Paper · May 24, 2019

Shrinking Ricci solitons with positive isotropic curvature

Keaton Naff

math.DGarXiv:1905.10305v2
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Abstract

We show that in dimensions , a non-flat complete gradient shrinking solitons with uniformly positive isotropic curvature (PIC) must be a quotient of either the round sphere or the cylinder . We also observe that in dimensions , a complete gradient shrinking soliton that is strictly PIC and weakly PIC2 must be a quotient of either the round sphere or the cylinder .

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Submitted
May 24, 2019
Last revised
May 29, 2019 (v2)
Subjects
math.DG

Comments: Citations updated

Ricci solitonsSphere theorems

Its neighbourhood

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 Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature (Xiaolong Li, Lei Ni and Kui Wang, 2016)1Li 2016 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)7Brendle 2020 Ancient solutions to the Ricci flow with isotropic curvature conditions (Jae Ho Cho and Yu Li, 2020)5Cho 2020 Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (Huai-Dong Cao and Junming Xie, 2022)3Cao 2022 Classification of compact manifolds with positive isotropic curvature (Hong Huang, 2023)6Huang 2023 Open manifolds with uniformly positive isotropic curvature (Hong Huang, 2023)8Huang 2023 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)2Chen 2025 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)4Brendle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

    Xiaolong Li, Lei Ni and Kui Wang

    Topics Sphere theorems, Ricci solitons · Shared terms “positive isotropic”, “isotropic curvature”, “shrinking soliton”

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

    Zhengnan Chen

    Topics Sphere theorems, Ricci solitons · Shared terms “positive isotropic”, “isotropic curvature”, “shrinking Ricci”

  3. Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

    Huai-Dong Cao and Junming Xie

    Topics Sphere theorems, Ricci solitons · Shared terms “isotropic curvature”, “positive isotropic”, “complete gradient”

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

    Simon Brendle and Raphael Tsiamis

    Topics Sphere theorems, Ricci solitons · Shared terms “positive isotropic”, “isotropic curvature”, “uniformly positive”

  5. Ancient solutions to the Ricci flow with isotropic curvature conditions

    Jae Ho Cho and Yu Li

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

  6. Classification of compact manifolds with positive isotropic curvature

    Hong Huang

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

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

    Simon Brendle and Keaton Naff

    Topics Sphere theorems · Same author Keaton Naff · Shared terms “PIC”, “quotient”

  8. Open manifolds with uniformly positive isotropic curvature

    Hong Huang

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “uniformly positive”

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 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)

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.