Radially Symmetric Solutions to the Graphic Willmore Surface Equation

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• 作者：Jingyi Chen ; Yuxiang Li
• 关键词：Willmore surface ; Classification of radially symmetric graphic solutions ; Isolated singularity ; Inverted catenoids
• 刊名：The Journal of Geometric Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：27
• 期：1
• 页码：671-688
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：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 US
• ISSN：1559-002X
• 卷排序：27

文摘

We show that a smooth radially symmetric solution u to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in \({\mathbb R}^3\). In particular, radially symmetric entire Willmore graphs in \({\mathbb R}^3\) must be flat. When u is a smooth radial solution over a punctured disk \(D(\rho )\backslash \{0\}\) and is in \(C^1(D(\rho ))\), we show that there exist a constant \(\lambda \) and a function \(\beta \) in \(C^0(D(\rho ))\) such that \(u''(r) =\frac{\lambda }{2}\log r+\beta (r)\); moreover, the graph of u is contained in a graphical region of an inverted catenoid which is uniquely determined by \(\lambda \) and \(\beta (0)\). It is also shown that a radial solution on the punctured disk extends to a \(C^1\) function on the disk when the mean curvature is square integrable.