Unstable Willmore surfaces of revolution subject to natural boundary conditions

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.

Boundary value problems for Willmore curves in $$\mathbb {R}^2$$ R 2

“…In Sect. 2 we review some preliminary facts about Willmore curves and Jacobi's elliptic functions appearing in the definition of κ a,b , see (4). Next, in Sect.…”

“…The curvature functions of such curves satisfy −κ +κ = 1 2κ 3 instead of (7) so that it is tempting to conjecture that boundary value problems for Willmore surfaces of revolution can be solved with a similar analysis as ours. We refer the interested reader to [2][3][4][5]8] for more information on recent advances concerning Willmore surfaces of revolution.…”

Noether’s theorem and the Willmore functional

“…where ⋆ is the usual Hodge-star operator and is the Levi-Civita symbol⁷ with components 11 = 0 = 22 and 12 = 1 = − 21 . Einstein's summation convention applies throughout.…”

“…In [21], the authors study Willmore surfaces of revolution. They use the invariances of the Willmore functional to recast the Willmore ODE in a form that is a special case of (1.3).…”

Rigidity and stability of spheres in the Helfrich model

“…The Willmore conjecture, proposed by Willmore (1965), was recently resolved by Marques and Neves (2014). Work on the Willmore functional continues to be a very active area, with recent progress made on quantisation (Bernard and Riviere, 2014), the gradient flow Schätzle, 2001, 2002), and boundary value problems (Alessandroni and Kuwert, 2014;Dall'Acqua, 2012;Dall'Acqua et al, 2013;Deckelnick and Grunau, 2009). There are many other works besides those mentioned here -the literature on analysis of the Willmore functional is vast.…”