Willmore functional and Lagrangian surfaces

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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.