Skip to main content
SpringerLink
Log in

On a non-local perimeter-preserving curve evolution problem for convex plane curves

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

This paper deals with a non-local evolution problem for closed convex plane curves which preserves the perimeter of the evolving curve but enlarges the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle in the C ∞ metric as the time t goes to infinity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews B.: Evolving convex curves. Calc. Var. PDE’s 7, 315–371 (1998)

    Article  MATH  Google Scholar 

  2. Athanassenas M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72, 52–66 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonnesen T., Fenchel W.: Theorie der Convexen Körper. Chelsea, New York (1948)

    Google Scholar 

  4. Chou K.S., Zhu X.P.: The Curve Shortening Problem. CRC Press, Boca Raton (2001)

    MATH  Google Scholar 

  5. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Science Press/American Mathematical Society, Beijing/Providence (2006)

  6. Ecker, K.: Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser, Boston (2004)

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Gage M.E.: Curve shortening makes convex curves circular. Invent. Math. 76, 357–364 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gage, M.E.: On an area-preserving evolution equation for plane curves. In: DeTurck, D.M. (ed.) Nonlinear Problems in Geometry, Contemp. Math, vol. 51, pp. 51–62 (1986)

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

    MATH  MathSciNet  Google Scholar 

  11. Grayson M.: The heat equation shrinks embedded plane curve to round points. J. Diff. Geom. 26, 285–314 (1987)

    MATH  MathSciNet  Google Scholar 

  12. Green M., Osher S.: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 3, 659–676 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Huisken G.: Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom. 20, 237–266 (1984)

    MATH  MathSciNet  Google Scholar 

  14. Huisken G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)

    MATH  MathSciNet  Google Scholar 

  15. Ma L., Chen D.Z.: Curve shortening in a Riemannian manifold. Ann. Mat. Pura Appl. 186, 663–684 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mayer U.F.: A singular example for the averaged mean curvature flow. Exp. Math. 10, 103–107 (2001)

    MATH  Google Scholar 

  17. Mayer U.F., Simonett G.: Self-intersections for the surface diffusion and the volume-preserving mean curvature flow. Diff. Integ. Equ. 13, 1189–1199 (2000)

    MATH  MathSciNet  Google Scholar 

  18. McCoy J.A.: The surface area preserving mean curvature flow. Asian J. Math. 7(1), 7–30 (2003)

    MATH  MathSciNet  Google Scholar 

  19. McCoy J.A.: Mixed volume preserving curvature flows. Calc. Var. PDEs 24(2), 131–154 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Osserman R.: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86, 1–29 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pan S.L.: A note on the general curve flows. J. Math. Stud. 33, 17–26 (2000)

    MATH  Google Scholar 

  22. Schneider R.: Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  23. Tsai D.H.: On the formation of singularities in the curve expanding flow. Calc. Var. Partial Diff. Equ. 14(3), 385–398 (2002)

    Article  MATH  Google Scholar 

  24. Tsai D.H.: Asymptotic closeness to limiting shapes for expanding embedded plane curves. Invent. Math. 162, 473–492 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Urbas J.I.E.: Convex curves moving homothetically by negative powers of their curvature. Asian J. Math. 3, 635–658 (1990)

    MathSciNet  Google Scholar 

  27. White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the ICM 2002, vol. I, pp. 525–538

  28. Zhu, X.P.: Lectures on mean curvature flows. AMS/IP Studies in Advanced Mathematics, vol. 32. American Mathematical Society/International Press, Providence/Somerville (2002)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East China Normal University, 200062, Shanghai, People’s Republic of China

    Shengliang Pan & Juanna Yang

Authors
  1. Shengliang Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juanna Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengliang Pan.

Additional information

The first author is supported in part by the National Science Foundation of China (No.10671066) and the Shanghai Leading Academic Discipline Project (No. B407).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pan, S., Yang, J. On a non-local perimeter-preserving curve evolution problem for convex plane curves. manuscripta math. 127, 469–484 (2008). https://doi.org/10.1007/s00229-008-0211-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-008-0211-x

Mathematics Subject Classification (2000)

Search

Navigation