Paper · Feb 1, 2024

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

Ziyi Zhao, Xiaohua Zhu

math.DGarXiv:2402.00316
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Abstract

Let be a complete noncompact -noncollapsed steady Ricci soliton with and away from a compact set of . We prove that there is no any -dimensional compact split limit Ricci flow of type I arising from the blow-down of , if there is an -dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that with must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows of converges subsequently to a family of shrinking quotient cylinders.

Comments: arXiv admin note: text overlap with arXiv:2310.12529

Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsingBoundaries & 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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. No compact split limit Ricci flow of type II from the blow-down

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics (Non)collapsing, Ancient solutions, Boundaries & non-compact · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “split limit”, “compact split”, “limit Ricci”

  2. steady gradient Ricci solitons with nonnegative curvature away from a compact set

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics (Non)collapsing, Boundaries & non-compact, Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “nonnegative curvature”, “compact set”, “operator away”

  3. Rigidity of the Bryant Ricci soliton

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “Bryant Ricci”, “steady Ricci”

  4. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng PDF

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

  5. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao PDF

    Topics (Non)collapsing, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “noncollapsed steady”

  6. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al. PDF

    Topics Ancient solutions, Singularities · Same author Ziyi Zhao · Shared terms “curvature operator”

  7. An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

    Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al. PDF

    Topics (Non)collapsing, Ancient solutions, Ricci solitons · Shared terms “steady gradient”, “noncollapsed steady”, “gradient Ricci”

  8. Singularity models in the three-dimensional Ricci flow

    S. Brendle PDF

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “steady gradient”, “noncollapsed steady”, “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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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.