Journal of the Mathematical Society of Japan

Stability and bifurcation for surfaces with constant mean curvature

Miyuki KOISO, Bennett PALMER, and Paolo PICCIONE

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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

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

First available in Project Euclid: 25 October 2017

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Zentralblatt MATH identifier

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] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90] 49R05: Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)

bifurcation constant mean curvature surfaces stability


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.

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  • 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,
  • T. I. Vogel, Stability and bifurcation of a surface of constant mean curvature in a wedge, Indiana Univ. Math. J., 41 (1992), 625–648.