Paper · Jul 16, 2018

Rigidity of positively curved shrinking Ricci solitons in dimension four

Giovanni Catino

math.DGarXiv:1807.05766v2
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Abstract

We classify four-dimensional shrinking Ricci solitons satisfying , where and denote the sectional and the scalar curvature, respectively. They are isometric to either (and quotients), , or with their standard metrics.

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Submitted
Jul 16, 2018
Last revised
Mar 28, 2019 (v2)
Subjects
math.DG
Journal
Geom. Flows 4 (2019), no. 1, 1-8
Ricci solitons4-manifoldsScalar curvatureStability & uniqueness

Its neighbourhood

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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 Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)6Munteanu 2016 Curvature decompositions on Einstein four-manifolds (Peng Wu, 2019)5Wu 2019 Isometry theorem of gradient Shrinking Ricci solitons (Absos Ali Shaikh and Chandan Kumar Mondal, 2020)3Shaikh 2020 Four-dimensional complete gradient shrinking Ricci solitons (Huai-Dong Cao, Ernani Ribeiro and Detang Zhou, 2020)2Cao 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)1Cheng 2021 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature I (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)8Li 2024 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature II (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2025)7Li 2025 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)4Wang 2026
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  1. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “shrinking Ricci”, “four dimensional”, “quotient”

  2. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “four dimensional”, “soliton satisfying”

  3. Isometry theorem of gradient Shrinking Ricci solitons

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “soliton satisfying”, “four dimensional”

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

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “dimensional shrinking”, “four dimensional”, “shrinking Ricci”

  5. Curvature decompositions on Einstein four-manifolds

    Peng Wu

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “positively curved”

  6. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “quotient”, “four dimensional”

  7. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature II

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “dimensional shrinking”, “quotient”, “four dimensional”

  8. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature I

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “dimensional shrinking”, “quotient”, “four dimensional”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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