Let be the projection map onto the second factor and let be a domain in such that for , every fiber is a smoothly bounded strongly pseudoconvex domain in and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solution on each fiber . This family of flows induces a smooth real (1,1)-form on the total space whose restriction to the fiber satisfies . In this paper, we prove that is positive for all in if is positive. As a corollary, we also prove that the fiberwise Kähler-Einstein metric is positive semi-definite on if is pseudoconvex in .
Cite this article
Young-Jun Choi, Sungmin Yoo, Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains. Doc. Math. 27 (2022), pp. 847–868DOI 10.4171/DM/886