Paper · Aug 11, 2026

Compactness and Rigidity of Complete Kähler Ricci Shrinkers

Tongxin Xu, Zhenlei Zhang

math.DGarXiv:2608.10953
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Abstract

In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.

Comments: 34 pages

Ricci solitonsKähler geometryLimit spaces & RCDStability & uniquenessAnalytic minimal model program

Its neighbourhood

Connected papers

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 K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers (Charles Cifarelli and Carlos Esparza, 2025)4Cifarelli 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)3Hallgren 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)2Xu 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026 Shrinking Kähler-Ricci solitons on toric fano fibrations (Tristan C. Collins, Genggeng Huang, Freid Tong et al., 2026)8Collins 2026 A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers (Carlos Esparza and Junsheng Zhang, 2026)6Esparza 2026 Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers (Yu Li and Junsheng Zhang, 2026)5Li 2026 Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “complete Kähler”, “Ricci shrinkers”

  2. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang

  3. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  4. K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

    Charles Cifarelli and Carlos Esparza PDF

    Topics Analytic minimal model program, Kähler geometry, Stability & uniqueness · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  5. Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  6. A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers

    Carlos Esparza and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Stability & uniqueness · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  7. Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang

  8. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al. PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Shared terms “Fano fibration”

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 geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) 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.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.