Paper · Nov 1, 2024

Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration

Hanci Chi

math.DGarXiv:2411.00581v4
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Abstract

We establish the existence of two 3-parameter families of non-Einstein, non-shrinking Ricci solitons: one on and one on . Each family includes a continuous 1-parameter subfamily of asymptotically paraboloidal (non-collapsed) steady Ricci solitons, with the Jensen sphere as the base. Additionally, we extend this result by proving the existence of a 2-parameter family on , which contains a 1-parameter subfamily of asymptotically paraboloidal steady Ricci solitons based on the Bourguignon–Karcher sphere.

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Submitted
Nov 1, 2024
Last revised
Mar 8, 2026 (v4)
Subjects
math.DG

Comments: We added a new section giving a proof of the positivity of sectional curvatures for the constructed solitons for sufficiently small parameters. Corollary 1.8 has been strengthened to Theorem 1.8

Ricci solitons(Non)collapsingHomogeneous spaces

Its neighbourhood

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    Topics (Non)collapsing, Homogeneous spaces · Shared terms “collapsed”, “fibration”

  6. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee

    Topics (Non)collapsing, Ricci solitons · Shared terms “steady Ricci”, “parameter subfamily”, “parameter family”

  7. Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

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    Topics Homogeneous spaces, Ricci solitons · Shared terms “cohomogeneity”, “parameter family”

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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) (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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.