Skip to main content
SpringerLink
Log in

A non-local area preserving curve flow

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in \(C^{\infty }\) sense as time goes into 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. Chou, K.S., Zhu, X.P.: The Curve Shortening Problem. CRC Press, Boca Raton (2001)

    Book  MATH  Google Scholar 

  3. Chow, B., Tsai, D.H.: Geometric expansion of convex plane curves. J. Differ. Geom. 44, 312–330 (1996)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

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

    MATH  MathSciNet  Google Scholar 

  8. Gage, M., Li, Y.: Evolving plane curves by curvature in relative geometries. II. Duke Math. J. 75(1), 79–98 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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 

  10. Grayson, M.: The heat equation shrinks embeded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Ma, L.: B-minimal sub-manifolds and their stability. Math. Nach 279, 1597–1601 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Ma, L., Yang, Y.: A remark on soliton equation of mean curvature flow. Anais Acad. Brasileira de Ciencias 76, 467–473 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ma, L., Zhu, A.Q.: On a length preserving curve flow. Monatsh Math. 165, 57–78 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. De Miguel, G.: Differentiation of integrals in Rn. In: Lectures in Mathematics, vol. 481. Springer (1977)

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

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

    MATH  Google Scholar 

  20. 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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhongyuan Institute of Mathematics, Henan Normal university, Xinxiang, 453007, People’s Republic of China

    Li Ma

  2. School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, People’s Republic of China

    Liang Cheng

Authors
  1. Li Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liang Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Ma.

Additional information

The research is partially supported by the National Natural Science Foundation of China No.11271111 and SRFDP 20090002110019.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, L., Cheng, L. A non-local area preserving curve flow. Geom Dedicata 171, 231–247 (2014). https://doi.org/10.1007/s10711-013-9896-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-013-9896-4

Keywords

Mathematics Subject Classification (1991)

Search

Navigation