Paper · Jun 20, 2019

Regularity estimates for the gradient flow of a spinorial energy functional

Fei He, Changliang Wang

math.DGarXiv:1906.08750
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Abstract

In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow.

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Submitted
Jun 20, 2019
Subjects
math.DG

Comments: 32 pages

Ricci flowSingularitiesLong-time existence & convergence

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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 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 Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature (Fei He, 2016)8He 2016 Blowup criteria for geometric flows on surfaces (Lothar Schiemanowski, 2018)4Schiemanowsk 2018 A gradient flow of isometric G₂ structures (Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis, 2019)7Dwivedi 2019 Parallel spinor flows on three-dimensional Cauchy hypersurfaces (Ángel Murcia and C. S. Shahbazi, 2021)3Murcia 2021 The spinorial energy for asymptotically Euclidean Ricci flow (Julius Baldauf and Tristan Ozuch, 2022)1Baldauf 2022 Perelman's functionals on manifolds with non-isolated conical singularities (Xianzhe Dai and Changliang Wang, 2022)6Dai 2022 A Spinorial Heat Flow Framework for Geometric Degeneration on 3-Manifolds (Ferhat Taş, 2025)2Taş 2025 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)5Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch

    Topics Ricci flow · Shared terms “spinorial energy”, “gradient flow”

  2. A Spinorial Heat Flow Framework for Geometric Degeneration on -Manifolds

    Ferhat Taş

    Topics Singularities · Shared terms “spinorial”, “spinor field”, “observation”

  3. Parallel spinor flows on three-dimensional Cauchy hypersurfaces

    Ángel Murcia and C. S. Shahbazi

    Topics Long-time existence & convergence · Shared terms “spinor flow”, “spinorial”, “initial data”

  4. Blowup criteria for geometric flows on surfaces

    Lothar Schiemanowski

    Topics Singularities · Shared terms “spinor flow”, “criteria”

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

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong

    Shared terms “spinorial”, “gradient flow”, “energy functional”

  6. Perelman's functionals on manifolds with non-isolated conical singularities

    Xianzhe Dai and Changliang Wang

    Topics Singularities · Same author Changliang Wang · Shared terms “lower bound”

  7. A gradient flow of isometric structures

    Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis

    Topics Long-time existence & convergence, Singularities · Shared terms “gradient flow”, “energy functional”, “initial data”

  8. Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature

    Fei He

    Topics Ricci flow · Same author Fei He · Shared terms “unbounded”, “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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 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.

  2. 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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.