Willmore functional and Lagrangian surfaces

威尔莫尔函数曲面和拉格朗日曲面

• 批准号: 339625802
• 负责人: Professor Dr. Ernst Kuwert
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339625802?language=en
• 关键词: Willmore; functional; Lagrangian; surfaces

The project studies the analysis of the Willmore functional for immersed surfaces in C^2 under a Lagrangian constraint. It is motivated by work of Minicozzi who proved the existence of a smooth minimizing Lagrangian torus. The goal is to investigate geometric properties of surfaces which are Willmore critical under Hamiltonian deformations, and of a Willmore gradient flow preserving the Lagrangian property. This was recently introduced by Luo and Wang. Specific questions include the rigidity of the Whitney sphere in the Lagrangian class, and stability properties of the flow near the Whitney sphere and the Clifford torus. For unconstrained Willmore surfaces with L^2 bounded second fundamental form one has strong bubbling results, in particular the asymptotic behavior at a point singularity or at infinity is understood. It is a challenging question if and how this generalizes to the Lagrangian Willmore case.

该项目研究拉格朗日约束下 C^2 浸没表面的 Willmore 泛函分析。 它是受到 Minicozzi 工作的启发，Minicozzi 证明了平滑最小化拉格朗日环面的存在。目标是研究在哈密顿变形下威尔莫尔临界的表面几何特性，以及保留拉格朗日特性的威尔莫尔梯度流。这是罗和王最近介绍的。具体问题包括拉格朗日类惠特尼球的刚性，以及惠特尼球和克利福德环附近流动的稳定性。对于具有 L^2 有界第二基本形式的无约束 Willmore 曲面，具有很强的冒泡结果，特别是可以理解奇点或无穷远点处的渐近行为。这是否以及如何推广到拉格朗日威尔莫尔案例是一个具有挑战性的问题。