# Stability Analysis for Variational Problems for Surfaces with Constraint

## Abstract

Surfaces with constant mean curvature (CMC surfaces) are critical points of the area functional among surfaces enclosing the same volume. Therefore, they are a simple example of solutions of variational problem with constraint. A CMC surface is said to be stable if the second variation of the area is nonnegative for all volume-preserving variations satisfying the given boundary condition. The purpose of this article is to show some fundamental methods to study the stability for CMC surfaces. Especially, we give a criterion on the stability for compact CMC surfaces with prescribed boundary. Another concept that is closely related to the stability for CMC surfaces is the so-called bifurcation. We give sufficient conditions on a one-parameter family of CMC surfaces so that there exists a bifurcation. Moreover, we give a criterion for CMC surfaces in the bifurcation branch to be stable.

## Keywords

Bifurcation Constant mean curvature Pitchfork bifurcation Stability Symmetry breaking Variational problem## Notes

### Acknowledgments

The author is partially supported by Grant-in-Aid for Scientific Research (B) No. 25287012 of the Japan Society for the Promotion of Science, and the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development.

## References

- 1.J.L. Barbosa, M. do Carmo, Stability of hypersurfaces of constant mean curvature. Math. Zeit.
**185**, 339–353 (1984)Google Scholar - 2.J.L. Barbosa, M. do Carmo, J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Zeit.
**197**, 123–138 (1988)Google Scholar - 3.M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Func. Anal.
**54**, 321–340 (1971)MathSciNetCrossRefGoogle Scholar - 4.M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal.
**52**, 161–180 (1973)MathSciNetCrossRefMATHGoogle Scholar - 5.N. Kapouleas, Complete constant mean curvature surfaces in euclidean three-space. Ann. Math.
**131**(2), 239–330 (1990)Google Scholar - 6.N. Kapouleas, Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math.
**119**, 443–518 (1995)MathSciNetCrossRefMATHGoogle Scholar - 7.H. Kielhöfer, in
*Bifurcation Theory: An Introduction with Applications to Partial Differential Equations*. Applied Mathematical Sciences, 2nd edn., vol. 156. (Springer, New York, 2012)Google Scholar - 8.M. Koiso, Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J.
**54**(2), 145–159 (2002)Google Scholar - 9.M. Koiso, B. Palmer, P. Piccione, in,
*Bifurcation and Symmetry Breaking of Nodoids with Fixed Boundary*. Advances in Calculus of Variation (to appear)Google Scholar - 10.M. Koiso, B. Palmer, P. Piccione, Stability and bifurcation for surfaces with constant mean curvature (in preparation)Google Scholar
- 11.M. Koiso, P. Piccione, T. Shoda, On bifurcation and local rigidity of triply periodic minimal surfaces in \({ R}^3\) (preprint)Google Scholar
- 12.N. Koiso,
*Variational Problem*(Kyoritsu, Tokyo, Japan, 1998) (In Japanese)Google Scholar - 13.J.H. Maddocks, Stability of nonlinearly elastic rods. Arch. Rat. Mech. Anal.
**85**, 311–354 (1984)MathSciNetCrossRefMATHGoogle Scholar - 14.J.H. Maddocks, Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles. SIAM J. Math. Anal.
**16**, 47–68 (1985)MathSciNetCrossRefMATHGoogle Scholar - 15.J.H. Maddocks, Stability and folds. Arch. Rat. Mech. Anal.
**99**, 301–328 (1987)MathSciNetCrossRefGoogle Scholar - 16.U. Patnaik, Volume constrained Douglas problem and the stability of liquid bridges between two coaxial tubes. Dissertation, University of Toledo, USA, 1994Google Scholar
- 17.S. Smale, On the Morse index theorem. J. Math. Mech.
**14**, 1049–1055 (1965)MathSciNetMATHGoogle Scholar - 18.T.I. Vogel, Stability of a liquid drop trapped between two parallel planes. SIAM J. Appl. Math.
**47**, 516–525 (1987)MathSciNetCrossRefMATHGoogle Scholar - 19.T.I. Vogel, Stability of a liquid drop trapped between two parallel planes II. General contact angles. SIAM J. Appl. Math.
**49**, 1009–1028 (1989)MathSciNetCrossRefMATHGoogle Scholar - 20.T.I. Vogel, On constrained extrema. Pac. J. Math.
**176**, 557–561 (1996)Google Scholar - 21.T.I. Vogel, Sufficient conditions for multiply constrained extrema. Pac. J. Math.
**180**, 377–383 (1997)CrossRefGoogle Scholar - 22.T. I. Vogel, Non-linear stability of a certain capillary surfaces. Dynam. Contin. Discrete Impuls. Syst.
**5**, 1–15 (1999)Google Scholar - 23.H.C. Wente, Counterexample to a conjecture of H. Hopf. Pac. J. Math.
**121**, 193–243 (1986)MathSciNetCrossRefMATHGoogle Scholar - 24.H.C. Wente, A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature. Pac. J. Math.
**147**, 375–379 (1991)MathSciNetCrossRefMATHGoogle Scholar