AI 中文总结
该研究针对具有常数量曲率的完备收缩梯度Ricci孤立子，在两个假设条件下证明其等距于$\mathbb{R}^2 \times \mathbb{S}^{n-2}$，且所用条件弱于已有径向平坦条件。
AI 中文摘要
设$(M^n, g, f)$为具有常数量曲率的完备收缩梯度Ricci孤立子，在以下假设下：(i) 在$M\setminus D$上满足$Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$，其中$D$是$M$上的紧集；(ii) $(M^n, g, f)$光滑收敛到$\mathbb{R}^2 \times \mathbb{S}^{n-2}$，我们得出$(M^n, g, f)$等距于$\mathbb{R}^2 \times \mathbb{S}^{n-2}$。值得注意的是，条件(i)比文献[Petersen-Wylie2]中的径向平坦条件更弱。
英文摘要
Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$, we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.
