. 453-471, 1989.
    @article{Ecker-Huisken, mrkey = {1025164},
      author = {Ecker, Klaus and Huisken, Gerhard},
      title = {Mean curvature evolution of entire graphs},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {130},
      year = {1989},
      number = {3},
      pages = {453--471},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53A10 (53C45)},
      mrnumber = {1025164},
      mrreviewer = {S. Walter Wei},
      doi = {10.2307/1971452},
      zblnumber = {0696.53036},
      }
  • [Ferrer-Martin-Meeks] Go to document L. Ferrer, F. Mart’in, and W. H. Meeks III, "Existence of proper minimal surfaces of arbitrary topological type," Adv. Math., vol. 231, iss. 1, pp. 378-413, 2012.
    @article{Ferrer-Martin-Meeks, mrkey = {2935393},
      author = {Ferrer, Leonor and Mart{\'ı}n, Francisco and Meeks III, William H.},
      title = {Existence of proper minimal surfaces of arbitrary topological type},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {231},
      year = {2012},
      number = {1},
      pages = {378--413},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {53A10},
      mrnumber = {2935393},
      mrreviewer = {Fei-Tsen Liang},
      doi = {10.1016/j.aim.2012.05.007},
      zblnumber = {1246.53006},
      }
  • [Huisken1] Go to document G. Huisken, "Asymptotic behavior for singularities of the mean curvature flow," J. Differential Geom., vol. 31, iss. 1, pp. 285-299, 1990.
    @article{Huisken1, mrkey = {1030675},
      author = {Huisken, Gerhard},
      title = {Asymptotic behavior for singularities of the mean curvature flow},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {31},
      year = {1990},
      number = {1},
      pages = {285--299},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {53A10 (35B99 53C45 58G11)},
      mrnumber = {1030675},
      mrreviewer = {Dennis M. DeTurck},
      url = {http://projecteuclid.org/euclid.jdg/1214444099},
      zblnumber = {0694.53005},
      }
  • [Huisken2] Go to document G. Huisken, "Local and global behaviour of hypersurfaces moving by mean curvature," in Differential Geometry: Partial Differential Equations on Manifolds, Providence, RI: Amer. Math. Soc., 1993, vol. 54, pp. 175-191.
    @incollection{Huisken2, mrkey = {1216584},
      author = {Huisken, Gerhard},
      title = {Local and global behaviour of hypersurfaces moving by mean curvature},
      booktitle = {Differential Geometry: Partial Differential Equations on Manifolds},
      venue = {{L}os {A}ngeles, {CA},
      1990},
      series = {Proc. Sympos. Pure Math.},
      volume = {54},
      pages = {175--191},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1993},
      mrclass = {58E15 (53A10 58G11)},
      mrnumber = {1216584},
      mrreviewer = {Li Ma},
      doi = {10.1090/pspum/054.1/1216584},
      zblnumber = {0791.58090},
      }
  • [Ilmanen] Go to document T. Ilmanen, Problems in mean curvature flow.
    @misc{Ilmanen,
      author = {Ilmanen, T.},
      title = {Problems in mean curvature flow},
      url = {http://people.math.ethz.ch/~ilmanen/classes/eil03/problems03.pdf},
      }
  • [Kapouleas-Kleene-Moller] Go to document N. Kapouleas, S. J. Kleene, and N. M. Møller, Mean curvature self-shrinkers of high genus: Non-compact examples.
    @misc{Kapouleas-Kleene-Moller,
      author = {Kapouleas, N. and Kleene, S. J. and Møller, N. M.},
      title = {Mean curvature self-shrinkers of high genus: {N}on-compact examples},
      doi = {10.1515/crelle-2015-0050},
      note = {published online 2015-10-14},
      }
  • [Lawson] Go to document B. H. Lawson Jr., "The unknottedness of minimal embeddings," Invent. Math., vol. 11, pp. 183-187, 1970.
    @article{Lawson, mrkey = {0287447},
      author = {Lawson, Jr., H. Blaine},
