Paper · Aug 14, 2026

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

Dongwoo Gang

math.DGarXiv:2608.14039
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Abstract

For a closed connected Riemannian manifold , the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding . The resulting family agrees with Ricci flow to first order in , but in general not to second order. We prove that where is quadratic in the full curvature tensor. The term also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is , and round spheres show that this estimate is sharp.

Comments: 20 pages

Ricci flowRicci-flat & Calabi–YauHeat kernelLimit spaces & RCD

Its neighbourhood

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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 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)4Choi 2021 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)6Huang 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)5Kunikawa 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)1Li 2023 Perelman's entropy and heat kernel bounds on RCD spaces (Camillo Brena, 2025)8Brena 2025 Dynamical functionals on ancient ARF Ricci flows (Isaac M. Lopez and Rio Schillmoeller, 2025)7Lopez 2025 Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three (Yu Li, 2026)2Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang PDF

    Topics Heat kernel, Limit spaces & RCD

  2. Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

    Yu Li PDF

    Topics Heat kernel, Limit spaces & RCD · Shared terms “Hausdorff”, “Gromov”, “Riemannian manifolds”

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

    Richard H BamlerThe canon PDF

    Topics Heat kernel, Limit spaces & RCD · Shared terms “pointwise”

  4. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer PDF

    Topics Heat kernel, Limit spaces & RCD

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

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Heat kernel

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

    Hong Huang PDF

    Topics Heat kernel

  7. Dynamical functionals on ancient ARF Ricci flows

    Isaac M. Lopez and Rio Schillmoeller PDF

    Topics Heat kernel, Ricci-flat & Calabi–Yau

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

    Camillo Brena PDF

    Topics Heat kernel, Limit spaces & RCD

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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.