Paper · Sep 6, 2025

A note on a diffeomorphism criterion via long-time Ricci flow

Shaochuang Huang, Zhuo Peng

math.DGarXiv:2509.05802
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Abstract

In this note, we give a diffeomorphism (to ) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in [Manifolds with small curvature concentration, Ann. PDE, 2024] by Chan, Lee and the first named author and [Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc., 2025] by A. Martens about manifolds with small curvature concentration can be improved to diffeomorphism in dimension .

Comments: 9 pages, all comments are welcome

Ricci flowScalar curvature

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.

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 Manifolds with small curvature concentration (Pak-Yeung Chan, Shaochuang Huang and Man-Chun Lee, 2022)1Chan 2022 Sharpening a gap theorem: nonnegative Ricci and small curvature concentration (Adam Martens, 2024)7Martens 2024 Removing scalar curvature assumption for Ricci flow smoothing (Adam Martens, 2024)3Martens 2024 A note on Ricci flow from small curvature concentration and a Morrey-type condition (Albert Chau and Adam Martens, 2026)8Chau 2026 On an invariant curvature cone along 4-dimensional Ricci flow (Hongting Ding, Shaochuang Huang and Zhuo Peng, 2026)6Ding 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)2Huang 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)5Huang 2026 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)4Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Manifolds with small curvature concentration

    Pak-Yeung Chan, Shaochuang Huang and Man-Chun Lee PDF

    Topics Scalar curvature · Same author Shaochuang Huang · Shared terms “small curvature”, “curvature concentration”

  2. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Scalar curvature · Same authors Shaochuang Huang, Zhuo Peng

  3. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens PDF

    Topics Scalar curvature · Shared terms “Removing scalar”, “flow smoothing”, “curvature concentration”

  4. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci flow · Same authors Shaochuang Huang, Zhuo Peng · Shared terms “affirmative”, “conclusion”

  5. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci flow · Same authors Shaochuang Huang, Zhuo Peng

  6. On an invariant curvature cone along 4-dimensional Ricci flow

    Hongting Ding, Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci flow · Same authors Shaochuang Huang, Zhuo Peng

  7. Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

    Adam Martens PDF

    Topics Ricci flow · Shared terms “curvature concentration”, “small curvature”, “time Ricci”

  8. A note on Ricci flow from small curvature concentration and a Morrey-type condition

    Albert Chau and Adam Martens PDF

    Topics Ricci flow · Shared terms “small curvature”, “curvature concentration”, “long 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 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.