Paper · Aug 26, 2024

Cohomogeneity two Ricci solitons with sub-Euclidean volume

Benjy Firester, Raphael Tsiamis

math.DGarXiv:2408.13982v2J Geom Anal 35, 407 (2025)
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Abstract

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge-Ampère equation to prove rigidity for these solitons.

Comments: To appear in Journal of Geometric Analysis

Ricci solitons(Non)collapsing4-manifoldsKähler geometryHomogeneous spacesStability & uniqueness

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  1. On a general class of free boundary Monge-Ampère equations

    Tristan C. Collins and Benjy Firester PDF

    Topics Kähler geometry, Ricci solitons · Same author Benjy Firester · Shared terms “Monge Ampère”, “Ampère equation”, “degenerate”

  2. -invariant steady gradient Ricci solitons on four-manifolds

    Timothy Buttsworth PDF

    Topics 4-manifolds, (Non)collapsing, Homogeneous spaces · Shared terms “cohomogeneity”, “four dimensional”

  3. Steady conformally Kähler Ricci solitons with isoparametric potential

    Ernani Ribeiro and Hung Tran PDF

    Topics 4-manifolds, (Non)collapsing, Homogeneous spaces · Shared terms “conformal factor”, “cohomogeneity”, “four dimensional”

  4. Classification of superpotentials for cohomogeneity one Ricci solitons

    Qiu Shi Wang PDF

    Topics Homogeneous spaces, Kähler geometry, Ricci solitons · Shared terms “cohomogeneity”, “steady solitons”

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

    Hanci Chi PDF

    Topics (Non)collapsing, Homogeneous spaces, Ricci solitons · Shared terms “cohomogeneity”, “Einstein”

  6. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”

  7. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics 4-manifolds, (Non)collapsing, Ricci solitons · Shared terms “steady solitons”

  8. Cohomogeneity one 4-dimensional gradient Ricci solitons

    Patrick Donovan PDF

    Topics 4-manifolds, Homogeneous spaces, Kähler geometry · Shared terms “cohomogeneity”, “four dimensional”

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.