Paper · Feb 1, 2016

Gradient Ricci solitons with vanishing conditions on Weyl

Giovanni Catino, Paolo Mastrolia, Dario Daniele Monticelli

math.DGarXiv:1602.00534v2
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Abstract

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any -dimensional () gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of , , the product of a Einstein manifold with the Gaussian shrinking soliton . The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton.

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Submitted
Feb 1, 2016
Last revised
Feb 2, 2016 (v2)
Subjects
math.DG
Journal
J. Math. Pures Appl. 108 (2017), no. 1, 1-13

Comments: Minor typos

Ricci solitonsAncient solutionsEinstein metrics3-manifolds

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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 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors (Seungsu Hwang and Gabjin Yun, 2016)3Hwang 2016 Rigidity of Gradient Shrinking Ricci Solitons (Fei Yang and Liangdi Zhang, 2017)2Yang 2017 A potential generalization of some canonical Riemannian metrics (Giovanni Catino and Paolo Mastrolia, 2017)6Catino 2017 Weyl scalars on compact Ricci solitons (Giovanni Catino and Paolo Mastrolia, 2018)1Catino 2018 A conformal Yamabe problem with potential on the euclidean space (Giovanni Catino, Filippo Gazzola and Paolo Mastrolia, 2019)8Catino 2019 Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons (Shun Maeta, 2019)7Maeta 2019 On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor (Huai-Dong Cao and Jiangtao Yu, 2020)4Cao 2020 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)5Lai 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weyl scalars on compact Ricci solitons

    Giovanni Catino and Paolo Mastrolia

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia · Shared terms “vanishing condition”, “Weyl tensor”, “divergence”

  2. Rigidity of Gradient Shrinking Ricci Solitons

    Fei Yang and Liangdi Zhang

    Topics Ricci solitons · Shared terms “order divergence”, “fourth order”, “divergence free”

  3. Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

    Seungsu Hwang and Gabjin Yun

    Topics Einstein metrics, Ricci solitons · Shared terms “Weyl tensor”, “soliton satisfying”, “gradient shrinking”

  4. On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

    Huai-Dong Cao and Jiangtao Yu

    Topics Ancient solutions, Ricci solitons · Shared terms “gradient steady”, “vanishing”, “complete gradient”

  5. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai

    Topics Ancient solutions, 3-manifolds, Ricci solitons · Shared terms “gradient Ricci”, “Bryant soliton”

  6. A potential generalization of some canonical Riemannian metrics

    Giovanni Catino and Paolo Mastrolia

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia · Shared terms “gradient Ricci”

  7. Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

    Shun Maeta

    Topics Ancient solutions, 3-manifolds, Ricci solitons · Shared terms “gradient Ricci”, “Bryant soliton”, “quotient”

  8. A conformal Yamabe problem with potential on the euclidean space

    Giovanni Catino, Filippo Gazzola and Paolo Mastrolia

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.