Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Mean curvature flow with generic initial data (2003.14344v2)

Published 31 Mar 2020 in math.DG and math.AP

Abstract: We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in $\mathbb{R}{4}$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (113)
  1. Unique asymptotics of compact ancient solutions to three-dimensional Ricci flow. Comm. Pure Appl. Math., 75(5):1032–1073, 2022.
  2. A computed example of nonuniqueness of mean curvature flow in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Comm. Partial Differential Equations, 20(11-12):1937–1958, 1995.
  3. Unique asymptotics of ancient convex mean curvature flow solutions. J. Differential Geom., 111(3):381–455, 2019.
  4. Uniqueness of two-convex closed ancient solutions to the mean curvature flow. Ann. of Math. (2), 192(2):353–436, 2020.
  5. Sigurd B. Angenent. Shrinking doughnuts. In Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), volume 7 of Progr. Nonlinear Differential Equations Appl., pages 21–38. Birkhäuser Boston, Boston, MA, 1992.
  6. Uniqueness of convex ancient solutions to mean curvature flow in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Invent. Math., 217(1):35–76, 2019.
  7. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. Geom. Topol., 25(5):2195–2234, 2021.
  8. Uniqueness of compact ancient solutions to three-dimensional Ricci flow. Invent. Math., 226(2):579–651, 2021.
  9. Mean curvature flow with surgery of mean convex surfaces in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Invent. Math., 203(2):615–654, 2016.
  10. Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J., 158(3):537–551, 2011.
  11. Gluing Eguchi-Hanson metrics and a question of Page. Comm. Pure Appl. Math., 70(7):1366–1401, 2017.
  12. On the multiplicity one conjecture for mean curvature flows of surfaces. https://arxiv.org/pdf/2312.02106.pdf, 2023.
  13. On the construction of closed nonconvex nonsoliton ancient mean curvature flows. Int. Math. Res. Not. IMRN, (1):757–768, 2021.
  14. Noncompact self-shrinkers for mean curvature flow with arbitrary genus. https://arxiv.org/abs/2110.06027, 2021.
  15. Kenneth A. Brakke. The motion of a surface by its mean curvature. ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–Princeton University.
  16. Simon Brendle. A sharp bound for the inscribed radius under mean curvature flow. Invent. Math., 202(1):217–237, 2015.
  17. Simon Brendle. Embedded self-similar shrinkers of genus 0. Ann. of Math. (2), 183(2):715–728, 2016.
  18. Simon Brendle. Ancient solutions to the Ricci flow in dimension 3. Acta Math., 225(1):1–102, 2020.
  19. Jacob Bernstein and Lu Wang. A sharp lower bound for the entropy of closed hypersurfaces up to dimension six. Invent. Math., 206(3):601–627, 2016.
  20. Jacob Bernstein and Lu Wang. A topological property of asymptotically conical self-shrinkers of small entropy. Duke Math. J., 166(3):403–435, 2017.
  21. Jacob Bernstein and Lu Wang. Hausdorff stability of the round two-sphere under small perturbations of the entropy. Math. Res. Lett., 25(2):347–365, 2018.
  22. Jacob Bernstein and Lu Wang. An integer degree for asymptotically conical self-expanders. https://arxiv.org/abs/1807.06494, 2018.
  23. Jacob Bernstein and Lu Wang. Topology of closed hypersurfaces of small entropy. Geom. Topol., 22(2):1109–1141, 2018.
  24. Jacob Bernstein and Lu Wang. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow. Int. Math. Res. Not. IMRN, (12):9016–9044, 2021.
  25. Jacob Bernstein and Lu Wang. The space of asymptotically conical self-expanders of mean curvature flow. Math. Ann., 380(1-2):175–230, 2021.
  26. Jacob Bernstein and Lu Wang. Closed hypersurfaces of low entropy in ℝ4superscriptℝ4\mathbb{R}^{4}blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT are isotopically trivial. Duke Math. J., 171(7):1531–1558, 2022.
  27. Jacob Bernstein and Lu Wang. Relative expander entropy in the presence of a two-sided obstacle and applications. Adv. Math., 399:Paper No. 108284, 48, 2022.
  28. Jacob Bernstein and Lu Wang. Topological uniqueness for self-expanders of small entropy. Camb. J. Math., 10(4):785–833, 2022.
  29. Mean curvature flow with generic low-entropy initial data. to appear in Duke Math. J., https://arxiv.org/pdf/2102.11978.pdf, 2021.
  30. Mean curvature flow with generic initial data II. https://arxiv.org/pdf/2302.08409.pdf, 2023.
  31. Mean curvature flow from conical singularities. https://arxiv.org/pdf/2312.00759.pdf, 2023.
  32. Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds. Comm. Pure Appl. Math., 74(4):865–905, 2021.
  33. Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness. Acta Math., 228(2):217–301, 2022.
