The linearized Monge-Ampere equation and its geometric applications

In this talk, we will introduce the linearized Monge-Ampere equation and discuss its boundary regularity in joint works with Ovidiu Savin and Truyen Nguyen. Linearized Monge-Ampere equation is an interesting combination of the linear elliptic equation and the Monge-Ampere equation. Though highly degenerate, linearized Monge-Ampere equation has the same regularity results as those of the Poisson equation. Though linear, it has the same challenging aspects of the fully nonlinear Monge-Ampere equation. We will also describe applications of our regularity results to fourth order, fully nonlinear geometric partial differential equations such as affine maximal surface and Abreu equations in affine and complex geometry.