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In this paper, we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addition, we prove a symmetry result for a subclass of Willmore surfaces satisfying symmetric Dirichlet boundary data.

Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution