Paper · Jun 2, 2023

Optimal Transport and Generalized Ricci Flow

Eva Kopfer, Jeffrey Streets

math.DGarXiv:2306.01649v2SIGMA 20 (2024), 003, 15 pages
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Abstract

We prove results relating the theory of optimal transport and generalized Ricci flow. We define an adapted cost functional for measures using a solution of the associated dilaton flow. This determines a formal notion of geodesics in the space of measures, and we show geodesic convexity of an associated entropy functional. Finally, we show monotonicity of the cost along the backwards heat flow, and use this to give a new proof of the monotonicity of the energy functional along generalized Ricci flow.

Ricci flowGeneralized & G₂ flowsLimit spaces & RCD

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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 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)4Gindi 2021 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)1Kopfer 2022 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)2Streets 2022 Ricci flow on Courant algebroids (Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach, 2024)7Streets 2024 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)6Erbar 2025 The canonical symmetry reduction of string backgrounds (Aaron Kennon and Jeffrey Streets, 2025)8Kennon 2025 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)5Streets 2025 Topology of low-dimensional generalized Ricci solitons and string backgrounds (Jeffrey Streets, 2026)3Streets 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same authors Eva Kopfer, Jeffrey Streets · Shared terms “determine”

  2. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  3. Topology of low-dimensional generalized Ricci solitons and string backgrounds

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Limit spaces & RCD · Same author Jeffrey Streets · Shared terms “determine”

  4. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  5. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang PDF

    Topics Generalized & G₂ flows, Limit spaces & RCD · Same author Jeffrey Streets

  6. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz PDF

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

  7. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  8. The canonical symmetry reduction of string backgrounds

    Aaron Kennon and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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.