Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid

Abstract

Let $(M^{2m},g,f,J)$ be a complete gradient Kähler--Ricci shrinker satisfying \[ \operatorname{Ric}+\nabla^2 f=\frac12 g. \] 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 \[ (M^{2m},g,J)\cong \bigl(N^{2k}\times\mathbb C^{m-k}, g_N+g_{\mathrm{Euc}},J_N\oplus J_0\bigr), \] where $(N^{2k},g_N,J_N)$ is Kähler--Einstein and \[ \operatorname{Ric}_{g_N}=\frac12 g_N. \] As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition ${\mathcal{L}}_{\nabla f}\mathrm{Ric}=0$ implies radial flatness and hence rigidity.