Area-preserving evolution of nonsimple symmetric plane curves

Abstract

The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles, while for Abresch–Langer type curves, the convergence to m-fold circles happens if and only if the enclosed algebraic area is positive.

This is a preview of subscription content, access via your institution.

References

  1. 1.

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

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Andrews B.: Evolving convex curves. Calc. Var. Partial Differential Equations 7, 315–371 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

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

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Angenent S.B., Velázquez J.J.L.: Asymptotic shape of cusp singularities in curve shortening. Duke Math. J. 77, 71–110 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

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

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Chou K.S.: A blow-up criterion for the curve shortening flow by surface diffusion. Hokkaido mathematical journal 32, 1–19 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

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

  8. 8.

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

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Escher J., Ito K.: Some dynamic properties of volume preserving curvature driven flows. Mathematische Annalen 333, 213–230 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    C.L. Epstein, M.E. Gage, The curve shortening flow, in: Wave Motion: Theory, Modelling, and Computation, Proc. Conf. Hon. 60th Birthday P.D.Lax, Publ. Math. Sci. Res. Inst. 7(1987) 15–59.

  11. 11.

    Gage M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50, 1225–1229 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    M.E. Gage, On an area-preserving evolution equation for plane curves, Nonlinear problems in geometry (Mobile, Ala., 1985), 51–62, Contemp. Math., 51, Amer. Math. Soc., Providence, RI, 1986.

  13. 13.

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

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Grayson M.A.: The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26, 285–314 (1987)

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Huisken G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 34–48 (1987)

    MathSciNet  Google Scholar 

  16. 16.

    Jiang L.S., Pan S.L.: On a non-local curve evolution problem in the plane. Comm. Anal. Geom. 16, 1–26 (2008)

    Article  MathSciNet  Google Scholar 

  17. 17.

    O. A. Ladyženskaja & V. A Solonnikov & N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967.

  18. 18.

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

  19. 19.

    Y.C. Lin, D.H. Tsai, Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves, J. of Evolution Equations, 2012.

  20. 20.

    Y.C. Lin, D.H. Tsai, On a simple maximum principle technique applied to equations on the circle, J. Diff. Eq. 245(2008) 377–391.

    Google Scholar 

  21. 21.

    Ma L., Zhu A.Q.: On a length preserving curve flow. Monatshefte Für Mathematik, 165, 57–78 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Chen X.Y., Matano H.: Convergence, asymptotic periodicity, and finite point blow-up in one-dimensional semilinear heat equations. J. Diff. Eq. 78, 160–190 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    McCoy J.: The surface area preserving mean curvature flow. Asian J. of Math. 7, 7–30 (2003)

    MATH  MathSciNet  Google Scholar 

  24. 24.

    Oaks J.A.: Singularities and self-intersections of curves evolving on surfaces. Indiana Univ. Math. J. 43, 959–981 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Pan S.L., Yang J.N.: On a non-local perimeter-preserving curve evolution problem for convex plane curves. Manuscripta Math. 127, 469–484 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Pan S.L., Zhang H.: On a curve expanding flow with a nonlocal term, Comm. Contemp. Math. 12, 815–829 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Sapiro G, Tannenbaum A.: Area and length preserving geometric invariant scale-spaces, Pattern Analysis and Machine Intelligence. IEEE Transactions, 17, 67–72 (1995)

    Google Scholar 

  28. 28.

    Tso K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math. 38, 867–882 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Wang X.L.: The stability of m-fold circles in the curve shortening problem. Manuscripta Math. 134, 493–511 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiao-Liu Wang.

Additional information

This work is partially supported by the National Natural Science Foundation of China 11101078, 11171064 and the Education Department Program of Liaoning Province L2010068.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wang, XL., Kong, LH. Area-preserving evolution of nonsimple symmetric plane curves. J. Evol. Equ. 14, 387–401 (2014). https://doi.org/10.1007/s00028-014-0219-5

Download citation

Mathematics Subject Classification (2010)

  • Primary 53C44
  • 35B40
  • Secondary 35K59
  • 37B25

Keywords

  • Mean curvature flow
  • nonlocal parabolic equation
  • asymptotic behavior