Paper · Nov 4, 2026

Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres

Reto Buzano, Shengwen Wang

math.DGarXiv:2611.06678
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Abstract

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

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Submitted
Nov 4, 2026
Subjects
math.DG
Ricci flowPerelman entropy3-manifoldsLimit spaces & RCDStability & uniquenessLong-time existence & convergence

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 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)5Li 2021 Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds (Robin Neumayer, 2022)1Neumayer 2022 Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds (Shouhei Honda, Christian Ketterer, Ilaria Mondello et al., 2023)3Honda 2023 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)2Rafik 2024 Perelman's entropy and heat kernel bounds on RCD spaces (Camillo Brena, 2025)6Brena 2025 On the structure of complete G₂-solitons (Haozhao Li, Yuanqing Ma and Kai Zheng, 2026)4Li 2026 Stochastic Thermodynamics on Time-Evolving Curved Spaces (Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al., 2026)8Nagase 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)7Angella 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

    Robin Neumayer

    Topics Perelman entropy, Limit spaces & RCD · Shared terms “Gromov Hausdorff”

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

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy · Shared terms “dynamical”, “perturbation”

  3. Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

    Shouhei Honda, Christian Ketterer, Ilaria Mondello et al.

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “Gromov Hausdorff”, “Hausdorff stability”, “Hausdorff closeness”

  4. On the structure of complete -solitons

    Haozhao Li, Yuanqing Ma and Kai Zheng

    Topics Perelman entropy, Limit spaces & RCD, Long-time existence & convergence · Shared terms “Gromov Hausdorff”

  5. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “Gromov Hausdorff”

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

    Camillo Brena

    Topics Perelman entropy, Limit spaces & RCD, Stability & uniqueness

  7. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

    Topics Limit spaces & RCD, Long-time existence & convergence · Shared terms “Gromov Hausdorff”, “normalised”

  8. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy

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) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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.