Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

@misc{horacio2026rigidity,
  title = {{Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature}},
  author = {Matheus Horácio and Wenqi Li and Jianyu Ou and Guoqiang Wu and Detang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11935},
  eprint = {2610.11935},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11935},
}

TY  - UNPB
TI  - Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature
AU  - Matheus Horácio
AU  - Wenqi Li
AU  - Jianyu Ou
AU  - Guoqiang Wu
AU  - Detang Zhou
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11935
UR  - https://arxiv.org/abs/2610.11935
AB  - Let (M^2m,g,f,J) be a complete nonsteady gradient Kähler-Ricci soliton satisfying Ric+∇^2 f=λg, λ≠ 0. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as N^2k×ℂ^m−k, where N^2k is Kähler-Einstein with Ric_g_N=λg_N. In the shrinking case the quotient is trivial; in the normalization λ=1/2 one has R≡ k. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if ℒ_∇ fRic 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 0≤Ric≤λg 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.
ER  -