Paper · Jan 27, 2020

Mixed Integral Norms for Ricci Flow

Gianmichele Di Matteo

math.DGarXiv:2001.09920
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Abstract

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.

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Submitted
Jan 27, 2020
Subjects
math.DG, math.AP

Comments: 16 pages, 1 figure, comments are welcome!

Ricci flowSingularitiesScalar curvature

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 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 Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)5Nomura 2016 Analysis of Type I Singularities in the Harmonic Ricci Flow (Gianmichele Di Matteo, 2018)7Di Matteo 2018 Heat flow on time-dependent manifolds (Beomjun Choi, Jianhui Gao, Robert Haslhofer et al., 2020)8Choi 2020 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature (Reto Buzano and Gianmichele Di Matteo, 2020)1Buzano 2020 Type of finite time singularities of the Ricci flow with bounded scalar curvature (Shota Hamanaka, 2021)3Hamanaka 2021 Some aspects of Ricci flow on the 4-sphere (Sun-Yung Alice Chang and Eric Chen, 2021)6Chang 2021 Volume estimates and convergence results for solutions to Ricci flow with L^(p) bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)4Liu 2024 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II (Reto Buzano and Gianmichele Di Matteo, 2026)2Buzano 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

    Reto Buzano and Gianmichele Di Matteo

    Topics Scalar curvature, Singularities · Same author Gianmichele Di Matteo · Shared terms “curvature tensor”

  2. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II

    Reto Buzano and Gianmichele Di Matteo

    Topics Scalar curvature, Singularities · Same author Gianmichele Di Matteo

  3. Type of finite time singularities of the Ricci flow with bounded scalar curvature

    Shota Hamanaka

    Topics Scalar curvature, Singularities · Shared terms “time singularity”, “finite time”

  4. Volume estimates and convergence results for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon

    Topics Scalar curvature · Shared terms “time integral”, “space time”, “curvature tensor”

  5. Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities

    Ryosuke Nomura

    Topics Scalar curvature, Singularities · Shared terms “time singularity”, “finite time”

  6. Some aspects of Ricci flow on the 4-sphere

    Sun-Yung Alice Chang and Eric Chen

    Topics Scalar curvature · Shared terms “curvature tensor”

  7. Analysis of Type I Singularities in the Harmonic Ricci Flow

    Gianmichele Di Matteo

    Topics Singularities · Same author Gianmichele Di Matteo

  8. Heat flow on time-dependent manifolds

    Beomjun Choi, Jianhui Gao, Robert Haslhofer et al.

    Topics Scalar curvature, Singularities · Shared terms “lower bound”

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) 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)

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. 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.