A Liouville theorem for the Neumann problem of Monge-Amp?re equations

In this paper, we study the Neumann problem of Monge-Ampere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension n >= 3, we show that the conclusion still holds if either the boundary value is zero or the viscosity convex solutions restricted on some n - 2 dimensional subspace is bounded from above by a quadratic function.(c) 2022 Elsevier Inc. All rights reserved.