Paper · Oct 19, 2023

steady gradient Ricci solitons with nonnegative curvature away from a compact set

Ziyi Zhao, Xiaohua Zhu

math.DGarXiv:2310.12529v3
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Abstract

In the paper, we analysis the asymptotic behavior of noncompact -noncollapsed steady gradient Ricci soliton with nonnegative curvature operator away from a compact set of . In particular, we prove: any noncompact -noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of , which converges subsequently to a family of shrinking quotient cylinders.

Comments: Proposition 4.1, Lemma 4.2 and Lemma 4.3 have been generalized for any dimension

Ricci solitons(Non)collapsing4-manifoldsBoundaries & non-compact

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    Ziyi Zhao and Xiaohua Zhu PDF

    Topics (Non)collapsing, Boundaries & non-compact, Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “nonnegative curvature”, “compact set”, “operator away”

  2. No compact split limit Ricci flow of type II from the blow-down

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics (Non)collapsing, Boundaries & non-compact, Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “curvature away”, “nonnegative curvature”, “compact set”

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    Yuxing Deng and Yuehan Hao PDF

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

  4. Rigidity of the Bryant Ricci soliton

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “Bryant Ricci”, “sectional curvature”

  5. 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 “noncollapsed steady”, “steady gradient”, “nonnegative sectional”

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

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

  7. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

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

  8. Steady conformally Kähler Ricci solitons with isoparametric potential

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

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.