Paper · Nov 24, 2021

-invariant steady gradient Ricci solitons on four-manifolds

Timothy Buttsworth

math.DGarXiv:2111.12807
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Abstract

Using center manifolds and topological degree theory, we construct a new family of complete, -invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram . We also provide simpler constructions of the existing -invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams for any , including Appleton's non-collapsed solitons for .

Comments: 16 pages, no figures

Ricci solitons(Non)collapsing4-manifoldsHomogeneous spacesBoundaries & non-compact

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    Topics 4-manifolds, (Non)collapsing, Boundaries & non-compact · Shared terms “steady gradient”, “gradient Ricci”

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    Topics 4-manifolds, Homogeneous spaces, Boundaries & non-compact · Shared terms “compact cohomogeneity”, “gradient Ricci”, “four dimensional”

  4. Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

    JeongHyeong Park and Wooseok Shin PDF

    Topics 4-manifolds, Homogeneous spaces, Boundaries & non-compact · Shared terms “four manifolds”, “four dimensional”, “gradient Ricci”

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    Benjy Firester and Raphael Tsiamis PDF

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

  6. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao PDF

    Topics (Non)collapsing, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”

  7. A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

    Huai-Dong Cao and Junming Xie PDF

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “four dimensional”, “complete gradient”

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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) 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.