Paper · Aug 19, 2025

Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu

math.DGarXiv:2508.13495v2
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Abstract

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

Ricci solitonsKähler–Ricci solitons3-manifolds4-manifoldsSphere theoremsKähler geometrySasakian & contact geometry

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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 2021, 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 Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2022)4Chang 2022 Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds (Shu-Cheng Chang, Chien Lin and Chin-Tung Wu, 2022)5Chang 2022 Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven (Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al., 2022)1Chang 2022 Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds (Roberto Mossa and Giovanni Placini, 2023)8Mossa 2023 On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2024)7Chang 2024 Transverse Rigidity of Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2025)3Chang 2025 Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor (Shu-Cheng Chang and Hongbing Qiu, 2025)2Chang 2025 On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian 5-manifold (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2026)6Chang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

    Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al. PDF

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler geometry · Same authors Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu · Shared terms “Sasaki Ricci”, “shrinking Sasaki”, “gradient shrinking”

  2. Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor

    Shu-Cheng Chang and Hongbing Qiu PDF

    Topics Sasakian & contact geometry, Ricci solitons · Same author Shu-Cheng Chang · Shared terms “shrinking Sasaki”, “Sasaki Ricci”, “gradient shrinking”

  3. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Ricci solitons · Same author Shu-Cheng Chang · Shared terms “Sasaki Ricci”, “shrinking Sasaki”, “transverse”

  4. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al. PDF

    Topics Sasakian & contact geometry · Same authors Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu · Shared terms “Sasaki Ricci”, “transverse”

  5. Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

    Shu-Cheng Chang, Chien Lin and Chin-Tung Wu PDF

    Topics Sasakian & contact geometry · Same authors Shu-Cheng Chang, Chin-Tung Wu · Shared terms “Sasaki Ricci”, “Sasaki analogue”, “Secondly”

  6. On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian -manifold

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Kähler geometry, Ricci solitons · Same authors Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu · Shared terms “Sasaki Ricci”, “transverse”, “Secondly”

  7. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Kähler geometry, Ricci solitons · Same authors Shu-Cheng Chang, Chin-Tung Wu · Shared terms “Sasaki Ricci”, “transverse”

  8. Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

    Roberto Mossa and Giovanni Placini PDF

    Topics Sasakian & contact geometry, Sphere theorems, Kähler–Ricci solitons · Shared terms “Sasaki Ricci”, “holomorphic”

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 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) 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)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  3. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.