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.

该项目研究在拉格朗日约束下C^2中浸没曲面的Willmore泛函的分析。这是由Minicozzi的工作所激发的，他证明了光滑最小化拉格朗日环面的存在。目的是研究在哈密顿变形下Willmore临界曲面的几何性质，以及保持拉格朗日性质的Willmore梯度流的几何性质。这是最近由罗和王引入的。具体问题包括拉格朗日类中惠特尼球体的刚性，以及惠特尼球体和Clifford环面附近流动的稳定性。对于具有L^2有界第二基本形式的无约束Willmore曲面，第一类曲面有很强的泡化结果，特别是在奇点或无穷远处的渐近行为被理解。这是否以及如何推广到拉格朗日·威尔莫尔案是一个具有挑战性的问题。