Paper · Jul 9, 2024

Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

Man-Chun Lee, Stephen Shang Yi Liu

math.DGarXiv:2407.06575v3Pacific J. Math. 340 (2026) 375-397
PDF Abstract page

As on arXiv

Abstract

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang.

Comments: 23 pages; abstract updated; references updated and revised according to reviewer comments. To appear, Pacific Journal of Mathematics

Ricci flowSingularitiesScalar curvatureWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick

Its neighbourhood

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 Ricci flow of W^(2,2)-metrics in four dimensions (Tobias Lamm and Miles Simon, 2021)4Lamm 2021 Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow (Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al., 2022)7Cartas-Fuent 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)1Lee 2022 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)3Burkhardt-Gu 2022 Ricci Flow under Kato-type curvature lower bound (Man-Chun Lee, 2023)6Lee 2023 Removing scalar curvature assumption for Ricci flow smoothing (Adam Martens, 2024)8Martens 2024 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)5Cai 2026 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)2Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Scalar curvature · Same author Man-Chun Lee · Shared terms “curvature lower”, “lower bound”

  2. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

    Topics Scalar curvature · Same author Man-Chun Lee · Shared terms “distributional scalar”, “flow smoothing”, “curvature lower”

  3. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim PDF

    Topics Harmonic maps & DeTurck trick, Scalar curvature

  4. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Scalar curvature

  5. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “flow starting”, “short time”, “lower bound”

  6. Ricci Flow under Kato-type curvature lower bound

    Man-Chun Lee PDF

    Topics Scalar curvature, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “existence theory”, “curvature lower”, “flow smoothing”

  7. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al. PDF

    Topics Harmonic maps & DeTurck trick, Scalar curvature

  8. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens PDF

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Singularities · Shared terms “flow smoothing”, “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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.