Paper · May 18, 2021

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

Shota Hamanaka

math.DGarXiv:2105.08250v5
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Abstract

In this paper, we study the Ricci flow on a closed manifold of dimension and finite time interval on which the scalar curvature are uniformly bounded. We prove that if such flow of dimension has finite time singularities, then every blow-up sequence of a locally Type I singularity has certain property. Here, locally Type I singularity is what Buzano and Di-Matteo defined.

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Submitted
May 18, 2021
Last revised
Mar 29, 2022 (v5)
Subjects
math.DG

Comments: 17 pages. arXiv admin note: text overlap with arXiv:2006.16227 by other authors

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 Convergence of Ricci flows with bounded scalar curvature (Richard H. Bamler, 2016)6Bamler 2016 Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)3Nomura 2016 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)2Buzano 2020 Ricci flow with bounded curvature integrals (Shota Hamanaka, 2021)4Hamanaka 2021 Limit theorems for the total scalar curvature (Shota Hamanaka, 2022)7Hamanaka 2022 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)1Li 2026 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)5Buzano 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)8Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li

    Topics Scalar curvature, Singularities · Shared terms “bounded scalar”

  2. 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 · Shared terms “bounded scalar”

  3. 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”

  4. Ricci flow with bounded curvature integrals

    Shota Hamanaka

    Topics Singularities · Same author Shota Hamanaka · Shared terms “time interval”, “finite time”, “closed manifold”

  5. 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 · Shared terms “bounded scalar”

  6. Convergence of Ricci flows with bounded scalar curvature

    Richard H. Bamler

    Topics Scalar curvature · Shared terms “bounded scalar”, “sequence”

  7. Limit theorems for the total scalar curvature

    Shota Hamanaka

    Topics Scalar curvature · Same author Shota Hamanaka

  8. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang

    Topics Scalar curvature · Shared terms “bounded scalar”, “uniformly bounded”

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.