Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution. (English) Zbl 1402.53002
Summary: 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.
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 53A30 | Conformal differential geometry (MSC2010) |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 49Q10 | Optimization of shapes other than minimal surfaces |
Keywords:
Willmore surfaces of revolution; Willmore boundary value problems; symmetry breaking; symmetry; explicit formulasReferences:
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