Tohoku Mathematical Journal

Deformation and stability of surfaces with constant mean curvature

Miyuki Koiso

Full-text: Open access

Abstract

For a CMC immersion from a two-dimensional compact smooth manifold with boundary into the Euclidean three-space, we give sufficient conditions under which it has a CMC deformation fixing the boundary. Moreover, we give a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtain a 'visible' way of judging the stability.

Article information

Source
Tohoku Math. J. (2), Volume 54, Number 1 (2002), 145-159.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113247184

Digital Object Identifier
doi:10.2748/tmj/1113247184

Mathematical Reviews number (MathSciNet)
MR1878932

Zentralblatt MATH identifier
1010.58007

Subjects
Primary: 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Citation

Koiso, Miyuki. Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J. (2) 54 (2002), no. 1, 145--159. doi:10.2748/tmj/1113247184. https://projecteuclid.org/euclid.tmj/1113247184


Export citation

References

  • J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), 339--353.
  • R. Böhme and F. Tomi, Zur Struktur der Lösungsmenge des Plateauproblems, Math. Z. 133 (1973), 1--29.
  • A. Duschek, Zur geometrischen Variationsrechnung, Math. Z. 40 (1936), 279--291.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Second Edition), Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983.
  • L. Hörmander, Linear Partial Differential Operators, Third Revised Printing, Springer-Verlag, Berlin-Heidelberg-New York, 1969.
  • N. Kapouleas, {Complete constant mean curvature surfaces in euclidean three-space, Ann. of Math. (2) 131 (1990), 239--330.
  • M. Koiso, The stability and the vision number of surfaces with constant mean curvature, Bull. Kyoto Univ. Ed. Ser. B 92 (1998), 1--11.
  • N. Koiso, Variational Problems (in Japanese),} Kyōritu Publ., Tokyo, Japan, 1998.
  • O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York and London, 1968.
  • S. Lang, Differential Manifolds, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1985.
  • J. H. Maddocks, Stability of nonlinearly elastic rods, Arch. Rat. Mech. Anal. 85 (1984), 311--354.
  • J. H. Maddocks, Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles, SIAM J. Math. Anal. 16 (1985), 47--68.
  • J. H. Maddocks, Stability and folds, Arch. Rat. Mech. Anal. 99 (1987), 301--328.
  • S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965), 1049--1055.
  • F. Tomi, A perturbation theorem for surfaces of constant mean curvature, Math. Z. 141 (1975), 253--264.
  • T. I. Vogel, Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl. Math. 147 (1987), 516--525.
  • T. I. Vogel, On constrained extrema, Pacific J. Math. 176 (1996), 557--561.
  • T. I. Vogel, Sufficient conditions for multiply constrained extrema, Pacific J. Math. 180 (1997), 377--383.
  • T. I. Vogel, Non-linear stability of a certain capillary surfaces, Differential equations and dynamical systems (Waterloo, ON, 1997), Dynam. Contin. Discrete Impuls. Systems 5 (1999), 1--15.