Paper · Nov 11, 2022

Ancient solutions of Ricci flow with Type I curvature growth

Stephen Lynch, Andoni Royo Abrego

math.DGarXiv:2211.06253v3
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Abstract

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a -noncollapsed Type I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean factor. It follows that a -noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.

Ricci flowAncient solutionsSingularities(Non)collapsingSphere theoremsBoundaries & 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.

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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)6Brendle 2018 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)8Brendle 2022 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)1Cogo 2023 No compact split limit Ricci flow of type II from the blow-down (Ziyi Zhao and Xiaohua Zhu, 2024)3Zhao 2024 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)4Chen 2025 κ-solutions with the round cylinder as an asymptotic shrinker (Aprameya Girish Hebbar, 2026)2Hebbar 2026 Pinching cones for positive isotropic curvature in dimensions seven and eight (Jae Ho Cho, 2026)5Cho 2026 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)7Brendle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows PDF

    Topics (Non)collapsing, Ancient solutions, Boundaries & non-compact · Same author Stephen Lynch

  2. -solutions with the round cylinder as an asymptotic shrinker

    Aprameya Girish Hebbar PDF

    Topics Sphere theorems, Ancient solutions, Boundaries & non-compact · Shared terms “uniformly PIC”

  3. 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 · Shared terms “noncollapsed”

  4. On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

    Zhengnan Chen PDF

    Topics Sphere theorems, Ancient solutions · Shared terms “PIC”, “higher dimensions”

  5. Pinching cones for positive isotropic curvature in dimensions seven and eight

    Jae Ho Cho PDF

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “PIC”, “noncollapsed”

  6. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon PDF

    Topics Ancient solutions, Boundaries & non-compact, Singularities

  7. Ricci flow on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis PDF

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “noncollapsed”

  8. Singularity models in the three-dimensional Ricci flow

    S. Brendle PDF

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “singularity formation”, “noncollapsed”

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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. 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.

  3. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.