Abstract:Let $(M, g, f)$ be a gradient Ricci soliton $\nabla ^2 f+Ric=\lambda g$ with $\lambda \in \{\frac {1}{2}, 0, -\frac {1}{2}\}$. Suppose there is a geodesic line $\gamma : (-\infty , \infty )\rightarrow M$ satisfying \begin{eqnarray*} \liminf _{t\rightarrow \infty }\int _0^{t}Ric(\gamma ’(s), \gamma ’(s))ds +\liminf _{t\rightarrow -\infty }\int _{t}^{0}Ric(\gamma ’(s), \gamma ’(s))ds \geq 0, \end{eqnarray*} then $(M, g, f)$ splits off a line isometrically.

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中文翻译：

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Ricci孤子的分裂定理

摘要：设 $(M, g, f)$ 是梯度 Ricci 孤子 $\nabla ^2 f+Ric=\lambda g$ ，其中 $\lambda \in \{\frac {1}{2}, 0, - \frac{1}{2}\}$。假设有一条测地线 $\gamma : (-\infty , \infty )\rightarrow M$ 满足 \begin{eqnarray*} \liminf _{t\rightarrow \infty }\int _0^{t}Ric(\gamma '(s), \gamma '(s))ds +\liminf _{t\rightarrow -\infty }\int _{t}^{0}Ric(\gamma '(s), \gamma '(s) )ds \geq 0, \end{eqnarray*} 然后 $(M, g, f)$ 等距分割一条线。

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