Monge-Ampère equation with bounded periodic data

Abstract

We consider the Monge-Ampère equation $\det (D^{2}u)=f$ in ${\mathbb{R}}^n$, where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f\equiv 1$, this is the classic result by Jörgens, Calabi and Pogorelov. For $f\in C^{\alpha }$, this was proved by Caffarelli and the first named author.