Paper · Apr 21, 2021

Ricci flow with bounded curvature integrals

Shota Hamanaka

math.DGarXiv:2104.10300v3
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Abstract

In this paper, we study the Ricci flow on a closed manifold and finite time interval on which certain integral curvature energies are finite. We prove that in dimension four, such flow converges to a smooth Riemannian manifold except for finitely many orbifold singularities. We also show that in higher dimensions, the same assertions hold for a closed Ricci flow satisfying another conditions of integral curvature bounds. Moreover, we show that such flows can be extended over by an orbifold Ricci flow.

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Submitted
Apr 21, 2021
Last revised
Aug 31, 2021 (v3)
Subjects
math.DG
Journal
Pacific J. Math. 314 (2021) 283-309
DOI
10.2140/pjm.2021.314.283

Comments: 26 pages. arXiv admin note: text overlap with arXiv:1501.01291 by other authors

Ricci flowSingularities4-manifolds

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 The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds (Max Hallgren, 2020)6Hallgren 2020 Global ε-regularity for 4-dimensional Ricci flow with integral scalar curvature bound (Wangjian Jian, 2020)7Jian 2020 Ricci Flow with Ricci Curvature and Volume Bounded Below (Max Hallgren, 2021)3Hallgren 2021 Type of finite time singularities of the Ricci flow with bounded scalar curvature (Shota Hamanaka, 2021)2Hamanaka 2021 Integral curvature estimates for solutions to Ricci flow with L^p bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)1Liu 2024 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)5Cao 2024 Volume estimates and convergence results for solutions to Ricci flow with L^(p) bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)4Liu 2024 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. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon

    Topics 4-manifolds · Shared terms “integral curvature”, “curvature integral”, “closed manifold”

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

    Shota Hamanaka

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

  3. Ricci Flow with Ricci Curvature and Volume Bounded Below

    Max Hallgren

    Topics 4-manifolds, Singularities · Shared terms “orbifold”, “higher dimensions”, “finite time”

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

    Jiawei Liu and Miles Simon

    Topics 4-manifolds · Shared terms “orbifold Ricci”, “time interval”, “extended”

  5. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran

    Topics 4-manifolds, Singularities · Shared terms “dimension four”

  6. The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

    Max Hallgren

    Topics 4-manifolds, Singularities · Shared terms “curvature bounds”, “flow satisfying”, “orbifold singularities”

  7. Global -regularity for 4-dimensional Ricci flow with integral scalar curvature bound

    Wangjian Jian

    Topics 4-manifolds · Shared terms “curvature bounds”

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

    Xianbin Huang and Wancheng Zhang

    Topics 4-manifolds · Shared terms “curvature bounds”

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 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.