Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

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• 作者：Sascha Eichmann
• 关键词：Dirichlet boundary conditions ; Nonuniqueness ; Willmore surface ; Surface of revolution ; Bernstein ; type theorem
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：26
• 期：4
• 页码：2563-2590
• 全文大小：737 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X
• 卷排序：26

文摘

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.