Paper · Oct 28, 2026

Pseudolocality of -Laplacian Flow

Weiyong He, Longhuan Yan

math.DGarXiv:2610.38408
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Abstract

In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local -entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the -Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Ricci flowGeneralized & G₂ flowsPerelman entropyHarnack inequalitiesHeat kernelScalar curvatureStability & uniqueness

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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)2Bamler 2020 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)7Kunikawa 2022 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)6Streets 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)3Li 2022 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)5Koirala 2026 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)4Chow 2026 Backward 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)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Backward 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. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon PDF

    Topics Heat kernel, Perelman entropy, Scalar curvature · Shared terms “pointed Nash”, “Nash entropy”, “basepoint”

  3. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

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

  4. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala PDF

    Topics Heat kernel, Perelman entropy, Stability & uniqueness · Shared terms “pointed Nash”, “Nash entropy”, “conjugate heat”

  5. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala PDF

    Topics Heat kernel, Perelman entropy, Stability & uniqueness · Shared terms “Hein”, “conjugate heat”, “kernel measure”

  6. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets PDF

    Topics Heat kernel, Generalized & G₂ flows, Scalar curvature · Shared terms “Nash entropy”, “pseudolocality”, “conjugate heat”

  7. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Heat kernel, Scalar curvature · Shared terms “kernel estimates”, “Bamler”, “geometric flows”

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

    Yu Li, Longhuan Yan and Wenjia Zhang PDF

    Topics Perelman entropy, Stability & uniqueness · Same author Longhuan Yan · Shared terms “quantitative”, “modified”

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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 12A.9
    PseudolocalityBook 12A · Entropy and Noncollapsing

    Almost-Euclidean regions stay controlled, and the Harnack inequality behind it.

  3. 12A.10
    The Proof of PseudolocalityBook 12A · Entropy and Noncollapsing

    Perelman's §10 in detail: the point-picking argument, the conjugate heat kernel at the bad point, the localised entropy, and the contradiction with the almost-Euclidean isoperimetric inequality.