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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some properties of the curve straightening flow in the plane

Author: Anders Linnér
Journal: Trans. Amer. Math. Soc. 314 (1989), 605-618
MSC: Primary 58E10; Secondary 53A04, 53C22, 58F17
MathSciNet review: 989580
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Abstract: We will explicitly compute the gradient of the total squared curvature functional on a space of closed curves. An example shows that the flow along the gradient trajectory may cause curves to develop self-intersections. We prove the existence of strictly convex curves that momentarily turn nonconvex. In conclusion we use computer graphics to illustrate how self-intersections come about.

References [Enhancements On Off] (What's this?)

  • [1] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology 24 (1985), 75-88. MR 790677 (86j:58023)
  • [2] M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geometry 23 (1986), 69-96. MR 840401 (87m:53003)
  • [3] M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geometry 26 (1987), 285-314. MR 906392 (89b:53005)
  • [4] M. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984), 357-364. MR 742856 (85i:52004)
  • [5] Å. Linder, Föreläsningar i statik, Institutionen för Mekanik, The Royal Institute of Technology, Stockholm, 1978,.
  • [6] R. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), pp. 19-30. MR 547524 (81c:58026)
  • [7] S. Buchin, Lectures on differential geometry, World Scientific, Singapore, 1980, 19-30. MR 664600 (83i:53002)

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Keywords: Curve straightening, total squared curvature, gradient flow
Article copyright: © Copyright 1989 American Mathematical Society

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