Paper · Oct 28, 2026

Backward Pseudolocality of -Laplacian Flow

Weiyong He, Longhuan Yan

math.DGarXiv:2610.38418
PDF Abstract page

As on arXiv

Abstract

In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the -Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

Ricci flowGeneralized & G₂ flowsPerelman 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 Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds (Robin Neumayer, 2022)7Neumayer 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)5Li 2022 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)8Chan 2023 On the structure of complete G₂-solitons (Haozhao Li, Yuanqing Ma and Kai Zheng, 2026)2Li 2026 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)6Li 2026 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)1He 2026 On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan PDF

    Topics Perelman entropy, Generalized & G₂ flows, Scalar curvature · Same authors Weiyong He, Longhuan Yan · Shared terms “Laplacian flow”, “pseudolocality”, “pointed Nash”

  2. On the structure of complete -solitons

    Haozhao Li, Yuanqing Ma and Kai Zheng PDF

    Topics Perelman entropy, Generalized & G₂ flows, Limit spaces & RCD · Shared terms “pseudolocality”, “Laplacian”, “compactness”

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

    Richard H BamlerThe canon PDF

    Topics Perelman entropy, Limit spaces & RCD, Scalar curvature · Shared terms “pointed Nash”, “Nash entropy”, “structure theory”

  4. On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

    Yu Li, Longhuan Yan and Wenjia Zhang PDF

    Topics Perelman entropy · Same author Longhuan Yan

  5. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    Topics Perelman entropy, Generalized & G₂ flows · Shared terms “Nash entropy”

  6. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li PDF

    Topics Generalized & G₂ flows, Scalar curvature · Shared terms “Laplacian flow”, “geometric flows”

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

    Robin Neumayer PDF

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

  8. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Perelman entropy · Shared terms “Nash entropy”, “Bamler”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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)

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.