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

Abstract

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