  34. Ancient asymptotically cylindrical flows and applications. Invent. Math., 229(1):139–241, 2022.
  35. Quantitative stratification and the regularity of mean curvature flow. Geom. Funct. Anal., 23(3):828–847, 2013.
  36. David L. Chopp. Computation of self-similar solutions for mean curvature flow. Experiment. Math., 3(1):1–15, 1994.
  37. Rigidity of generic singularities of mean curvature flow. Publ. Math. Inst. Hautes Études Sci., 121:363–382, 2015.
  38. The round sphere minimizes entropy among closed self-shrinkers. J. Differential Geom., 95(1):53–69, 2013.
  39. Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I: generic singularities. Ann. of Math. (2), 175(2):755–833, 2012.
  40. Tobias H. Colding and William P. Minicozzi, II. Smooth compactness of self-shrinkers. Comment. Math. Helv., 87(2):463–475, 2012.
  41. Tobias H. Colding and William P. Minicozzi, II. Uniqueness of blowups and łojasiewicz inequalities. Ann. of Math. (2), 182(1):221–285, 2015.
  42. Tobias H. Colding and William P. Minicozzi, II. The singular set of mean curvature flow with generic singularities. Invent. Math., 204(2):443–471, 2016.
  43. Ancient gradient flows of elliptic functionals and Morse index. Amer. J. Math., 144(2):541–573, 2022.
  44. Generic regularity for minimizing hypersurfaces in dimensions 9 and 10. https://arxiv.org/pdf/2302.02253.pdf, 2023.
  45. Mean curvature flow with generic low-entropy initial data II. https://arxiv.org/pdf/2309.03856.pdf, 2023.
  46. Uniqueness of asymptotically conical tangent flows. Duke Math. J., 170(16):3601–3657, 2021.
  47. Bing-Long Chen and Le Yin. Uniqueness and pseudolocality theorems of the mean curvature flow. Comm. Anal. Geom., 15(3):435–490, 2007.
  48. Type II ancient compact solutions to the Yamabe flow. J. Reine Angew. Math., 738:1–71, 2018.
  49. Sulle singolarità eliminabili delle ipersuperficie minimali. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 38:352–357, 1965.
  50. Classification of compact ancient solutions to the curve shortening flow. J. Differential Geom., 84(3):455–464, 2010.
  51. Classification of ancient compact solutions to the Ricci flow on surfaces. J. Differential Geom., 91(2):171–214, 2012.
  52. Klaus Ecker. Logarithmic Sobolev inequalities on submanifolds of Euclidean space. J. Reine Angew. Math., 522:105–118, 2000.
  53. Klaus Ecker. Regularity theory for mean curvature flow, volume 57 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2004.
  54. Mean curvature evolution of entire graphs. Ann. of Math. (2), 130(3):453–471, 1989.
  55. Interior estimates for hypersurfaces moving by mean curvature. Invent. Math., 105(3):547–569, 1991.
  56. Mean convex mean curvature flow with free boundary. Comm. Pure Appl. Math., 75(4):767–817, 2022.
  57. Motion of level sets by mean curvature. I. J. Differential Geom., 33(3):635–681, 1991.
  58. Motion of level sets by mean curvature. IV. J. Geom. Anal., 5(1):77–114, 1995.
  59. Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.
  60. Robert Haslhofer and Or Hershkovits. Ancient solutions of the mean curvature flow. Comm. Anal. Geom., 24(3):593–604, 2016.
  61. Robert Haslhofer and Or Hershkovits. Singularities of mean convex level set flow in general ambient manifolds. Adv. Math., 329:1137–1155, 2018.
  62. Morris W. Hirsch. Differential topology. Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33.
  63. Mean curvature flow of mean convex hypersurfaces. Comm. Pure Appl. Math., 70(3):511–546, 2017.
  64. Mean curvature flow with surgery. Duke Math. J., 166(9):1591–1626, 2017.
  65. Area minimizing hypersurfaces with isolated singularities. J. Reine Angew. Math., 362:102–129, 1985.
  66. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math., 183(1):45–70, 1999.
  67. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations, 8(1):1–14, 1999.
  68. Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math., 175(1):137–221, 2009.
  69. Convex ancient solutions of the mean curvature flow. J. Differential Geom., 101(2):267–287, 2015.
  70. Gerhard Huisken. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom., 20(1):237–266, 1984.
  71. Gerhard Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 31(1):285–299, 1990.
  72. Gerhard Huisken. Local and global behaviour of hypersurfaces moving by mean curvature. In Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages 175–191. Amer. Math. Soc., Providence, RI, 1993.
  73. Or Hershkovits and Brian White. Sharp entropy bounds for self-shrinkers in mean curvature flow. Geom. Topol., 23(3):1611–1619, 2019.
  74. Or Hershkovits and Brian White. Nonfattening of mean curvature flow at singularities of mean convex type. Comm. Pure Appl. Math., 73(3):558–580, 2020.
  75. Or Hershkovits and Brian White. Avoidance for set-theoretic solutions of mean-curvature-type flows. Comm. Anal. Geom., 31(1):31–67, 2023.
