Paper · Sep 1, 2024

Notes on scalar curvature lower bounds of steady gradient Ricci solitons

Shota Hamanaka

math.DGarXiv:2409.00583v4
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Abstract

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose -Bakry–Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use -bubbles introduced by Gromov.

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Submitted
Sep 1, 2024
Last revised
May 21, 2026 (v4)
Subjects
math.DG

Comments: 20 pages. I have made some minor modifications in the proof of Proposition 5.1

Ricci solitonsScalar curvatureLimit spaces & RCD

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 On scalar curvature lower bounds and scalar curvature measure (John Lott, 2021)6Lott 2021 Limit theorems for the total scalar curvature (Shota Hamanaka, 2022)4Hamanaka 2022 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)5Chan 2022 Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds (Robin Neumayer, 2022)7Neumayer 2022 Parabolic frequency monotonicity on the conformal Ricci flow (Abimbola Abolarinwa and Shahroud Azami, 2023)8Abolarinwa 2023 Bounds of Scalar curvature, S-curvature and distortion on ∞-Einstein Finsler manifolds (Bin Shen, 2024)2Shen 2024 Steady soliton with 𝓛¹ decay curvature (Ming Hsiao, 2026)1Hsiao 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Steady soliton with decay curvature

    Ming Hsiao

    Topics Limit spaces & RCD, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “lower bound”

  2. Bounds of Scalar curvature, S-curvature and distortion on -Einstein Finsler manifolds

    Bin Shen

    Topics Limit spaces & RCD, Scalar curvature, Ricci solitons · Shared terms “lower bound”, “gradient Ricci”, “Gromov”

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

    Shaochuang Huang and Zhuo Peng

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “curvature lower”, “lower bound”, “Gromov”

  4. Limit theorems for the total scalar curvature

    Shota Hamanaka

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

  5. Lower bounds for the scalar curvatures of Ricci flow singularity models

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

    Topics Limit spaces & RCD, Scalar curvature, Ricci solitons · Shared terms “lower bound”

  6. On scalar curvature lower bounds and scalar curvature measure

    John Lott

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

  7. Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

    Robin Neumayer

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “lower bound”, “upper bound”, “Riemannian manifolds”

  8. Parabolic frequency monotonicity on the conformal Ricci flow

    Abimbola Abolarinwa and Shahroud Azami

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “Émery Ricci”, “Bakry”, “Riemannian manifolds”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 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)

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.