Paper · Jul 16, 2017

On the classification of four-dimensional gradient Ricci solitons

Fei Yang, Liangdi Zhang

math.DGarXiv:1707.04846v2
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Abstract

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with , we show that it is either Einstein or a finite quotient of , or . The same result can be obtained under the condition of . We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.

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Submitted
Jul 16, 2017
Last revised
Apr 11, 2019 (v2)
Subjects
math.DG

Comments: 23 pages

Ricci solitons4-manifoldsBoundaries & non-compact

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

    Huai-Dong Cao and Junming Xie

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

  2. Rigidity of four-dimensional Gradient shrinking Ricci solitons

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

  3. On curvature estimates for four-dimensional gradient Ricci solitons

    Huai-Dong Cao

    Topics 4-manifolds, Ricci solitons · Shared terms “dimensional gradient”, “four dimensional”, “gradient Ricci”

  4. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics 4-manifolds, Ricci solitons · Shared terms “four dimensional”, “dimensional complete”, “gradient shrinking”

  5. 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 “four dimensional”, “gradient Ricci”, “finite quotient”

  6. Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

    Huai-Dong Cao and Junming Xie

    Topics 4-manifolds, Ricci solitons · Shared terms “dimensional gradient”, “four dimensional”, “gradient steady”

  7. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang

    Topics 4-manifolds, Ricci solitons · Shared terms “four dimensional”, “dimensional gradient”, “steady Ricci”

  8. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu

    Topics 4-manifolds, Ricci solitons · Shared terms “four dimensional”, “gradient expanding”, “dimensional complete”

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) 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) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 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.