Paper · Jan 22, 2025

Deriving Perelman's entropy from Colding's monotonic volume

Ignacio Bustamante, Martin Reiris

math.DGarXiv:2501.12949v3
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As on arXiv

Abstract

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed -space, which becomes Ricci-flat as , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's -space.

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Submitted
Jan 22, 2025
Last revised
Jul 15, 2025 (v3)
Subjects
math.DG, math.AP
DOI
10.1515/crelle-2025-0034

Comments: Final version. To appear in Crelle's journal

Ricci-flat & Calabi–YauPerelman entropyReduced volume

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 Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds (Robin Neumayer, 2022)8Neumayer 2022 Wrinkling of fluid deformable surfaces (Veit Krause and Axel Voigt, 2024)5Krause 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)4Rafik 2024 An ε-regularity theorem for Perelman's reduced volume (Liang Cheng and Yongjia Zhang, 2025)1Cheng 2025 On a parabolic curvature lower bound generalizing Ricci flows (Marco Flaim and Erik Hupp, 2025)2Flaim 2025 Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type (Haoyuan Sun, 2026)6Sun 2026 Stochastic Thermodynamics on Time-Evolving Curved Spaces (Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al., 2026)7Nagase 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)3Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An -regularity theorem for Perelman's reduced volume

    Liang Cheng and Yongjia Zhang

    Topics Reduced volume · Shared terms “Perelman”

  2. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp

    Topics Reduced volume, Perelman entropy

  3. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Reduced volume, Perelman entropy

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

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy

  5. Wrinkling of fluid deformable surfaces

    Veit Krause and Axel Voigt

    Topics Reduced volume

  6. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    Haoyuan Sun

    Topics Reduced volume · Shared terms “Gromov”, “Perelman”

  7. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy

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

    Robin Neumayer

    Topics Perelman entropy · Shared terms “Gromov”

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-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Reduced volume Perelman's ℒ-length is a space-time distance adapted to Ricci flow. The reduced volume built from it is monotone, playing the role Huisken's… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 12A.5
    Reduced Distance and Reduced VolumeBook 12A · Entropy and Noncollapsing

    A space-time Bishop–Gromov theorem.

  2. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.

  3. 12A.7
    Monotonicity of the Reduced VolumeBook 12A · Entropy and Noncollapsing

    The 𝓛-exponential map and its Jacobian, the injectivity domain, the limit at τ = 0, barrier inequalities on complete flows, and min ℓ ≤ n/2, following Morgan and Tian's chapters 6 and 7.