Paper · Apr 29, 2024

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

Ziyi Zhao, Xiaohua Zhu

math.DGarXiv:2404.18494v2
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Abstract

By Perelman's -geodesic theory, we study the blow-down solutions on a noncompact -noncollapsed steady gradient Ricci soliton with nonnegative curvature operator and positive Ricci curvature away from a compact set of . We prove that any compact split ancient solution of codimension one from the blow-down of is of type I. The result is a generalization of our previous work from to any dimension.

Comments: To appear in Math of Crelle's Journal

Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsingBoundaries & non-compact

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

    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 “curvature away”, “nonnegative curvature”, “compact set”

  3. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego PDF

    Topics (Non)collapsing, Ancient solutions, Boundaries & non-compact · Shared terms “noncollapsed”

  4. Rigidity of the Bryant Ricci soliton

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu

  5. 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”, “Perelman”

  6. Singularity models in the three-dimensional Ricci flow

    S. Brendle PDF

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

  7. Classification of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Ziyi ZhaoWider flows PDF

    Topics (Non)collapsing, Ancient solutions · Same author Ziyi Zhao · Shared terms “noncollapsed”

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

    Yuxing Deng 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.