Advertisement

Journal of Experimental and Theoretical Physics

, Volume 91, Issue 6, pp 1279–1285 | Cite as

Minimal surfaces and fluctuations of membranes with nontrivial topology

  • E. I. Kats
  • M. I. Monastyrskii
Miscellaneous

Abstract

A new geometric approach to the description of phase transitions and fluctuations in membranes with nontrivial topology is proposed. The method is based on the possibility of representing real membranes and vesicles, defined in the space R3, as minimal surfaces embedded in S3. A change in the genus of the physical membrane corresponds to the formation of holes in the minimal surface. In the framework of mean field theory a model is constructed for a phase transition that can be characterized as the crystallization of holes in S3. In real membranes this corresponds to a phase transition from a cubic phase to a sponge.

Keywords

Spectroscopy Crystallization Phase Transition State Physics Field Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Physics of Amphiphilic Layers, Ed. by J. Meuner, D. Langevin, and N. Boccara (Springer-Verlag, Berlin, 1987); Springer Proceedings in Physics, Vol. 21.Google Scholar
  2. 2.
    S. A. Safran and N. A. Clark, Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987).Google Scholar
  3. 3.
    D. Nelson, T. Pvian, and S. Weinberg, Statistical Mechanics of Membranes and Surfaces (World Scientific, New York, 1989).Google Scholar
  4. 4.
    A. M. Belloog, J. Piaif, and P. Bothorel, Adv. Colloid Interface Sci. 20, 167 (1984).Google Scholar
  5. 5.
    G. Porte, J. Phys.: Condens. Matter 4, 8649 (1992).CrossRefADSGoogle Scholar
  6. 6.
    B. Fourcade, M. Mutz, and D. Bensimon, Phys. Rev. Lett. 68, 2251 (1992).ADSGoogle Scholar
  7. 7.
    F. Julicher, U. Seifert, and R. Lipowsky, Phys. Rev. Lett. 71, 452 (1993).ADSGoogle Scholar
  8. 8.
    E. I. Kats and V. V. Lebedev, Pis’ma Zh. Éksp. Teor. Fiz. 61, 57 (1995) [JETP Lett. 61, 59 (1995)].Google Scholar
  9. 9.
    W. Helfrich, Z. Naturforsch. B 103, 67 (1975).Google Scholar
  10. 10.
    T. J. Willmore, Riemannian Geometry (Clarendon Press, Oxford, 1993).Google Scholar
  11. 11.
    J. L. Weiner, Indiana Univ. Math. J. 27, 19 (1978).CrossRefADSMATHMathSciNetGoogle Scholar
  12. 12.
    U. Pinkal, Invent. Math. 81, 379 (1985).ADSMathSciNetGoogle Scholar
  13. 13.
    H. B. Lawson, Ann. Math. 92, 335 (1970).MATHMathSciNetGoogle Scholar
  14. 14.
    H. Karcher, U. Pinkal, and I. Sterling, J. Diff. Geom. 28, 169 (1988).Google Scholar
  15. 15.
    P. Li and S. T. Yau, Invent. Math. 69, 269 (1982).CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999).CrossRefADSGoogle Scholar
  17. 17.
    J. M. Ziman, Principles of the Theory of Solids (Cambridge Univ. Press, London, 1972, 2nd ed.; Mir, Moscow, 1966).Google Scholar
  18. 18.
    E. I. Kats, V. V. Lebedev, and A. R. Muratov, Phys. Rep. 228, 1 (1993).CrossRefADSGoogle Scholar
  19. 19.
    J. C. Lang and R. D. Morgan, J. Chem. Phys. 73, 5849 (1980).CrossRefADSGoogle Scholar
  20. 20.
    V. L. Golo, E. I. Kats, and G. Porte, Pis’ma Zh. Éksp. Teor. Fiz. 64, 575 (1996) [JETP Lett. 64, 631 (1996)].Google Scholar
  21. 21.
    A. I. Bobenko, Usp. Mat. Nauk 46(4), 3 (1991).MATHMathSciNetGoogle Scholar
  22. 22.
    U. Pinkal, Topology 24, 421 (1985).MathSciNetGoogle Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2000

Authors and Affiliations

  • E. I. Kats
    • 1
    • 2
  • M. I. Monastyrskii
    • 3
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Laue-Langevin InstituteGrenobleFrance
  3. 3.Instituteof Theoretical and Experimental PhysicsMoscowRussia

Personalised recommendations