Paper · Aug 3, 2022

Limit theorems for the total scalar curvature

Shota Hamanaka

math.DGarXiv:2208.01865v18
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Abstract

We study some preservation phenomena for lower bound of total scalar curvatures on a smooth manifold. In particular, we prove that the lower bound of the weighted total scalar curvature (which is known as Perelman's -functional) on a closed -manifold is preserved under the -convergence of Riemannian metrics and uniformly -convergence of potential functions, provided that each scalar curvature is nonnegative. In the proof, we used a certain stability of the Ricci flow and the heat flow with the Ricci flow background. We also give some examples that may provide clues to identify the weakest topology for such a preservation phenomenon of the lower bound.

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Submitted
Aug 3, 2022
Last revised
Sep 12, 2025 (v18)
Subjects
math.DG

Comments: 36 pages

Ricci flowScalar curvatureStability & uniqueness

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Jian Ge, Chuanhuan Li and Ronggang Li

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

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

    Shota Hamanaka

    Topics Scalar curvature · Same author Shota Hamanaka · Shared terms “lower bound”

  3. On scalar curvature lower bounds and scalar curvature measure

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    Topics Scalar curvature · Shared terms “lower bound”

  4. Weak scalar curvature lower bounds along Ricci flow

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

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

    Shaochuang Huang and Zhuo Peng

    Topics Scalar curvature, Stability & uniqueness · Shared terms “lower bound”, “preservation”

  6. Ricci Flow under Kato-type curvature lower bound

    Man-Chun Lee

    Topics Scalar curvature, Stability & uniqueness · Shared terms “lower bound”, “smooth manifold”

  7. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Scalar curvature, Stability & uniqueness · Shared terms “heat flow”, “lower bound”

  8. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Scalar curvature, Stability & uniqueness · Shared terms “potential function”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.