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

We study entire continuous plurisubharmonic solutions of the complex Monge-Ampère equation (i∂∂ u)ⁿ=2ⁿ n!μ in Cⁿ, in the Bedford-Taylor sense, where μ≡ 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.