Paper · Jun 18, 2022

The spinorial energy for asymptotically Euclidean Ricci flow

Julius Baldauf, Tristan Ozuch

math.DGarXiv:2206.09198
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Abstract

This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.

Comments: 26 pages

Ricci flowBoundaries & non-compactMinimal & CMC surfaces

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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 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 Spinors and mass on weighted manifolds (Julius Baldauf and Tristan Ozuch, 2022)1Baldauf 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)8Carron 2022 Harmonic spinors in the Ricci flow (Julius Baldauf, 2022)4Baldauf 2022 The Smale conjecture and min-max theory (Daniel Ketover and Yevgeny Liokumovich, 2023)3Ketover 2023 Semi-Continuity of the Morse Index for Ricci Shrinkers (Louis Yudowitz, 2024)7Yudowitz 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)5Chu 2024 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)6Chen 2025 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)2Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch PDF

    Topics Ricci flow · Same authors Julius Baldauf, Tristan Ozuch · Shared terms “asymptotically Euclidean”, “spinorial”, “Dirichlet energy”

  2. A Spinorial Perelman's Functional: Critical Points and Gradient Flow

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong PDF

    Shared terms “spinorial”, “critical points”, “spin manifolds”

  3. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich PDF

    Topics Minimal & CMC surfaces · Shared terms “min max”

  4. Harmonic spinors in the Ricci flow

    Julius Baldauf PDF

    Topics Ricci flow · Same author Julius Baldauf · Shared terms “gradient flow”, “Perelman”

  5. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “min max”

  6. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “critical points”, “gradient flow”

  7. Semi-Continuity of the Morse Index for Ricci Shrinkers

    Louis Yudowitz PDF

    Topics Minimal & CMC surfaces, Boundaries & non-compact

  8. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows PDF

    Topics Boundaries & non-compact · Shared terms “asymptotically Euclidean”, “Euclidean manifolds”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.