Paper · Oct 8, 2026

Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

Matheus Horácio, Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou

math.DGarXiv:2610.11935
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Abstract

Let be a complete nonsteady gradient Kähler-Ricci soliton satisfying , . We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as , where is Kähler-Einstein with . In the shrinking case the quotient is trivial; in the normalization one has . The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

Comments: 18 pages

Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsCurvature pinchingKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability

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    Topics Kähler–Einstein & K-stability, Einstein metrics, Scalar curvature · Same authors Wenqi Li, Jianyu Ou, Guoqiang Wu · Shared terms “constant scalar”, “complete nonsteady”, “nonsteady gradient”

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

    Matheus Andrade Ribeiro de Moura Horácio PDF

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

  3. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

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    Topics Einstein metrics, Scalar curvature, Stability & uniqueness · Same authors Wenqi Li, Jianyu Ou, Guoqiang Wu · Shared terms “constant scalar”, “universal cover”, “gradient Ricci”

  4. Some rigidity results on shrinking gradient Ricci soliton

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    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “constant scalar”, “Petersen Wylie”, “gradient Ricci”

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

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

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    Topics Scalar curvature, Stability & uniqueness, Ricci solitons · Same author Guoqiang Wu · Shared terms “constant scalar”, “Petersen Wylie”, “radial flatness”

  7. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

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    Topics Stability & uniqueness, Ricci solitons · Same authors Jianyu Ou, Guoqiang Wu · Shared terms “gradient Ricci”, “quotient”

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

    Matheus Andrade Ribeiro de Moura Horácio PDF

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.