A Geometric Flow towards Hamiltonian Stationary Submanifolds

In this talk, I will first give a brief survey on some recent advances on Hamiltonian stationary Lagrangian submanifolds. Then I will discuss a joint work with Micah Warre in which we introduce a geometric flow for Lagrangian submanifolds in a Kahler manifold which stays in its initial Hamiltonian isotopy class and is a gradient flow for volume, and the stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. We establish short time existence, uniqueness and higher order estimates for compact initial data.