Willmore functional and Lagrangian surfaces

本课题研究了拉格朗日约束下C^2中浸入曲面的Willmore泛函分析。它是由Minicozzi的工作推动的，他证明了光滑最小化拉格朗日环面的存在性。目的是研究在哈密顿变形下Willmore临界曲面的几何性质，以及Willmore梯度流保持拉格朗日性质的几何性质。这是罗和王最近介绍的。具体问题包括拉格朗日类中惠特尼球的刚性，以及惠特尼球和克利福德环面附近流动的稳定性。对于具有L^2有界第二基本形式1的无约束Willmore曲面，具有较强的冒泡结果，特别是在奇点和无穷远处的渐近行为得到了理解。这是一个具有挑战性的问题，如果以及如何推广到拉格朗日威尔莫情况。

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.