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The stability of m-fold circles in the curve shortening problem

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Abstract

The stability of m-fold circles in the curve shortening problem (CSP) is studied in this paper. It turns out that a suitable perturbation of m-fold circle will shrink to a point asymptotically like an m-fold circle under the curve shortening flow.

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References

  1. Abresch U., Langer J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196 (1986)

    MATH  MathSciNet  Google Scholar 

  2. Andrews B.: Evolving convex curves. Calc. Var. Partial Differ. Equ. 7, 315–371 (1998)

    Article  MATH  Google Scholar 

  3. Andrews B.: Classification of limiting shapes for isotropic curve flows. J. Am. Math. Soc. 16, 443–459 (2003)

    Article  MATH  Google Scholar 

  4. Angenent S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33, 601–633 (1991)

    MATH  MathSciNet  Google Scholar 

  5. Au T.K.: On the saddle point property of Abresch–Langer curves under the curve shortening flow. Commun. Anal. Geom. 18, 1–21 (2010)

    MATH  MathSciNet  Google Scholar 

  6. Chou K.S., Zhu X.P.: The Curve Shortening Problem. Chapman & Hall/CRC, Boca Raton, FL (2001)

    Book  MATH  Google Scholar 

  7. Epstein C.L., Weinstein M.I.: A stable manifold theorem for the curve shortening equation. Commun. Pure Appl. Math. 40, 119–139 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gage M., Hamilton R.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)

    MATH  MathSciNet  Google Scholar 

  9. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

  10. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, RI (1967)

  11. Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge, NJ (1996)

    MATH  Google Scholar 

  12. Lin T.C., Poon C.C., Tsai D.H.: Expanding convex immersed closed plane curves. Calc. Var. Partial Differ. Equ. 34, 153–178 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Xiao-Liu Wang.

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Wang, XL. The stability of m-fold circles in the curve shortening problem. manuscripta math. 134, 493–511 (2011). https://doi.org/10.1007/s00229-010-0410-0

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  • DOI: https://doi.org/10.1007/s00229-010-0410-0

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