Paper · May 25, 2020

Ancient solutions to the Ricci flow with isotropic curvature conditions

Jae Ho Cho, Yu Li

math.DGarXiv:2005.11866
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Abstract

We show that every -dimensional, -noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for or has weakly PIC and bounded curvature. Combining this with earlier results, we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, -noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.

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Submitted
May 25, 2020
Subjects
math.DG
Ricci flowKähler–Ricci flowAncient solutions(Non)collapsingSurfacesSphere theoremsKähler geometryBoundaries & 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 Ricci flow with surgery on manifolds with positive isotropic curvature (S. Brendle, 2017)7Brendle 2017 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)6Brendle 2018 Shrinking Ricci solitons with positive isotropic curvature (Keaton Naff, 2019)8Naff 2019 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)2Brendle 2020 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)1Li 2020 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)5Lynch 2022 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)4Chen 2025 Pinching cones for positive isotropic curvature in dimensions seven and eight (Jae Ho Cho, 2026)3Cho 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient solutions to the Kähler Ricci flow

    Yu Li

    Topics Kähler–Ricci flow, (Non)collapsing, Ancient solutions · Same author Yu Li · Shared terms “complete ancient”, “noncollapsed”

  2. Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

    Simon Brendle and Keaton Naff

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “weakly PIC”, “quotient thereof”, “uniformly PIC”

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

    Jae Ho Cho

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Same author Jae Ho Cho · Shared terms “isotropic curvature”, “weakly PIC”, “uniformly PIC”

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

    Zhengnan Chen

    Topics Sphere theorems, Ancient solutions · Shared terms “isotropic curvature”, “complete ancient”, “weakly PIC”

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

    Stephen Lynch and Andoni Royo Abrego

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

  6. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “quotient thereof”, “shrinking cylinder”, “Bryant soliton”

  7. Ricci flow with surgery on manifolds with positive isotropic curvature

    S. Brendle

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “isotropic curvature”, “weakly PIC”, “uniformly PIC”

  8. Shrinking Ricci solitons with positive isotropic curvature

    Keaton Naff

    Topics Sphere theorems · Shared terms “isotropic curvature”, “weakly PIC”, “quotient”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) (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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.