Paper · Apr 7, 2021

A no expanding breather theorem for noncompact Ricci flows

Liang Cheng, Yongjia Zhang

math.DGarXiv:2104.02834
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Abstract

In this note we show that, under certain curvature positivity conditions (the weak condition or the nonnegative bisectional curvature condition), a complete and noncompact expanding breather of the Ricci flow is also an expanding gradient Ricci soliton. This is the first no expanding breather theorem in the noncompact category.

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Submitted
Apr 7, 2021
Subjects
math.DG

Comments: 11 pages

Ricci flowRicci solitonsSphere theoremsBoundaries & non-compact

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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 A note on Perelman's no shrinking breather theorem (Yongjia Zhang, 2018)3Zhang 2018 On the κ-solutions of the Ricci flow on noncompact 3-manifolds (Liang Cheng and Anqiang Zhu, 2018)7Cheng 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)8Li 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)4Cheng 2021 Nontrivial breathers for Ricci flow (Peter M. Topping, 2021)6Topping 2021 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds (Liang Cheng and Yongjia Zhang, 2021)5Cheng 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 Boundaries & non-compact, Ricci solitons · Same authors Liang Cheng, Yongjia Zhang · Shared terms “breather theorem”, “noncompact Ricci”, “gradient Ricci”

  2. 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”, “noncompact category”

  3. A note on Perelman's no shrinking breather theorem

    Yongjia Zhang

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

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

    Liang Cheng and Yongjia Zhang

    Topics Sphere theorems, Boundaries & non-compact · Same authors Liang Cheng, Yongjia Zhang

  5. 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”

  6. Nontrivial breathers for Ricci flow

    Peter M. Topping

    Topics Boundaries & non-compact · Shared terms “breather”

  7. On the solutions of the Ricci flow on noncompact 3-manifolds

    Liang Cheng and Anqiang Zhu

    Topics Sphere theorems, Boundaries & non-compact · Same author Liang Cheng

  8. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Sphere theorems, Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  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.