Paper · Nov 13, 2025

Dynamical functionals on ancient ARF Ricci flows

Isaac M. Lopez, Rio Schillmoeller

math.DGarXiv:2511.10460
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Abstract

We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary -functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.

Comments: 17 pages

Ricci flowAncient solutionsRicci-flat & Calabi–YauHeat kernelStability & uniquenessEigenvalues & spectrum

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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 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)1Huang 2021 Eigenvalue estimates on shrinkers (S. Brendle and R. Tsiamis, 2024)6Brendle 2024 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)2Lopez 2024 Convergence of Ricci flow and long-time existence of Harmonic map heat flow (Kyeongsu Choi and Yi Lai, 2025)7Choi 2025 Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula (M. J. Luo, 2025)4Luo 2025 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)8Koirala 2026 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)3Chow 2026 Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow (Dongwoo Gang, 2026)5Gang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”, “upper bound”

  2. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions · Same author Isaac M. Lopez

  3. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala PDF

    Topics Heat kernel, Eigenvalues & spectrum, Stability & uniqueness · Shared terms “conjugate heat”

  4. Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula

    M. J. Luo PDF

    Topics Heat kernel · Shared terms “conjugate heat”

  5. Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow

    Dongwoo Gang PDF

    Topics Heat kernel, Ricci-flat & Calabi–Yau

  6. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis PDF

    Topics Eigenvalues & spectrum · Shared terms “eigenvalue estimate”

  7. Convergence of Ricci flow and long-time existence of Harmonic map heat flow

    Kyeongsu Choi and Yi Lai PDF

    Topics Eigenvalues & spectrum, Ancient solutions, Stability & uniqueness · Shared terms “heat flow”

  8. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala PDF

    Topics Heat kernel, Stability & uniqueness · 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) 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)

In the guidebook

  1. 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.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 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.