      title = {The unknottedness of minimal embeddings},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {11},
      year = {1970},
      pages = {183--187},
      issn = {0020-9910},
      mrclass = {53.04 (57.00)},
      mrnumber = {0287447},
      mrreviewer = {H. B. Griffiths},
      doi = {10.1007/BF01404649},
      zblnumber = {0205.52002},
      }
  • [Meeks-Yau] Go to document W. W. Meeks III and S. T. Yau, "The existence of embedded minimal surfaces and the problem of uniqueness," Math. Z., vol. 179, iss. 2, pp. 151-168, 1982.
    @article{Meeks-Yau, mrkey = {0645492},
      author = {Meeks, III, William W. and Yau, Shing Tung},
      title = {The existence of embedded minimal surfaces and the problem of uniqueness},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {179},
      year = {1982},
      number = {2},
      pages = {151--168},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {53C42 (49F10 53A10 58E12)},
      mrnumber = {0645492},
      mrreviewer = {R. Osserman},
      doi = {10.1007/BF01214308},
      zblnumber = {0479.49026},
      }
  • [Ros] Go to document A. Ros, "A two-piece property for compact minimal surfaces in a three-sphere," Indiana Univ. Math. J., vol. 44, iss. 3, pp. 841-849, 1995.
    @article{Ros, mrkey = {1375352},
      author = {Ros, Antonio},
      title = {A two-piece property for compact minimal surfaces in a three-sphere},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
      volume = {44},
      year = {1995},
      number = {3},
      pages = {841--849},
      issn = {0022-2518},
      coden = {IUMJAB},
      mrclass = {53A10},
      mrnumber = {1375352},
      doi = {10.1512/iumj.1995.44.2011},
      zblnumber = {0861.53009},
      }
  • [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, mrkey = {0634285},
      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},
      coden = {CPAMAT},
      mrclass = {49F22 (53C42 58E15)},
      mrnumber = {0634285},
      mrreviewer = {F. J. Almgren, Jr.},
      doi = {10.1002/cpa.3160340603},
      zblnumber = {0497.49034},
      }
  • [Wang1] Go to document L. Wang, "Uniqueness of self-similar shrinkers with asymptotically conical ends," J. Amer. Math. Soc., vol. 27, iss. 3, pp. 613-638, 2014.
    @article{Wang1, mrkey = {3194490},
      author = {Wang, Lu},
      title = {Uniqueness of self-similar shrinkers with asymptotically conical ends},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {27},
      year = {2014},
      number = {3},
      pages = {613--638},
      issn = {0894-0347},
      mrclass = {53C44 (35B60 35J93 53C24)},
      mrnumber = {3194490},
      mrreviewer = {Zhiyuan Xu},
      doi = {10.1090/S0894-0347-2014-00792-X},
      zblnumber = {1298.53069},
      }
  • [Wang2] Go to document X. Wang, "Convex solutions to the mean curvature flow," Ann. of Math., vol. 173, iss. 3, pp. 1185-1239, 2011.
    @article{Wang2, mrkey = {2800714},
      author = {Wang, Xu-Jia},
      title = {Convex solutions to the mean curvature flow},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {3},
      pages = {1185--1239},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {53C44 (35J60 35J93)},
      mrnumber = {2800714},
      mrreviewer = {James McCoy},
      doi = {10.4007/annals.2011.173.3.1},
      zblnumber = {1231.53058},
      }
  • [White] Go to document B. White, Boundary behavior in mean curvature flow.
    @misc{White,
      author = {White, B.},
      title = {Boundary behavior in mean curvature flow},
      note = {{O}berwolfach {R}eport 33/2014, pp.~1853--1855, {E}uropean {M}ath. {S}oc.},
      url = {http://www.ems-ph.org/journals/abstract/OWR/2014-011-003/2014-011-003-03.pdf},
     }

Authors

Simon Brendle

Stanford University, Stanford, CA

Current address:

Columbia University, New York, NY