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

### Young-Jun Choi

Department of Mathematics, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, Republic of Korea### Sungmin Yoo

Center for Complex Geometry, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea

## Abstract

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 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 $\omega(t)\vert_{D_y}=\omega_y(t)$. 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 Kähler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in $\mathbb{C}^{n+1}$.

## Cite this article

Young-Jun Choi, Sungmin Yoo, Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains. Doc. Math. 27 (2022), pp. 847–868

DOI 10.4171/DM/886