Expanding convex immersed closed plane curves



We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.

Mathematics Subject Classification (2000)

35K15 35K55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, B.: Evolving convex curves. Cal. Var. PDEs. 7(4), 315–371 (1998)MATHCrossRefGoogle Scholar
  2. 2.
    Andrews, B.: Classification of limiting shapes for isotropic curve flows. J. AMS 16(2), 443–459 (2003)MATHGoogle Scholar
  3. 3.
    Angenent, S.: The zero set of a solution of a parabolic equation. J. die Reine Angewandte Math. 390, 79–96 (1988)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33, 601–633 (1991)MATHMathSciNetGoogle Scholar
  5. 5.
    Chen, X.Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ. 78(1), 160–190 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cortazar, C., Del Pino, M., Elgueta, M.: On the blow-up set for \({\partial_{t}u = \bigtriangleup u^{m} + u^{m}, m > 1, }\) . Indiana Univ. Math. J. 47, 541–561 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chow, B., Tsai, D.H.: Geometric expansion of convex plane curves. J. Differ. Geom. 44, 312–330 (1996)MATHMathSciNetGoogle Scholar
  8. 8.
    Chou, K.S., Zhu, X.-P.: The Curve Shortening Problem. Chapman and Hall/CRC, London (2000)Google Scholar
  9. 9.
    Feireisl, E., Simondon, F.: Convergence for degenerate parabolic equations. J. Differ. Equ. 152, 439–466 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)MATHMathSciNetGoogle Scholar
  11. 11.
    Lin, Y.-C., Poon, C.C., Tsai, D.H.: Contracting convex immersed closed plane curves with fast speed (in preparation) (2008)Google Scholar
  12. 12.
    Lin, Y.-C., Poon, C.C., Tsai, D.H.: Contracting convex immersed closed plane curves with slow speed (in preparation) (2008)Google Scholar
  13. 13.
    Matano, H.: Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18(2), 221–227 (1978)MATHMathSciNetGoogle Scholar
  14. 14.
    Tsai, D.H.: Blowup and convergence of expanding immersed convex plane curves. Comm. Anal. Geom. 8(4), 761–794 (2000)MATHMathSciNetGoogle Scholar
  15. 15.
    Tsai, D.H.: Behavior of the gradient for solutions of parabolic equations on the circle. Cal. Var. PDEs. 23, 251–270 (2005)MATHCrossRefGoogle Scholar
  16. 16.
    Tsai, D.H.: Blowup behavior of an equation arising from plane curves expansion. Differ. Integ. Eq. 17(7–8), 849–872 (2004)MATHGoogle Scholar
  17. 17.
    Urbas, J.: An expansion of convex hypersurfaces. J. Differ. Geom. 33, 91–125 (1991)MATHMathSciNetGoogle Scholar
  18. 18.
    Urbas, J.: Convex curves moving homothetically by negative powers of their curvature. Asian J. Math. 3(3), 635–658 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan
  2. 2.Department of MathematicsNational Chung Cheng UniversityChiayiTaiwan
  3. 3.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan

Personalised recommendations