Paper · Oct 14, 2021

On scalar curvature lower bounds and scalar curvature measure

John Lott

math.DGarXiv:2110.07551v4
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Abstract

We relate the (non)existence of lower scalar curvature bounds to the existence of certain distance-decreasing maps. We also give a sufficient condition for the existence of a limiting scalar curvature measure in the backward limit of a Ricci flow solution.

Comments: Addendum about (3.19) put at end

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 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)1Jiang 2021 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)6Lee 2022 Limit theorems for the total scalar curvature (Shota Hamanaka, 2022)3Hamanaka 2022 Ricci Flow under Kato-type curvature lower bound (Man-Chun Lee, 2023)5Lee 2023 Notes on scalar curvature lower bounds of steady gradient Ricci solitons (Shota Hamanaka, 2024)2Hamanaka 2024 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)8Lee 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)4Huang 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)7Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang PDF

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

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

    Shota Hamanaka PDF

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

  3. Limit theorems for the total scalar curvature

    Shota Hamanaka PDF

    Topics Scalar curvature · Shared terms “lower bound”

  4. 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”

  5. Ricci Flow under Kato-type curvature lower bound

    Man-Chun Lee PDF

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

  6. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

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

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

    Xianbin Huang and Wancheng Zhang PDF

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

  8. 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 “curvature lower”, “lower bound”

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.