  76. Tom Ilmanen. The level-set flow on a manifold. In Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages 193–204. Amer. Math. Soc., Providence, RI, 1993.
  77. Tom Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc., 108(520):x+90, 1994.
  78. Tom Ilmanen. Lectures on mean curvature flow and related equations (Trieste notes). https://people.math.ethz.ch/~ilmanen/papers/notes.ps, 1995.
  79. Tom Ilmanen. Singularities of mean curvature flow of surfaces. https://people.math.ethz.ch/~ilmanen/papers/sing.ps, 1995.
  80. T. Ilmanen. A strong maximum principle for singular minimal hypersurfaces. Calc. Var. Partial Differential Equations, 4(5):443–467, 1996.
  81. Tom Ilmanen. Problems in mean curvature flow. https://people.math.ethz.ch/~ilmanen/classes/eil03/problems03.ps, 2003.
  82. On short time existence for the planar network flow. J. Differential Geom., 111(1):39–89, 2019.
  83. Sharp lower bounds on density for area-minimizing cones. Camb. J. Math., 3(1-2):1–18, 2015.
  84. Daniel Ketover. Self-shrinking platonic solids. https://arxiv.org/abs/1602.07271, 2016.
  85. Mean curvature self-shrinkers of high genus: non-compact examples. J. Reine Angew. Math., 739:1–39, 2018.
  86. Singular Ricci flows I. Acta Math., 219(1):65–134, 2017.
  87. Barry F. Knerr. Parabolic interior Schauder estimates by the maximum principle. Arch. Rational Mech. Anal., 75(1):51–58, 1980/81.
  88. Longzhi Lin. Mean curvature flow of star-shaped hypersurfaces. Comm. Anal. Geom., 28(6):1315–1336, 2020.
  89. Min-max theory and the Willmore conjecture. Ann. of Math. (2), 179(2):683–782, 2014.
  90. Xuan Hien Nguyen. Construction of complete embedded self-similar surfaces under mean curvature flow, Part III. Duke Math. J., 163(11):2023–2056, 2014.
  91. Felix Schulze. Uniqueness of compact tangent flows in mean curvature flow. J. Reine Angew. Math., 690:163–172, 2014.
  92. Natasa Sesum. Rate of convergence of the mean curvature flow. Comm. Pure Appl. Math., 61(4):464–485, 2008.
  93. Leon Simon. Schauder estimates by scaling. Calc. Var. Partial Differential Equations, 5(5):391–407, 1997.
  94. Nathan Smale. Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds. Comm. Anal. Geom., 1(2):217–228, 1993.
  95. Knut Smoczyk. Starshaped hypersurfaces and the mean curvature flow. Manuscripta Math., 95(2):225–236, 1998.
  96. Ao Sun. Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces. J. Differential Geom., 124(1):169–198, 2023.
  97. A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals. Indiana Univ. Math. J., 38(3):683–691, 1989.
  98. A local regularity theorem for mean curvature flow with triple edges. J. Reine Angew. Math., 758:281–305, 2020.
  99. Ao Sun and Zhichao Wang. Compactness of self-shrinkers in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT with fixed genus. Adv. Math., 367:107110, 39, 2020.
  100. Xu-Jia Wang. Convex solutions to the mean curvature flow. Ann. of Math. (2), 173(3):1185–1239, 2011.
  101. Lu Wang. Uniqueness of self-similar shrinkers with asymptotically conical ends. J. Amer. Math. Soc., 27(3):613–638, 2014.
  102. Lu Wang. Asymptotic structure of self-shrinkers. https://arxiv.org/abs/1610.04904, 2016.
  103. Brian White. The topology of hypersurfaces moving by mean curvature. Comm. Anal. Geom., 3(1-2):317–333, 1995.
  104. Brian White. Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math., 488:1–35, 1997.
  105. Brian White. The size of the singular set in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 13(3):665–695, 2000.
  106. Brian White. Evolution of curves and surfaces by mean curvature. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 525–538. Higher Ed. Press, Beijing, 2002.
  107. Brian White. The nature of singularities in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 16(1):123–138, 2003.
  108. Brian White. A local regularity theorem for mean curvature flow. Ann. of Math. (2), 161(3):1487–1519, 2005.
  109. Brian White. Currents and flat chains associated to varifolds, with an application to mean curvature flow. Duke Math. J., 148(1):41–62, 2009.
  110. Brian White. Topological change in mean convex mean curvature flow. Invent. Math., 191(3):501–525, 2013.
  111. Brian White. Subsequent singularities in mean-convex mean curvature flow. Calc. Var. Partial Differential Equations, 54(2):1457–1468, 2015.
  112. Brian White. Mean curvature flow with boundary. Ars Inven. Anal., pages Paper No. 4, 43, 2021.
  113. Jonathan J. Zhu. On the entropy of closed hypersurfaces and singular self-shrinkers. J. Differential Geom., 114(3):551–593, 2020.
Citations (46)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets