Paper · Sep 17, 2024

Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

math.DGarXiv:2409.116082025, Advances in Mathematics 480A, 110465

As on arXiv

Abstract

We consider the Kähler-Ricci flow $(X, ω (t))_{t \in [0, T)}$ on a compact manifold where the time of singularity, $T$ , is finite. We assume the existence of a holomorphic map from the Kähler manifold $X$ to some analytic variety $Y$ which admits a Kähler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[ω_{0}] \in H^{1, 1} (X, Q)$ . Under these assumptions we prove an $L^{4}$ -like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^{2}$ -sense.

Comments: 28 pages

Ricci flow Kähler–Ricci flow Singularities Kähler geometry

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Fano Fibrations and Twisted Kähler-Einstein Metrics I
Alexander Bednarek2025 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Alexander Bednarek · Shared terms “analytic variety”, “finite time”, “rational”
The modified Kähler-Ricci flow, II
Haotian Wu and Zhou Zhang2022 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”
Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I
Charles Cifarelli, Ronan Conlon, Max Hallgren et al.2026 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”
Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I
Tongxin Xu and Zhenlei Zhang2026 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”
Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
Alexander Bednarek2025 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Alexander Bednarek · Shared terms “finite time”, “rational”
Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow
Junsheng Zhang2025 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”
Finite-Time Singularities of the Kähler–Ricci Flow on a $C P^{m}$ -Bundle over a Product of Kähler–Einstein Manifolds
Yifan Xiao2026 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”
Type I estimates for the Kähler-Ricci flow II
Wangjian Jian and Jian Song2026 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “limiting cohomology”, “analytic variety”, “finite time”

Read around it

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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.
11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters