Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation

Abstract

We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Ampère equation $\det (u_{i\bar{j}})=1$ with homogeneous Dirichlet data under the condition that, for some constant $\kappa \ge 1$, $(\kappa u_{i\bar{j}}-u_{is}{u}^{s\bar{t}}{u}_{\overline{tj}})$ is non-negative definite. For $\kappa =1$, this gives the estimate for real convex solutions. The estimate depends only on the dimension, $\kappa$, and the $C^0$ and $C^1$ norms of the solution. As an application, every smooth entire convex solution of $\det (u_{i\bar{j}})=1$ on ${\mathbb{C}}^n$ is a real quadratic polynomial. The same conclusion holds if real convexity is assumed only outside a compact subset of ${\mathbb{C}}^n$.