## Journal of the Mathematical Society of Japan

### Stability and bifurcation for surfaces with constant mean curvature

#### Abstract

We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in $\mathbb R^3$, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids in ${\mathbb R}^3$.

#### Article information

Source
J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1519-1554.

Dates
First available in Project Euclid: 25 October 2017

https://projecteuclid.org/euclid.jmsj/1508918567

Digital Object Identifier
doi:10.2969/jmsj/06941519

Mathematical Reviews number (MathSciNet)
MR3715814

Zentralblatt MATH identifier
1382.58012

#### Citation

KOISO, Miyuki; PALMER, Bennett; PICCIONE, Paolo. Stability and bifurcation for surfaces with constant mean curvature. J. Math. Soc. Japan 69 (2017), no. 4, 1519--1554. doi:10.2969/jmsj/06941519. https://projecteuclid.org/euclid.jmsj/1508918567

#### References

• J. Arroyo, M. Koiso and B. Palmer, Stability of non liquid bridges, Mat. Contemp., 40 (2011), 243–260.
• J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z., 185 (1984), 339–353.
• R. G. Bettiol and P. Piccione, Delaunay-type hypersurfaces in cohomogeneity one manifolds, Int. Math. Res. Not., IMRN 2016, no.,10, 3124–3162.
• M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Analysis, 54 (1971), 321–340.
• M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161–180.
• C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures et Appl. Ser 1, 6 (1841), 309–320.
• J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53–57.
• M. Koiso, Deformation and stability of surfaces with constant mean curvature, Tohoku Math. J. (2), 54 (2002), 145–159.
• M. Koiso and B. Palmer, Anisotropic capillary surfaces with wetting energy, Calculus of Variations and Partial Differential Equations, 29 (2007), 295–345.
• M. Koiso and B. Palmer, Higher order variations of constant mean curvature surface, to appear in Calculus of Variations and PDE's.
• M. Koiso, B. Palmer and P. Piccione, Bifurcation and symmetry breaking of nodoids with fixed boundary, Adv. Calc. Var., 8 (2015), 337–370.
• M. Koiso, P. Piccione and T. Shoda, On bifurcation and local rigidity of triply periodic minimal surfaces in $\mathbb R^3$, arXiv:1408.0953.
• R. López, Bifurcation of cylinders for wetting and dewetting models with striped geometry, SIAM J. Math. Anal., 44 (2012), 946–965.
• R. Mazzeo and F. Pacard, Bifurcating nodoids, Contemp. Math., 314 (2002), 169–186.
• U. Patnaik, Volume constrained Douglas problem and the stability of liquid bridges between two coaxial tubes, Dissertation, University of Toledo, USA, 1994.
• J. Plateau, Experimental and theoretical statics of liquids subject to molecular forces only, Translated by Kenneth A. Brakke, http://www.susqu.edu/brakke/aux/downloads/plateau-eng.pdf
• T. I. Vogel, Stability and bifurcation of a surface of constant mean curvature in a wedge, Indiana Univ. Math. J., 41 (1992), 625–648.