Paper · Aug 17, 2017

The rigidity of under ancient Ricci flow

Yongjia Zhang

math.DGarXiv:1708.05459v2
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Abstract

In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I -solutions, namely, those whose asymptotic shrinkers are the standard cylinder . We use this stability result to prove a rigidity theorem: if a four-dimensional Type I -solution with nonnegative curvature operator has the standard cylinder as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.

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Submitted
Aug 17, 2017
Last revised
Oct 14, 2017 (v2)
Subjects
math.DG
Ricci flowRicci solitonsAncient solutionsSingularities4-manifoldsStability & uniqueness

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 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)8Zhang 2017 On three-dimensional Type I κ-solutions to the Ricci flow (Yongjia Zhang, 2017)1Zhang 2017 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)3Li 2018 Perelman's entropy on ancient Ricci flows (Zilu Ma and Yongjia Zhang, 2021)4Ma 2021 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 Ancient Ricci flows with asymptotic solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)7Chan 2021 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)6Deng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On three-dimensional Type I -solutions to the Ricci flow

    Yongjia Zhang

    Topics 4-manifolds, Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “dimensional Type”, “four dimensional”

  2. On the noncollapsedness of positively curved Type I ancient Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang · Shared terms “curved Type”

  3. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Ancient solutions, Ricci solitons · Same author Yongjia Zhang · Shared terms “standard cylinder”, “curvature operator”, “nonnegative curvature”

  4. Perelman's entropy on ancient Ricci flows

    Zilu Ma and Yongjia Zhang

    Topics Ancient solutions, Ricci solitons · Same author Yongjia Zhang · Shared terms “curvature operator”, “nonnegative curvature”

  5. Ancient Ricci flows with asymptotic solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang

  6. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang

  7. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ancient solutions, Singularities · Same author Yongjia Zhang

  8. Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics Ancient solutions · Same author Yongjia Zhang · Shared terms “nonnegative curvature”, “curvature operator”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.