The Level-Set Flow of the Topologist’s Sine Curve is Smooth

  • Casey Lam
  • Joseph Lauer


In this note we prove that the level-set flow of the topologist’s sine curve is a smooth closed curve. In Lauer (Geom Funct Anal 23(6): 1934–1961, 2013) it was shown by the second author that under the level-set flow, a locally connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we give the first example of a domain whose boundary is not locally connected for which the level-set flow is instantaneously smooth. Our methods also produce an example of a nonpath-connected set that instantly evolves into a smooth closed curve.


Curve shortening flow Level-set flow Mean curvature flow 

Mathematics Subject Classification




Casey Lam was supported by the Marianna Polonsky Slocum Memorial Fund.


  1. 1.
    Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33(3), 601–633 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brakke, K.: Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton, NJ (1978)zbMATHGoogle Scholar
  3. 3.
    Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749–786 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Clutterbuck, J.: Parabolic equations with continuous initial data, Ph.D. thesis (2004)Google Scholar
  5. 5.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. (2) 130(3), 453–471 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by their mean curvature. Invent. Math. 105, 547–569 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635–681 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Falconer, K.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)Google Scholar
  9. 9.
    Gage, M., Hamilton, R.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hershkovits, O.: Mean curvature flow of Reifenberg sets. arXiv:1412.4799v3
  11. 11.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108(520), x+90 (1994)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lauer, J.: A new length estimate for curve shortening flow and rough initial data. Geom. Funct. Anal. 23(6), 1934–1961 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsM.I.T.CambridgeUSA
  2. 2.Department of AstronomyUniversity of California at BerkeleyBerkeleyUSA
  3. 3.Fachbereich Mathematik und InformatikFreie UniversitätBerlinGermany
  4. 4.Department of MathematicsWellesley CollegeWellesleyUSA

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