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

Let π:Cn×CC\pi:\mathbb{C}^n\times\mathbb{C}\rightarrow \mathbb{C} be the projection map onto the second factor and let DD be a domain in Cn+1\mathbb{C}^{n+1} such that for yπ(D)y\in\pi(D), every fiber Dy:=Dπ1(y)D_y:=D\cap\pi^{-1}(y) is a smoothly bounded strongly pseudoconvex domain in Cn\mathbb{C}^n and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solution ωy(t)\omega_y(t) on each fiber XyX_y. This family of flows induces a smooth real (1,1)-form ω(t)\omega(t) on the total space DD whose restriction to the fiber DyD_y satisfies ω(t)Dy=ωy(t)\omega(t)\vert_{D_y}=\omega_y(t). In this paper, we prove that ω(t)\omega(t) is positive for all t>0t>0 in DD if ω(0)\omega(0) is positive. As a corollary, we also prove that the fiberwise Kähler-Einstein metric is positive semi-definite on DD if DD is pseudoconvex in Cn+1\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