Fiberwise K\"ahler-Ricci flows on families of bounded strongly pseudoconvex domains

TL;DR

This paper studies fiberwise Kähler-Ricci flows on families of strongly pseudoconvex domains, proving positivity of the induced form over time and semi-definiteness of fiberwise Kähler-Einstein metrics.

Contribution

It establishes the positivity preservation of the fiberwise Kähler-Ricci flow and the semi-definiteness of fiberwise Kähler-Einstein metrics on pseudoconvex domains.

Findings

The induced form remains positive for all time if initially positive.

Fiberwise Kähler-Einstein metrics are positive semi-definite on pseudoconvex domains.

Long-term existence of the flow on each fiber is utilized in the analysis.

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 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 $\mathbb{C}^{n+1}$.