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

Young-Jun Choi; Sungmin Yoo

Paper · May 26, 2020

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

Young-Jun Choi, Sungmin Yoo

math.CVarXiv:2005.12645

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Abstract

Let $π : C^{n} \times C \to C$ be the projection map onto the second factor and let $D$ be a domain in $C^{n + 1}$ such that for $y \in π (D)$ , every fiber $D_{y} := D \cap π^{- 1} (y)$ 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 $D_{y}$ . 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 $ω (t) ∣_{D_{y}} = ω_{y} (t)$ . 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 $C^{n + 1}$ .

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Submitted: May 26, 2020
Subjects: math.CV, math.DG

Comments: 18 pages

Ricci flow Kähler–Ricci flow Einstein metrics Kähler geometry Long-time existence & convergence Kähler–Einstein & K-stability

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[3] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

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

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