Paper · Oct 23, 2021

Weak scalar curvature lower bounds along Ricci flow

Wenshuai Jiang, Weimin Sheng, Huaiyu Zhang

math.DGarXiv:2110.12157v2
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Abstract

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is for some . As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in distributional sense, then the manifold is isometric to a Ricci flat manifold.

Comments: 25 pages

Ricci flowRicci-flat & Calabi–YauScalar curvature

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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 On scalar curvature lower bounds and scalar curvature measure (John Lott, 2021)2Lott 2021 Limit theorems for the total scalar curvature (Shota Hamanaka, 2022)8Hamanaka 2022 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)1Chan 2022 Notes on scalar curvature lower bounds of steady gradient Ricci solitons (Shota Hamanaka, 2024)4Hamanaka 2024 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)5Kim 2025 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)7Lee 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)6Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al. PDF

    Topics Ricci-flat & Calabi–Yau, Scalar curvature · Shared terms “lower bound”

  2. On scalar curvature lower bounds and scalar curvature measure

    John Lott PDF

    Topics Scalar curvature · Shared terms “curvature lower”, “lower bound”

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

    Shaochuang Huang and Zhuo Peng PDF

    Topics Scalar curvature · Shared terms “curvature lower”, “lower bound”, “along Ricci”

  4. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka PDF

    Topics Scalar curvature · Shared terms “curvature lower”, “lower bound”

  5. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau · Shared terms “along Ricci”

  6. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci-flat & Calabi–Yau · Shared terms “initial metric”

  7. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

    Topics Scalar curvature · Shared terms “distributional”, “curvature lower”, “lower bound”

  8. Limit theorems for the total scalar curvature

    Shota Hamanaka PDF

    Topics Scalar curvature · Shared terms “lower bound”, “preserved”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.