Paper · Mar 9, 2018

A note on Perelman's no shrinking breather theorem

Yongjia Zhang

math.DGarXiv:1803.03660v2
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Abstract

As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's -geometry. Our proof follows the argument in Lu and Zheng of constructing an ancient solution, and removes a technical assumption made by them.

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Submitted
Mar 9, 2018
Last revised
Apr 26, 2018 (v2)
Subjects
math.DG
Ricci solitonsAncient solutionsBoundaries & non-compact

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)6Zhang 2017 New proofs of Perelman's theorem on shrinking Breathers in Ricci flow (Peng Lu and Yu Zheng, 2017)2Lu 2017 On three-dimensional Type I κ-solutions to the Ricci flow (Yongjia Zhang, 2017)7Zhang 2017 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)5Li 2018 Perelman-type no breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2020)1Cheng 2020 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)8Cheng 2021 A no expanding breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)4Cheng 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Perelman-type no breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “shrinking breather”, “breather theorem”, “shrinking gradient”

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

    Peng Lu and Yu Zheng

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

  3. A no expanding breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “breather theorem”, “gradient Ricci”

  4. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows

    Topics Boundaries & non-compact · Same author Yongjia Zhang · Shared terms “breather theorem”

  5. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “shrinking gradient”, “gradient Ricci”

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

    Yongjia Zhang

    Topics Ancient solutions · Same author Yongjia Zhang · Shared terms “Perelman”

  7. On three-dimensional Type I -solutions to the Ricci flow

    Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact · Same author Yongjia Zhang · Shared terms “Perelman”

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

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact · Same author Yongjia Zhang

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 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

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

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.