Paper · May 17, 2018

Super Ricci flows for weighted graphs

Matthias Erbar, Eva Kopfer

math.DGarXiv:1805.06703
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Abstract

We present a notion of super Ricci flow for time-dependent finite weighted graphs. A challenging feature is that these flows typically encounter singularities where the underlying graph structure changes. Our notion is robust enough to allow the flow to continue past these singularities. As a crucial tool for this purpose we study the heat flow on such singular time-dependent weighted graphs with changing graph structure. We then give several equivalent characterizations of super Ricci flows in terms of a discrete dynamic Bochner inequality, gradient and transport estimates for the heat flow, and dynamic convexity of the entropy along discrete optimal transport paths. The latter property can be used to show that our notion of super Ricci flow is consistent with classical super Ricci flows for manifolds (or metric measure spaces) in a discrete to continuum limit.

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Submitted
May 17, 2018
Subjects
math.DG, math.PR

Comments: 40 pages; Comments welcome!

Ricci flowSingularitiesWeak & singular flowsLimit spaces & RCDDiscrete & graph Ricci

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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 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 Smoothing and non-smoothing via a flow tangent to the Ricci flow (Matthias Erbar and Nicolas Juillet, 2016)8Erbar 2016 Super-Ricci Flows for Metric Measure Spaces (Karl-Theodor Sturm, 2016)5Sturm 2016 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)1Kopfer 2016 Functional inequalities for the heat flow on time-dependent metric measure spaces (Eva Kopfer and Karl-Theodor Sturm, 2019)4Kopfer 2019 Optimal Transport and Generalized Ricci Flow (Eva Kopfer and Jeffrey Streets, 2023)6Kopfer 2023 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)2Erbar 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)3Erbar 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)7Flaim 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 Eva Kopfer · Shared terms “super Ricci”, “time dependent”, “heat flow”

  2. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

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

  3. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

    Topics Limit spaces & RCD, Weak & singular flows · Same author Matthias Erbar · Shared terms “time dependent”, “heat flow”, “optimal transport”

  4. 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 Eva Kopfer · Shared terms “time dependent”, “super Ricci”, “heat flow”

  5. Super-Ricci Flows for Metric Measure Spaces

    Karl-Theodor Sturm

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

  6. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets

    Topics Limit spaces & RCD · Same author Eva Kopfer · Shared terms “optimal transport”, “heat flow”

  7. 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 “super Ricci”, “time dependent”, “metric measure”

  8. Smoothing and non-smoothing via a flow tangent to the Ricci flow

    Matthias Erbar and Nicolas Juillet

    Topics Limit spaces & RCD, Weak & singular flows, Singularities · Same author Matthias Erbar · Shared terms “transport”, “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.

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.