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The shape of a figure-eight under the curve shortening flow. (English) Zbl 0669.53003
Main Theorem: The isoperimetric ratio $L\sp 2/A$ converges to infinity under the curve shortening flow if and only if the loops of the original figure-eight curve bound regions of equal area.
Reviewer: D.Ferus

53A04Curves in Euclidean space
Full Text: DOI EuDML
[1] Abresch, U., Länger, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom.23, 175-196 (1986) · Zbl 0592.53002
[2] Angenent, S.: The zeroset of a solution of a parabolic equation. Preprint · Zbl 0644.35050
[3] Epstein, C., Weinstein, M.: A stable manifold theorem for the curve shortening equation. Commun. Pure Appl. Math.40, 119-139 (1987) · Zbl 0602.34026 · doi:10.1002/cpa.3160400106
[4] Gage, M.: An isoperimetric inequality with application to curve shortening. Duke Math. J.50, 1225-1229 (1983) · Zbl 0534.52008 · doi:10.1215/S0012-7094-83-05052-4
[5] Gage, M.: Curve shortening makes convex curves circular. Invent. Math.76, 357-364 (1984) · Zbl 0542.53004 · doi:10.1007/BF01388602
[6] Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom.23, 69-96 (1986) · Zbl 0621.53001
[7] Grayson, M.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom.26, 285-314 (1987) · Zbl 0667.53001
[8] Grayson, M.: Shortening embedded curves. Ann. Math. (to appear) · Zbl 0686.53036