Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li’s result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $(1-u_t)\det (D_x^{2}u+I)=f$ in ${\mathbb{R}}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.