Paper · Feb 25, 2021

Small curvature concentration and Ricci flow smoothing

Pak-Yeung Chan, Eric Chen, Man-Chun Lee

math.DGarXiv:2102.12840v3
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Abstract

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times. As applications, we obtain under similar assumptions a compactness result and a gap theorem for complete noncompact manifolds with nonnegative Ricci Curvature.

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Submitted
Feb 25, 2021
Last revised
Feb 5, 2022 (v3)
Subjects
math.DG

Comments: 29 pages, reference updated, result added, minor mistakes fixed

Ricci flowLimit spaces & RCDBoundaries & non-compact

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.

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 On the uniqueness of Ricci flow (Man-Chun Lee, 2017)6Lee 2017 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)5Lee 2019 Manifolds with small curvature concentration (Pak-Yeung Chan, Shaochuang Huang and Man-Chun Lee, 2022)1Chan 2022 Gap Theorem on Riemannian manifolds using Ricci flow (Pak-Yeung Chan and Man-Chun Lee, 2023)2Chan 2023 Sharpening a gap theorem: nonnegative Ricci and small curvature concentration (Adam Martens, 2024)4Martens 2024 Removing scalar curvature assumption for Ricci flow smoothing (Adam Martens, 2024)7Martens 2024 Higher-dimensional flying wing Steady Ricci Solitons (Pak-Yeung Chan, Yi Lai and Man-Chun Lee, 2025)3Chan 2025 A note on Ricci flow from small curvature concentration and a Morrey-type condition (Albert Chau and Adam Martens, 2026)8Chau 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

    Topics Ricci flow · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “small curvature”, “curvature concentration”, “Ricci lower”

  2. Gap Theorem on Riemannian manifolds using Ricci flow

    Pak-Yeung Chan and Man-Chun Lee

    Topics Boundaries & non-compact · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “gap theorem”

  3. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee

    Topics Boundaries & non-compact · Same authors Pak-Yeung Chan, Man-Chun Lee · Shared terms “flow smoothing”

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

    Adam Martens

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

  5. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Limit spaces & RCD, Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “nonnegative Ricci”, “complete noncompact”

  6. On the uniqueness of Ricci flow

    Man-Chun Lee

    Topics Boundaries & non-compact · Same author Man-Chun Lee · Shared terms “noncompact manifolds”, “complete noncompact”

  7. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens

    Topics Limit spaces & RCD · Shared terms “flow smoothing”, “curvature concentration”, “local entropy”

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

    Albert Chau and Adam Martens

    Topics Boundaries & non-compact · Shared terms “small curvature”, “curvature concentration”, “bounded curvature”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.