Paper · Sep 29, 2026

Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou

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Abstract

Let $(M^{2 m}, g, f, J)$ be a complete gradient Kähler–Ricci shrinker satisfying $Ric + \nabla^{2} f = \frac{1}{2} g .$ We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer $k \in {0, \dots, m}$ such that $R \equiv k$ and $(M^{2 m}, g, J) ≅ (N^{2 k} \times C^{m - k}, g_{N} + g_{Euc}, J_{N} \oplus J_{0}),$ where $(N^{2 k}, g_{N}, J_{N})$ is Kähler–Einstein and $Ric_{g_{N}} = \frac{1}{2} g_{N} .$ As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition $L_{\nabla f} Ric = 0$ implies radial flatness and hence rigidity.

Ricci solitons Einstein metrics Kähler geometry Scalar curvature Stability & uniqueness Kähler–Einstein & K-stability

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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) 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) 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

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.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

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