Paper · Jun 9, 2026

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

Chuanhuan Li, Yi Li

math.DGarXiv:2606.10354Surveys in Differential Geometry(2026)
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Abstract

The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler-Ricci flow coupled with -forms, and the Laplacian flow.

Comments: 21 pages. This survey focuses on geometric flows with scalar curvature

Ricci flowGeneralized & G₂ flowsSingularitiesKähler geometryScalar curvatureLong-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 Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures (Chuanhuan Li, Yi Li and Kairui Xu, 2022)2Li 2022 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)5Li 2023 Integral curvature estimates for solutions to Ricci flow with L^p bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)8Liu 2024 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)1Wondo 2024 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)3Gallo 2026 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II (Reto Buzano and Gianmichele Di Matteo, 2026)4Buzano 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)6Huang 2026 Backward Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)7He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity Types for Long-Time Chern-Ricci Flow

    Hosea Wondo PDF

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

  2. Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures

    Chuanhuan Li, Yi Li and Kairui Xu PDF

    Topics Ricci flow · Same authors Chuanhuan Li, Yi Li

  3. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “geometric flows”

  4. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II

    Reto Buzano and Gianmichele Di Matteo PDF

    Topics Scalar curvature, Singularities · Shared terms “bounded scalar”

  5. Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

    Chuanhuan Li, Yi Li, Kairui Xu et al. PDF

    Topics Ricci flow · Same authors Chuanhuan Li, Yi Li

  6. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang PDF

    Topics Scalar curvature · Shared terms “bounded scalar”

  7. Backward Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan PDF

    Topics Generalized & G₂ flows, Scalar curvature · Shared terms “Laplacian flow”, “geometric flows”

  8. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon PDF

    Topics Scalar curvature, Kähler geometry · Shared terms “bounded scalar”

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) 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) 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

  1. 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.

  2. 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.

  3. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.