Paper · Nov 14, 2025

Synthetic approaches to Ricci flows

Matthias Erbar, Marco Flaim, Eric Hupp, Zhenhao Li, Timo Schultz, Karl-Theodor Sturm

math.DGarXiv:2511.11477
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Abstract

We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes. Each notion equivalently characterises (weighted) Ricci flow for smooth families of weighted Riemannian manifolds. We discuss the features of the different notions on various examples.

Ricci flowWeak & singular flowsLimit spaces & RCD

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz PDF

    Topics Limit spaces & RCD, Weak & singular flows · Same authors Matthias Erbar, Zhenhao Li, Timo Schultz · Shared terms “synthetic”, “different notions”, “metric measure”

  2. 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 · Same authors Marco Flaim, Karl-Theodor Sturm · Shared terms “synthetic”, “metric measure”, “time dependent”

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

    Karl-Theodor Sturm PDF

    Topics Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “synthetic Ricci”, “metric measure”, “measure spaces”

  4. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp PDF

    Topics Limit spaces & RCD, Weak & singular flows · Same author Marco Flaim · Shared terms “optimal transport”, “heat flow”, “Riemannian manifolds”

  5. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Limit spaces & RCD · Shared terms “optimal transport”, “heat flow”

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

    Xiang-Dong Li PDF

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

  7. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer PDF

    Topics Limit spaces & RCD, Weak & singular flows

  8. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows PDF

    Topics Limit spaces & RCD, Weak & singular flows

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) 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) 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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.