Paper · Oct 30, 2022

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

Robin Neumayer

math.DGarXiv:2210.16852v2
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Abstract

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-Hölder and bi- homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.

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Submitted
Oct 30, 2022
Last revised
Nov 8, 2022 (v2)
Subjects
math.DG

Comments: Missing reference added

Perelman entropyScalar 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 Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)3Bamler 2020 Notes on scalar curvature lower bounds of steady gradient Ricci solitons (Shota Hamanaka, 2024)1Hamanaka 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)7Rafik 2024 Bounds of Scalar curvature, S-curvature and distortion on ∞-Einstein Finsler manifolds (Bin Shen, 2024)6Shen 2024 Perelman's entropy and heat kernel bounds on RCD spaces (Camillo Brena, 2025)8Brena 2025 On the structure of complete G₂-solitons (Haozhao Li, Yuanqing Ma and Kai Zheng, 2026)2Li 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)5Huang 2026 Backward Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)4He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka

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

  2. On the structure of complete -solitons

    Haozhao Li, Yuanqing Ma and Kai Zheng

    Topics Perelman entropy, Limit spaces & RCD, Scalar curvature · Shared terms “almost Euclidean”, “Gromov Hausdorff”, “lower bound”

  3. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon

    Topics Perelman entropy, Limit spaces & RCD, Scalar curvature · Shared terms “compactness”

  4. Backward Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan

    Topics Perelman entropy, Limit spaces & RCD, Scalar curvature · Shared terms “compactness”

  5. 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 “lower bound”, “Gromov Hausdorff”

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

    Bin Shen

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

  7. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy

  8. Perelman's entropy and heat kernel bounds on RCD spaces

    Camillo Brena

    Topics Perelman entropy, Limit spaces & RCD

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.

Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.