Paper · Nov 30, 2020

Perelman-type no breather theorem for noncompact Ricci flows

Liang Cheng, Yongjia Zhang

math.DGarXiv:2011.14973
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Abstract

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete -dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient – a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.

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Submitted
Nov 30, 2020
Subjects
math.DG

Comments: 28 pages

Ricci flowRicci solitonsAncient solutionsBoundaries & non-compact

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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.

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 New proofs of Perelman's theorem on shrinking Breathers in Ricci flow (Peng Lu and Yu Zheng, 2017)4Lu 2017 A compactness theorem for four-dimensional shrinking gradient Ricci solitons (Yongjia Zhang, 2017)8Zhang 2017 A note on Perelman's no shrinking breather theorem (Yongjia Zhang, 2018)1Zhang 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)7Li 2018 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)5Cheng 2021 A no expanding breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds (Liang Cheng and Yongjia Zhang, 2021)6Cheng 2021
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, Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “shrinking breather”, “breather theorem”, “shrinking gradient”

  2. A no expanding breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

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

  3. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows

    Topics Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang · Shared terms “breather theorem”

  4. 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”, “shrinking gradient”, “gradient Ricci”

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

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang

  6. Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang · Shared terms “noncompact Ricci”

  7. 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 “asymptotic shrinking”, “shrinking gradient”, “gradient Ricci”

  8. A compactness theorem for four-dimensional shrinking gradient Ricci solitons

    Yongjia Zhang

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

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) 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.