Paper · Nov 28, 2025

Monotonicity of Perelman -Entropy of Mean Curvature Flow

Xiang-Dong Li, Qi Yan

math.DGarXiv:2511.23074
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Abstract

In this paper, we study Perelman' s entropy for mean curvature flow in . Analogously to Perelman's -entropy defined for Ricci flow, K. Ecker in defined a functional for the mean curvature flow in and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined -entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this -entropy.

Comments: 8 pages

Ricci flowMean curvature flowHarnack inequalitiesStability & uniqueness

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

  3. Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

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  4. Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

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  5. Mean curvature flow in an extended Ricci flow background

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    Topics Harnack inequalities · Shared terms “Perelman”, “Hamilton”

  6. Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

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    Topics Harnack inequalities · Shared terms “Hamilton”

  7. Existence of martingale solutions for a stochastic weighted mean curvature flow of graphs

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  8. The Harnack inequality without convexity for curve shortening flow

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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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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. 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.