FIBERWISE KÄHLER-RICCI FLOWS ON FAMILIES OF BOUNDED STRONGLY PSEUDOCONVEX DOMAINS

Abstract

© 2022Let π: Cn × C → C be the projection map onto the second factor and let D be a domain in Cn+1 such that for y ∈ π(D), every fiber Dy: = D ∩ π−1(y) is a smoothly bounded strongly pseudoconvex domain in Cn and is diffeomorphic to each other. By Chau’s theorem, the Kähler-Ricci flow has a long time solution ωy(t) on each fiber Xy. This family of flows induces a smooth real (1,1)-form ω(t) on the total space D whose restriction to the fiber Dy satisfies ω(t)|Dy = ωy(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 Cn+1.