Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Möbius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.