Let $\pi:\mathbb{ C}^n\times\mathbb{ C}\rightarrow\mathbb{ C}$ be the projection map onto the second factor and let $D$ be a domain in $\mathbb{ C}^{ n+1}$ such that for $y\in\pi(D)$, every fiber $D_y:=D\cap\pi^{ -1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $\mathbb{ C}^n$ and is diffeomorphic to each other. By Chau's theorem, the K\"ahler-Ricci flow has a long time solution $\omega_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $\omega(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $\omega(t)\vert_{ D_y}=\omega_y(t)$. In this paper, we prove that $\omega(t)$ is positive for all $t>0$ in $D$ if $\omega(0)$ is positive. As a corollary, we also prove that the fiberwise K\"ahler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in $\mathbb{ C}^{ n+1}$.
Categories: math.CV math.DG