Paper · Mar 7, 2016

Super-Ricci Flows for Metric Measure Spaces

Karl-Theodor Sturm

math.DGarXiv:1603.02193v2
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Abstract

We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces . The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of the synthetic lower Ricci bounds of Lott-Sturm-Villani for static metric measure spaces, the defining property for super-Ricci flows is the `dynamic convexity' of the Boltzmann entropy regarded as a functions on the time-dependent geodesic space . For Ricci flows, in addition a nearly dynamic concavity of the Boltzmann entropy is requested. Alternatively, super-Ricci flows will be studied in the framework of the -calculus of Bakry-Émery-Ledoux and equivalence to gradient estimates will be derived. For both notions of super-Ricci flows, also enforced versions involving an `upper dimension bound' will be presented.

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Submitted
Mar 7, 2016
Last revised
Aug 9, 2017 (v2)
Subjects
math.DG, math.FA, math.MG
Ricci flowWeak & singular flowsLimit spaces & RCD

Its neighbourhood

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 2016, 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 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)1Kopfer 2016 Super Ricci flows for weighted graphs (Matthias Erbar and Eva Kopfer, 2018)7Erbar 2018 Functional inequalities for the heat flow on time-dependent metric measure spaces (Eva Kopfer and Karl-Theodor Sturm, 2019)2Kopfer 2019 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)5Sturm 2024 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)6Erbar 2025 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)8Li 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)4Erbar 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)3Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

    Eva Kopfer and Karl-Theodor Sturm

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

  2. Functional inequalities for the heat flow on time-dependent metric measure spaces

    Eva Kopfer and Karl-Theodor Sturm

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

  3. 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 · Same author Karl-Theodor Sturm · Shared terms “super Ricci”, “metric measure”, “measure spaces”

  4. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

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

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

    Karl-Theodor Sturm

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

  6. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

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

  7. Super Ricci flows for weighted graphs

    Matthias Erbar and Eva Kopfer

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

  8. 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 “super Ricci”, “metric measure”, “measure spaces”

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.

Foundational paper in the Weak & singular flows guide: Sturm: super-Ricci flows for metric measure spaces.

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.