Paper · Jan 13, 2025

Synthetic notions of Ricci flow for metric measure spaces

Matthias Erbar, Zhenhao Li, Timo Schultz

math.DGarXiv:2501.07175
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Abstract

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.

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Submitted
Jan 13, 2025
Subjects
math.DG, math.MG

Comments: 62 pages, Comments welcome

Ricci flowWeak & singular flowsLimit spaces & RCD

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  1. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

    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

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

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

    Karl-Theodor Sturm

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

  4. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets

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

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

    Xiang-Dong Li

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

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

    Ali Taheri and Vahideh Vahidifar

    Topics Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”, “time dependent”

  7. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “optimal transport”, “along Wasserstein”, “Wasserstein geodesics”

  8. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

    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.