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

  • Casey Lam
  • Joseph Lauer
Article
  • 18 Downloads

Abstract

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.

Keywords

Curve shortening flow Level-set flow Mean curvature flow 

Mathematics Subject Classification

53C44 

Notes

Acknowledgements

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

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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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