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

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Abstract

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.

Keywords

Willmore surfaces of revolution Willmore boundary value problems Symmetry breaking Symmetry Explicit formulas 

Mathematics Subject Classification

Primary 53C42 Secondary 34B15 49Q10 

Notes

Acknowledgements

The author expresses his gratitude to the anonymous reviewer for numerous helpful suggestions and his great effort leading to an improved, completed and corrected version of the manuscript. Moreover, the author would like to thank the Scuola Normale Superiore di Pisa for the hospitality during his stay there. The work on this project was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Grant Number MA 6290/2-1.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

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