Paper · Oct 30, 2024

A rigidity result for axisymmetric toric Ricci solitons

Shiqiao Zhang

math.DGarXiv:2411.00861
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Abstract

We examine a non-axisymmetric perturbation of a family of axisymmetric toric Einstein manifolds and Ricci solitons studied in Firester-Tsiamis (2024). We establish a rigidity result stating that these axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases. For these new cases, our result leads to an explicit description of the Einstein metrics and a classification of the Ricci solitons under a volume-collapsing ansatz.

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Submitted
Oct 30, 2024
Subjects
math.DG

Comments: 7 pages

Ricci solitonsEinstein metrics(Non)collapsingStability & uniqueness

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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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  2. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  3. Cohomogeneity two Ricci solitons with sub-Euclidean volume

    Benjy Firester and Raphael Tsiamis

    Topics (Non)collapsing, Stability & uniqueness, Ricci solitons · Shared terms “volume collapsing”, “Einstein”

  4. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  5. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang

    Topics (Non)collapsing, Stability & uniqueness, Ricci solitons

  6. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  7. Infinitely many non-collapsed steady Ricci solitons on complex line bundles

    Hanci Chi

    Topics (Non)collapsing, Einstein metrics, Ricci solitons

  8. The Stability of Generalized Ricci Solitons

    Kuan-Hui Lee

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.