Paper · Jun 10, 2017

A compactness theorem for four-dimensional shrinking gradient Ricci solitons

Yongjia Zhang

math.DGarXiv:1706.03163v2
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Abstract

Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for noncompact four-dimensional shrinking gradient Ricci solitons with a topological restriction and a noncollapsing assumption, that is, we consider Ricci shrinkers that can be embedded in a closed four-manifold with vanishing second homology group over every field and are strongly -noncollapsed with respect to a universal . In particular, we do not need any curvature assumption and the limit is still a smooth nonflat shrinking gradient Ricci soliton.

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Submitted
Jun 10, 2017
Last revised
Jul 18, 2017 (v2)
Subjects
math.DG, math.AP
Ricci solitons(Non)collapsing4-manifoldsLimit spaces & RCDBoundaries & non-compact

Its neighbourhood

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  1. -Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

    Shaosai Huang

    Topics 4-manifolds, (Non)collapsing, Boundaries & non-compact · Shared terms “dimensional shrinking”, “four dimensional”, “closed four”

  2. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

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

  3. On the classification of four-dimensional gradient Ricci solitons

    Fei Yang and Liangdi Zhang

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

  4. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics 4-manifolds, (Non)collapsing, Ricci solitons · Same author Yongjia Zhang · Shared terms “gradient Ricci”, “noncollapsed”

  5. Four-Dimensional Steady Gradient Ricci Solitons with -Cylindrical Tangent Flows at Infinity

    Richard Bamler, Bennett Chow, Yuxing Deng et al.

    Topics 4-manifolds, Ricci solitons · Same author Yongjia Zhang · Shared terms “four dimensional”, “gradient Ricci”

  6. Perelman-type no breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact, Ricci solitons · Same author Yongjia Zhang · Shared terms “shrinking gradient”, “dimensional shrinking”, “gradient Ricci”

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

    Timothy Buttsworth

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

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

    JeongHyeong Park and Wooseok Shin

    Topics 4-manifolds, Limit spaces & RCD, Boundaries & non-compact · Shared terms “four dimensional”, “four manifolds”, “compactness”

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 the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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.