Liouville theorem and sharp solvability for solutions of the parabolic Monge-Amp\`ere equation with periodic data

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Amp\`ere 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 $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ 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.