Paper · May 20, 2026

Sharp Gaussian Isoperimetry along a Ricci Flow

Robert Koirala

math.DGarXiv:2605.21193
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Abstract

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's -Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's -centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

Comments: 25 pages, comments welcome

Ricci flowPerelman entropyHeat kernelStability & uniquenessGeometric inequalities

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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)3Bamler 2020 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)8Huang 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)5Kunikawa 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)4Li 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)7Li 2023 Perelman's entropy and heat kernel bounds on RCD spaces (Camillo Brena, 2025)6Brena 2025 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)1Chow 2026 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)2He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala PDF

    Topics Heat kernel, Perelman entropy, Stability & uniqueness · Same author Robert Koirala · Shared terms “conjugate heat”, “kernel measure”, “Poincaré inequality”

  2. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan PDF

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

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

    Richard H BamlerThe canon PDF

    Topics Heat kernel, Perelman entropy · Shared terms “Hein”, “conjugate heat”, “Poincaré inequality”

  4. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    Topics Heat kernel, Perelman entropy

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

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Heat kernel · Shared terms “Gaussian”, “Bamler”

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

    Camillo Brena PDF

    Topics Heat kernel, Perelman entropy, Stability & uniqueness

  7. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang PDF

    Topics Heat kernel · Shared terms “Bamler”

  8. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”

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. 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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 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.

  2. 6D.4
    Fokker–Planck, Time Reversal and Diffusion ModelsBook 6D · Diffusions, Finance and Computation

    How densities of diffusions evolve, relative entropy and convergence to equilibrium, reversing a diffusion in time with the score, and the generative models built on it, with Perelman's conjugate heat equation as the geometric twin.

  3. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.