On the Ricci curvature of Kahler-Ricci Flow

Cheuk Yan Fung

Paper · Nov 24, 2021

On the Ricci curvature of Kahler-Ricci Flow

Cheuk Yan Fung

math.DGarXiv:2111.12415v2

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Abstract

In this paper, we consider $n$ -dimensional compact K $a ¨$ hler manifold with semi-ample canonical line bundle under the long time solution of K $a ¨$ hler Ricci Flow. In particular, if the Kodaira dimension is one, Ricci curvature converge to negative of generalized K $a ¨$ hler Einstein metric $ω_{B}$ locally away from singular set in $C_{l oc}^{0} (ω (t))$ topology.

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Submitted: Nov 24, 2021
Last revised: Feb 20, 2026 (v2)
Subjects: math.DG

Comments: We have to add an additional assumption: the fibres of X over the canonical model X_can of X are biholomorphic to each other

Ricci flow Kähler–Ricci flow Einstein metrics Kähler geometry Long-time existence & convergence Analytic minimal model program

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Background reading

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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… 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.

On the Ricci curvature of Kahler-Ricci Flow

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