Journal of Experimental and Theoretical Physics

, Volume 113, Issue 6, pp 1063–1070 | Cite as

Mean curvature flow of a hyperbolic surface

  • Yu. N. OvchinnikovEmail author
  • I. M. Sigal
Statistical, Nonlinear, and Soft Matter Physics


A four-parameter family of self-similar solutions is obtained to the mean curvature flow equation for a surface. This family is shown to be stable with respect to a small deformation of a hyperbolic surface. At time instant t*, a singular point is formed within a finite time interval, that is accompanied by a change in the topology of the surface. The solution is continued beyond the singular point. A relationship between the parameters describing the hyperbolic surface before and after the change in the surface topology is obtained. A particular case is analyzed when the unperturbed surface is a cylinder. A cylindrical surface is weakly unstable with respect to a perturbation in the form of a “wide neck.” At the final stage of the development of the neck when its transverse size becomes much less than the cylinder radius at large distances from the neck, the surface flow in a wide region in the neighborhood of the neck is described by a universal two-parameter family of self-similar solutions. These solutions are stable with respect to small perturbations of the surface.


Theoretical Physic Singular Point Curvature Flow Similar Solution Parameter Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Eggers and M. A. Fontelos, Nonlinearity 22, R1 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    S. Allen and J. W. Cahn, Acta. Metall. 27, 1084 (1979).Google Scholar
  3. 3.
    M. Kowalczyk, Ann. Mat. Pura Appl. 184, 17 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1963; Butterworth-Heinemann, Oxford, 1981).Google Scholar
  5. 5.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Fizmatgiz, Moscow, 1962; Academic, New York, 1980).Google Scholar
  6. 6.
    J. W. Cahn and J. E. Hillard, J. Chem. Phys. 28, 258 (1958).CrossRefADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Max-Planck Institute for Physics of Complex SystemsDresdenGermany
  2. 2.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations