Paper · Aug 16, 2020

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

Wangjian Jian

math.DGarXiv:2008.06881
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Abstract

Ge-Jiang (Geom Funct Anal 27:1231-1256, 2017) proved global -regularity for 4-dimensional Ricci flow with bounded scalar curvature. In this note, we extend this result to 4-dimensional Ricci flow with integral bound on the scalar curvature.

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Submitted
Aug 16, 2020
Subjects
math.DG
Journal
Pacific J. Math. 319 (2022) 333-342
DOI
10.2140/pjm.2022.319.333

Comments: 9 pages, all comments are welcome

Ricci flow4-manifoldsScalar curvature

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang

    Topics 4-manifolds, Scalar curvature · Shared terms “curvature bounds”, “bounded scalar”, “Jiang”

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

    Jiawei Liu and Miles Simon

    Topics 4-manifolds, Scalar curvature · Shared terms “integral bound”, “bounded scalar”, “dimensional Ricci”

  3. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon

    Topics 4-manifolds, Scalar curvature · Shared terms “bounded scalar”, “dimensional Ricci”

  4. Geometric Structure of Ends of Ricci Shrinkers

    Alessandro Bertellotti and Reto Buzano

    Topics 4-manifolds, Scalar curvature · Shared terms “curvature bounds”

  5. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Scalar curvature · Shared terms “integral scalar”

  6. On the improved no-local-collapsing theorem of Ricci flow

    Wangjian Jian

    Topics Scalar curvature · Same author Wangjian Jian · Shared terms “curvature bounds”

  7. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics 4-manifolds, Scalar curvature

  8. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature

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) 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) 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. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.