Paper · Nov 29, 2016

Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature

Antoine Song

math.DGarXiv:1611.09747v2
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As on arXiv

Abstract

We construct examples of spherical space forms with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at : a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of singularities which can occur for metrics with these symmetries.

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Submitted
Nov 29, 2016
Last revised
May 3, 2017 (v2)
Subjects
math.DG
Journal
Geom. Topol. 23 (2019) 3501-3535
DOI
10.2140/gt.2019.23.3501

Comments: v2: title changed, first part will appear elsewhere, second part about appearance of stable spheres is extended substantially

Ricci flowSingularitiesCurvature pinchingSphere theoremsScalar curvatureMinimal & CMC surfaces

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 Minimal two-spheres in three-spheres (Robert Haslhofer and Daniel Ketover, 2017)5Haslhofer 2017 Some sharp differential sphere theorems for nonnegative scalar curvature manifolds (Qing Cui and Linlin Sun, 2017)3Cui 2017 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)6Lotay 2018 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)7Baker 2020 Surfaces pinched by normal curvature for mean curvature flow in space forms (Dong Pu, Jingjing Su and Hongwei Xu, 2020)8Pu 2020 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)4Mramor 2024 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)1Sun 2025 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)2Jiang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu

    Topics Minimal & CMC surfaces, Scalar curvature · Shared terms “stable minimal”

  2. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete

    Topics Minimal & CMC surfaces, Sphere theorems, Scalar curvature

  3. Some sharp differential sphere theorems for nonnegative scalar curvature manifolds

    Qing Cui and Linlin Sun

    Topics Curvature pinching, Sphere theorems, Scalar curvature · Shared terms “following”

  4. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Minimal & CMC surfaces, Singularities · Shared terms “stable minimal”, “embedded minimal”

  5. Minimal two-spheres in three-spheres

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal spheres”, “embedded minimal”

  6. Consequences of strong stability of minimal submanifolds

    Jason D. Lotay and Felix SchulzeWider flows

    Topics Minimal & CMC surfaces, Singularities · Shared terms “stable minimal”

  7. Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces

  8. Surfaces pinched by normal curvature for mean curvature flow in space forms

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Curvature pinching, Sphere theorems · Shared terms “space form”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 12B.6
    Bounded Curvature at Bounded DistanceBook 12B · κ-Solutions and Surgery

    The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 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.