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Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains

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