Paper · May 6, 2025

On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces

Xiang-Dong Li

math.DGarXiv:2505.03202
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Abstract

In this paper, we extend Perelman's -entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the -entropy on RCD spaces. Finally, we use the -entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.

Comments: arXiv admin note: text overlap with arXiv:2504.01864

Ricci flow(Non)collapsingHarnack inequalitiesWeak & singular flowsLimit spaces & RCD

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  1. Monotonicity of Perelman -Entropy of Mean Curvature Flow

    Xiang-Dong Li and Qi Yan PDF

    Topics Harnack inequalities · Same author Xiang-Dong Li · Shared terms “Perelman”, “Hamilton”

  2. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows PDF

    Topics Harnack inequalities · Same author Xiang-Dong Li · Shared terms “Yau Hamilton”

  3. Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

    Ali Taheri and Vahideh Vahidifar PDF

    Topics Harnack inequalities, Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”, “Perelman”

  4. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz PDF

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “metric measure”, “measure spaces”, “concavity”

  5. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm PDF

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “super Ricci”, “metric measure”, “measure spaces”

  6. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm PDF

    Topics Harnack inequalities, Limit spaces & RCD, Weak & singular flows · Shared terms “metric measure”, “measure spaces”, “equivalence”

  7. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu PDF

    Topics Harnack inequalities, Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”, “Yau”

  8. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp PDF

    Topics Harnack inequalities, Limit spaces & RCD, Weak & singular flows · Shared terms “super Ricci”, “Perelman Harnack”, “equivalence”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 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.