Unstable Willmore surfaces of revolution subject to natural boundary conditions

  • Anna Dall’Acqua
  • Klaus Deckelnick
  • Glen Wheeler


In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.

Mathematics Subject Classification (2000)

35J40 35B38 58E99 49J45 49Q10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 2003(10), 553–576 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bergner M., Dall’Acqua A., Fröhlich S.: Symmetric Willmore surfaces of revolution satisfying natural boundary conditions. Calc. Var. Partial Differ. Equ. 39(3–4), 361–378 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bergner, M., Dall’Acqua, A., Fröhlich, S.: Willmore surfaces of revolution with two prescribed boundary circles. J. Geom. Anal. (2012), On-line firstGoogle Scholar
  4. 4.
    Bernard Y., Riviere T.: Local Palais-Smale sequences for the Willmore functional. Comm. Anal. Geom. 19(3), 563–600 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blaschke, W.: Vorlesungen über Differentialgeometrie, vol. I–III, (1929)Google Scholar
  6. 6.
    Bryant R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20(1), 23–53 (1984)zbMATHGoogle Scholar
  7. 7.
    Bryant R., Griffiths P.: Reduction for constrained variational problems and \({\int \frac{1}{2}k^{2} ds}\). Am. J. Math. 108(3), 525–570 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dacorogna B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dall’Acqua, A.: Uniqueness for the homogeneous Dirichlet Willmore boundary value problem. Ann. Glob. Anal. Geom. On-line first.Google Scholar
  10. 10.
    Dall’Acqua A., Deckelnick K., Grunau H.-C.: Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv. Calc. Var. 1(4), 379–397 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dall’Acqua A., Fröhlich S., Grunau H.-C., Schieweck F.: Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv. Calc. Var. 4(1), 1–81 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deckelnick K., Grunau H.-C.: A Navier boundary value problem for Willmore surfaces of revolution. Analysis 29(3), 229–258 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grunau, H.-C.: The asymptotic shape of a boundary layer of symmetric willmore surfaces of revolution. In: Bandle, C. et al. (eds.) Inequalities and Applications 2010, Volume 161 of International Series of Numerical Mathematics, pp. 19–29. Springer, Basel (2012)Google Scholar
  14. 14.
    Hertrich-Jeromin U., Pinkall U.: Ein Beweis der Willmoreschen Vermutung für Kanaltori. J. Reine Angew. Math. 430, 21–34 (1992)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kuwert, E.: The Willmore functional. Lectures held at ETH in 2007Google Scholar
  16. 16.
    Kuwert, E., Li, Y.: W 2,2-conformal immersions of a closed Riemann surface into \({\mathbb{R}^n}\). Arvxiv peprint arXiv:1007.3967 (2010)Google Scholar
  17. 17.
    Kuwert, E., Schätzle, R.: Closed surfaces with bounds on their Willmore energy. To appear in Annali Sc. Norm. Sup. PisaGoogle Scholar
  18. 18.
    Langer J., Singer D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nitsche J.C.C.: Boundary value problems for variational integrals involving surface curvatures. Q. Appl. Math. 51(2), 363–387 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Palmer B.: The conformal Gauss map and the stability of Willmore surfaces. Ann. Glob. Anal. Geom. 9(3), 305–317 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Palmer B.: Uniqueness theorems for Willmore surfaces with fixed and free boundaries. Indiana Univ. Math. J. 49(4), 1581–1602 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rivière T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1–45 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schätzle R.: The Willmore boundary problem. Calc. Var. Partial Differ. Equ. 37(3–4), 275–302 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–326 (1993)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Struwe, M.: Plateau’s Problem and the Calculus Of Variations. Mathematical Notes, vol. 35. Princeton University Press, PrincetonGoogle Scholar
  26. 26.
    Thomsen G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. (German). Abh. Math. Sem. Hamburg. 3, 31–56 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weiner J.L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Willmore T.J.: Riemannian Geometry. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Anna Dall’Acqua
    • 1
  • Klaus Deckelnick
    • 1
  • Glen Wheeler
    • 1
  1. 1.Otto-von-Guericke-Universität, Universitätsplatz 2MagdeburgGermany

Personalised recommendations