PreprintCC BY 4.0arXiv
Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid
Abstract
Let (M2m,g,f,J) be a complete gradient Kähler--Ricci shrinker satisfying Ric+∇2 f=1/2 g. We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer k∈{0,…,m} such that R≡ k and (M2m,g,J)≅ (N2k×ℂm−k, gN+gEuc,JN⊕ J0), where (N2k,gN,JN) is Kähler--Einstein and RicgN=1/2 gN. 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∇ fRic=0 implies radial flatness and hence rigidity.
Related
- Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
- Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set
- Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature
- Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature
- Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature
BibTeX
@misc{li2026kählerricci,
title = {{Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid}},
author = {Wenqi Li and Jianyu Ou and Guoqiang Wu and Detang Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36418},
eprint = {2609.36418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36418},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kähler--Ricci Shrinkers with Constant Scalar Curvature Are Rigid
AU - Wenqi Li
AU - Jianyu Ou
AU - Guoqiang Wu
AU - Detang Zhou
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36418
UR - https://arxiv.org/abs/2609.36418
AB - Let (M^2m,g,f,J) be a complete gradient Kähler--Ricci shrinker satisfying Ric+∇^2 f=1/2 g. We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer k∈{0,…,m} such that R≡ k and (M^2m,g,J)≅ (N^2k×ℂ^m−k, g_N+g_Euc,J_N⊕ J_0), where (N^2k,g_N,J_N) is Kähler--Einstein and Ric_g_N=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_∇ fRic=0 implies radial flatness and hence rigidity.
ER -