Paper · Aug 23, 2024

Singularity Types for Long-Time Chern-Ricci Flow

Hosea Wondo

math.DGarXiv:2408.12797
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Abstract

We extend some results known for the Kähler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly bounded torsion and curvature, then any other solution starting from an initial metric of the same class will also exhibit uniform bounds on torsion and curvature.

Comments: 13 pages, any comments welcomed

Ricci flowKähler–Ricci flowGeneralized & G₂ flowsSingularitiesKähler geometryLong-time existence & convergence

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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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)2Tosatti 2021 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)3Wondo 2021 Calabi Symmetry and the Continuity Method (Hosea Wondo, 2022)8Wondo 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)5Dang 2023 Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows (Hosea Wondo and Zhou Zhang, 2023)1Wondo 2023 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)7Li 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)6Huang 2026 Chern holonomy under the first and second Chern–Ricci flows (Yuan Liao, 2026)4Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

    Hosea Wondo and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Hosea Wondo · Shared terms “singularity type”, “independence”, “solution starting”

  2. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove PDF

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”

  3. Curvature Estimates for the Continuity Method

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Hosea Wondo

  4. Chern holonomy under the first and second Chern–Ricci flows

    Yuan Liao PDF

    Topics Generalized & G₂ flows, Long-time existence & convergence, Singularities · Shared terms “Chern Ricci”

  5. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang PDF

    Topics Generalized & G₂ flows, Kähler geometry, Singularities · Shared terms “Chern Ricci”

  6. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “initial metric”

  7. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li PDF

    Topics Generalized & G₂ flows, Long-time existence & convergence, Kähler geometry

  8. Calabi Symmetry and the Continuity Method

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Hosea Wondo

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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)

In the guidebook

  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.