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

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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.

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Acknowledgements

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

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Author notes

  1. Casey Lam

    Present address: Department of Astronomy, University of California at Berkeley, Berkeley, CA, 94720-3411, USA

  2. Joseph Lauer

    Present address: Department of Mathematics, Wellesley College, Wellesley, MA, 02481, USA

Authors and Affiliations

  1. Department of Mathematics, M.I.T., Cambridge, MA, 02139, USA

    Casey Lam

  2. Fachbereich Mathematik und Informatik, Freie Universität, 14195, Berlin, Germany

    Joseph Lauer

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  1. Casey Lam
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  2. Joseph Lauer
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Correspondence to Joseph Lauer.

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Lam, C., Lauer, J. The Level-Set Flow of the Topologist’s Sine Curve is Smooth. J Geom Anal 29, 1019–1031 (2019). https://doi.org/10.1007/s12220-017-9868-2

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  • DOI: https://doi.org/10.1007/s12220-017-9868-2

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