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

Abstract

Let $\pi :{\mathbb{C}}^n\times \mathbb{C}\to \mathbb{C}$ 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 Kähler-Ricci flow has a long time solution ${\omega }_y(t)$ on each fiber $X_y$. 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 $\omega (t){|}_{D_y}={\omega }_y(t)$. 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 Kähler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in ${\mathbb{C}}^{n+1}$.