Singular limits of Kähler-Ricci flow on Fano G-manifolds

Yan Li; Gang Tian; Xiaohua Zhu

Paper · Jul 24, 2018

Singular limits of Kähler-Ricci flow on Fano $G$ -manifolds

Yan Li, Gang Tian, Xiaohua Zhu

math.DGarXiv:1807.09167v5

View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of $SO_{4} (C)$ and one Fano compactification of $Sp_{4} (C)$ , on which the Kähler-Ricci flow will develop singularities of type II. To the authors' knowledge, these are the first examples of Ricci flow with singularities of type II on Fano manifolds in the literature.

View PDF View HTML arXiv page

Submitted: Jul 24, 2018
Last revised: Dec 22, 2021 (v5)
Subjects: math.DG

Comments: We added an appendix in Section 7 to prove Theorem 7.3 which is a generalization of Theorem 1.1 for any complex reductive Lie group $G$

Ricci flow Kähler–Ricci flow Einstein metrics Singularities Kähler geometry Homogeneous spaces Kähler–Einstein & K-stability

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 2016, plus the canon.

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

K-stability of Gorenstein Fano group compactifications with rank two
Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo2022

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “compactification”, “Fano”, “semisimple complex”
Horosymmetric limits of Kähler-Ricci flow on Fano $G$ -manifolds
Gang Tian and Xiaohua Zhu2022

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano”, “Lie”
Kähler-Ricci flow for deformed complex structures
Gang Tian, Liang Zhang and Xiaohua Zhu2021

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano”, “Lie”
Equivariant $R$ -test configurations and semistable limits of $Q$ -Fano group compactifications
Yan Li and ZhenYe Li2021

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Kähler geometry · Same author Yan Li · Shared terms “compactification”, “Fano”
Laplace comparison on Kähler Ricci flow and convergence
Gang Tian, Qi S. Zhang, Zhenlei Zhang et al.2025

Topics Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Xiaohua Zhu · Shared terms “Fano”, “literature”
Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold
Feng Wang and Xiaohua Zhu2020

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Xiaohua Zhu · Shared terms “Fano”
On the Kahler-Ricci flow on Fano manifolds
Bin Guo, Duong H. Phong and Jacob Sturm2020

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Fano”
Tian's partial $C^{0}$ -estimate implies Hamilton-Tian's conjecture
Feng Wang and Xiaohua Zhu2020

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Xiaohua Zhu · Shared terms “Fano”

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) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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.

[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] 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.

[3] 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.

Singular limits of Kähler-Ricci flow on Fano $G$ -manifolds

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters