A Liouville theorem for plurisubharmonic functions with periodic complex Monge-Ampère measure

Abstract

We study entire continuous plurisubharmonic solutions of the complex Monge-Ampère equation $(i\partial\bar\partial u)^n=2^n n!μ$ in $\mathbb{C}^n$, in the Bedford-Taylor sense, where $μ\not\equiv 0$ is a nonnegative locally finite Borel measure periodic with respect to a lattice of full real rank. We assume that $μ$ is locally dominated by complex Monge-Ampère measures of Hölder continuous plurisubharmonic functions, a condition under which the existence theorem of Demailly et al. yields Hölder continuous periodic correctors. Under this assumption, we prove that every such solution with positive two-sided quadratic growth can be written uniquely, up to an additive constant, as the sum of a real quadratic polynomial and a Hölder continuous periodic function.