Paper · Feb 16, 2024

Ricci flow on Courant algebroids

Jeffrey Streets, Charles Strickland-Constable, Fridrich Valach

math.DGarXiv:2402.11069
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Abstract

We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow.

Comments: 38 pages, 1 figure

Ricci flowGeneralized & G₂ flowsScalar curvatureStability & uniquenessHarmonic maps & DeTurck trickPhysics

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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 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)6Kopfer 2022 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)1Streets 2022 Optimal Transport and Generalized Ricci Flow (Eva Kopfer and Jeffrey Streets, 2023)5Kopfer 2023 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)4Apostolov 2024 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)7Fusi 2024 Pluriclosed flow and the Hull-Strominger system (Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets, 2024)3Garcia-Ferna 2024 Courant Algebroid Relations, T-Dualities and Generalised Ricci Flow (Thomas C. De Fraja, Vincenzo Emilio Marotta and Richard J. Szabo, 2025)2De Fraja 2025 BRIDGES Lectures: Flows of geometric structures, especially G₂-structures (Spiro Karigiannis, 2025)8Karigiannis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Physics, Scalar curvature · Same author Jeffrey Streets · Shared terms “curvature monotonicity”, “monotonicity formula”, “Perelman”

  2. Courant Algebroid Relations, T-Dualities and Generalised Ricci Flow

    Thomas C. De Fraja, Vincenzo Emilio Marotta and Richard J. Szabo PDF

    Topics Generalized & G₂ flows, Stability & uniqueness · Shared terms “Courant algebroids”, “isometry”

  3. Pluriclosed flow and the Hull-Strominger system

    Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “algebroids”, “Courant”, “constructing solutions”

  4. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Scalar curvature, Stability & uniqueness · Same author Jeffrey Streets · Shared terms “Courant algebroids”

  5. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  6. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “yielding”

  7. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

    Topics Generalized & G₂ flows · Shared terms “Courant algebroids”

  8. BRIDGES Lectures: Flows of geometric structures, especially -structures

    Spiro Karigiannis PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows, Stability & uniqueness · Shared terms “geometric flows”, “short time”, “time existence”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.