Paper · Sep 24, 2026

Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

Matheus Andrade Ribeiro de Moura Horácio

math.DGarXiv:2609.29359
PDF Abstract page

As on arXiv

Abstract

We prove that every complete nonsteady gradient Kähler-Ricci soliton with constant scalar curvature is rigid. This establishes Cao's rigidity conjecture for Kähler-Ricci solitons in arbitrary dimension and gives the corresponding result for expanding solitons, without additional curvature assumptions. The proof uses a rigidity criterion for gradient Ricci solitons with constant scalar curvature, expressed by the vanishing of the Lie derivative of the Ricci tensor along the soliton vector field. In the Kähler case, this vanishing follows from constant scalar curvature, the closedness of the Ricci form, and the classical real holomorphicity of the soliton vector field.

Comments: 5 pages

Ricci solitonsKähler–Ricci solitonsKähler geometryScalar curvatureStability & uniqueness

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 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)7Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)5Bamler 2022 The Invariant Szegő metric on strongly pseudoconvex domains (Anjali Bhatnagar and Jiliang Fan, 2026)6Bhatnagar 2026 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)8Li 2026 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu et al., 2026)3Li 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)2Conlon 2026 Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature (Matheus Horácio, Wenqi Li, Jianyu Ou et al., 2026)1Horácio 2026 On Ricci solitons whose level hypersurfaces have parallel second fundamental form (Matheus Andrade Ribeiro de Moura Horácio, 2026)4Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

    Matheus Horácio, Wenqi Li, Jianyu Ou et al. PDF

    Topics Kähler–Ricci solitons, Scalar curvature, Kähler geometry · Same author Matheus Andrade Ribeiro de Moura Horácio · Shared terms “nonsteady gradient”, “constant scalar”, “complete nonsteady”

  2. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “soliton vector”, “gradient Kähler”, “vector field”

  3. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al. PDF

    Topics Scalar curvature, Kähler geometry, Stability & uniqueness · Shared terms “constant scalar”, “nonsteady gradient”, “rigidity criterion”

  4. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio PDF

    Topics Scalar curvature, Ricci solitons · Same author Matheus Andrade Ribeiro de Moura Horácio · Shared terms “nonsteady gradient”, “constant scalar”, “complete nonsteady”

  5. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al. PDF

    Topics Kähler–Ricci solitons, Scalar curvature, Kähler geometry · Shared terms “gradient Kähler”

  6. The Invariant Szegő metric on strongly pseudoconvex domains

    Anjali Bhatnagar and Jiliang Fan PDF

    Topics Kähler–Ricci solitons, Scalar curvature, Kähler geometry · Shared terms “constant scalar”, “gradient Kähler”, “vanishing”

  7. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Scalar curvature, Kähler geometry · Shared terms “soliton vector”, “gradient Kähler”, “vector field”

  8. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Shared terms “constant scalar”, “gradient Ricci”

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) 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) 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)

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.