Lagrangian mean curvature equations and flows

COLLOQUIUM TALK

Title: Lagrangian mean curvature equations and flows

Abstract: In this talk, we will introduce the special Lagrangian equation and the Lagrangian mean curvature flow. We will discuss interior Hessian estimates for shrinkers and expanders of the Lagrangian mean curvature flow, and further extend this result to a broader class of Lagrangian mean curvature type equations. We assume the Lagrangian phase to be hypercritical, which results in the convexity of the potential of the initial Lagrangian submanifold. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds.

Speaker Bio: Arunima Bhattacharya is a faculty member in the Department of Mathematics at the University of North Carolina, Chapel Hill, United States. She worked at the Simons Laufer Mathematical Sciences Institute in Berkeley and the University of Washington in Seattle. She earned her Ph.D. in Mathematics from the University of Oregon. Her research focuses on nonlinear partial differential equations and geometric analysis.