Paper · May 23, 2017

New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

Peng Lu, Yu Zheng

math.DGarXiv:1705.08325
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Abstract

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being shrinking gradient Ricci solitons.

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Submitted
May 23, 2017
Subjects
math.DG

Comments: 6 pages

Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsing

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 Ancient Ricci Flow Solutions on Bundles (Peng Lu and Y. K. Wang, 2016)4Lu 2016 Ancient Solutions on Bundles with Non-abelian Structural Group (Peng Lu and Y. K. Wang, 2016)3Lu 2016 A note on Perelman's no shrinking breather theorem (Yongjia Zhang, 2018)1Zhang 2018 Perelman-type no breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2020)2Cheng 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)6Brendle 2021 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)5Brendle 2022 Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2024)7Zhao 2024 No compact split limit Ricci flow of type II from the blow-down (Ziyi Zhao and Xiaohua Zhu, 2024)8Zhao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on Perelman's no shrinking breather theorem

    Yongjia Zhang

    Topics Ancient solutions, Ricci solitons · Shared terms “shrinking breather”, “Perelman”, “shrinking gradient”

  2. Perelman-type no breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Ricci solitons · Shared terms “shrinking breather”, “shrinking gradient”, “gradient Ricci”

  3. Ancient Solutions on Bundles with Non-abelian Structural Group

    Peng Lu and Y. K. Wang

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Peng Lu

  4. Ancient Ricci Flow Solutions on Bundles

    Peng Lu and Y. K. Wang

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Peng Lu · Shared terms “collapsed”

  5. Singularity models in the three-dimensional Ricci flow

    S. Brendle

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “singularity models”, “Perelman”, “gradient Ricci”

  6. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “Perelman”

  7. Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

    Ziyi Zhao and Xiaohua Zhu

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “gradient Ricci”, “rescaled”

  8. No compact split limit Ricci flow of type II from the blow-down

    Ziyi Zhao and Xiaohua Zhu

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “gradient Ricci”, “Perelman”

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.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.