Paper · Oct 11, 2024

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

Jiawei Liu, Miles Simon

math.DGarXiv:2410.08667v2
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Abstract

In this paper we study -dimensional Ricci flows where is a potentially singular time, and for which the spatial norm, , of the scalar curvature is uniformly bounded on In the case that is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper of the authors, as well as non-inflating estimates (already known due to works of Bamler), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to , then the solution convergences to an orbifold as and that the flow can be extended using the Orbifold Ricci flow to the time interval for some We also prove local versions of many of the results mentioned above.

Comments: Version 2: Included some missing references

Ricci flow(Non)collapsing4-manifoldsScalar curvature

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    Jiawei Liu and Miles Simon PDF

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

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    Topics 4-manifolds, Scalar curvature · Shared terms “bounded scalar”, “Bamler”, “uniformly bounded”

  3. On the Hamilton-Lott conjecture in higher dimensions

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    Topics (Non)collapsing, Scalar curvature · Same author Miles Simon · Shared terms “dimensional Ricci”

  4. Ricci flow of -metrics in four dimensions

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  5. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

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    Topics 4-manifolds, (Non)collapsing, Scalar curvature · Shared terms “bounded scalar”, “dimensional Ricci”, “singular time”

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

    Wangjian Jian PDF

    Topics (Non)collapsing, Scalar curvature · Shared terms “improved”

  7. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

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

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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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.