M. E. Taylor Analysis and PDE Seminar – Micah Warren (UOregon)

A geometric flow towards Hamiltonian stationary submanifolds

Abstract. In joint work with Jingyi Chen, we introduce a geometric flow for Lagrangian submanifolds in a Kähler manifold which stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. The flow is not strictly parabolic but it corresponds to a fourth order strictly parabolic scalar equation in the cotangent bundle of the submanifold via Weinstein’s Lagrangian neighborhood theorem. For any compact initial Lagrangian immersion, we establish short-time existence, uniqueness, and higher order estimates when the second fundamental forms are uniformly bounded up to time T.

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