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Unstable Willmore surfaces of revolution subject to natural boundary conditions

  • Anna Dall’Acqua
  • Klaus Deckelnick
  • Glen Wheeler
Article

Abstract

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 

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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

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