发表机构
Universidade de Brasília; East China Normal University; Xiamen University; Zhejiang Sci-Tech University; Universidade Federal Fluminense（巴西利亚大学; 华东师范大学; 厦门大学; 浙江理工大学; 弗鲁米嫩塞联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对满足特定条件的完备非稳定梯度凯勒-里奇孤立子，证明其标量曲率为常数时必为刚性，即通用覆盖可分解为凯勒-爱因斯坦因子与平凡因子的乘积，且该结论基于里奇孤立子的刚性准则及相关分析方法。
AI 中文摘要
设$(M^{2m},g,f,J)$为满足$\mathrm{Ric}+\nabla^2 f=\lambda g$、$\lambda\neq 0$的完备非稳定梯度凯勒-里奇孤立子。我们证明常数标量曲率迫使该孤立子具有刚性。更确切地说，通用覆盖可全纯且等距地分解为$N^{2k}\times\mathbb{C}^{m-k}$，其中$N^{2k}$是具有$\mathrm{Ric}_{g_N}=\lambda g_N$的凯勒-爱因斯坦流形。在收缩情形下，商是平凡的；在归一化$\lambda=1/2$中，有$R\equiv k$。该凯勒结果基于一个黎曼刚性准则：在完备非稳定梯度里奇孤立子上，若$\mathcal{L}_{\nabla f}\mathrm{Ric}$处处非负或非正，则其必为零，且孤立子具有刚性；无需对标量曲率做任何假设。沿该孤立子生成的里奇流，这意味着随时间单调的里奇张量在时间上是常数，且这一性质迫使孤立子具有刚性。我们还给出一个直接证明，即挤压条件$0\leq\mathrm{Ric}\leq\lambda g$迫使标量曲率为常数且径向平坦，通过彼得森-怀利特征刻画得出刚性结论。该证明结合了加权截断论证与里奇自同态的部分科达齐对称性。
英文摘要
Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient Kähler-Ricci soliton satisfying $\mathrm{Ric}+\nabla^2 f=λg$, $λ\neq 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times\mathbb{C}^{m-k}$, where $N^{2k}$ is Kähler-Einstein with $\mathrm{Ric}_{g_N}=λg_N$. In the shrinking case the quotient is trivial; in the normalization $λ=1/2$ one has $R\equiv k$. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if $\mathcal{L}_{\nabla f}\mathrm{Ric}$ 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 $0\leq\mathrm{Ric}\leqλg$ 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.
Comments18 pages
