Partial differential equations and volume-minimization problems for Lagrangian submanifolds

Abstract

We study several problems relating to calibrated geometry and volume-minimizing Lagrangian submanifolds. We prove a rigidity theorem for calibrated submanifolds with flat normal bundles, showing that a submanifold which is calibrated by a parallel form and has commuting shape operators is totally geodesic. In joint work with Yu Yuan, we derive constant rank theorems for the special Lagrangian equation, an application of which is a rigidity result for special Lagrangian cones, and the quadratic Hessian equation. In joint work with Arunima Bhattacharya, we consider regularity of Hamiltonian stationary Lagrangian graphs, proving that a Hamiltonian stationary graph which is Hölder continuous with exponent larger than 1/3 and has supercritical Lagrangian phase is smooth. We then construct singular solutions showing that 1/3 is the optimal Hölder exponent. Finally, in joint work with Micah Warren, we construct a properly immersed translating solution to curve diffusion flow, which is a fourth order analogue of curve shortening flow, and the one-dimensional case of the gradient flow for volume within a Hamiltonian isotopy class.