The Level-Set Flow of the Topologist’s Sine Curve is Smooth
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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.
KeywordsCurve shortening flow Level-set flow Mean curvature flow
Mathematics Subject Classification53C44
Casey Lam was supported by the Marianna Polonsky Slocum Memorial Fund.
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