Geometry of shrinking Ricci solitons
Geometry of shrinking Ricci solitons
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DOI：
10.1112/s0010437x15007496
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发表时间：
2014-10
影响因子：
1.8
通讯作者：
Ovidiu Munteanu;Jiaping Wang
中科院分区：
文献类型：
--
作者：
Ovidiu Munteanu;Jiaping Wang
本文主要研究了四维收缩梯度Ricci孤子的曲率行为。对于这样一个标量曲率S有界的孤子M，证明了M的曲率算子Rm满足估计|\text{Rm}|\leqslant cS$ for some constant $c$.此外，曲率算子$\text{Rm}$在无穷远处渐近非负，并且有一个下界$\text{Rm}\geqslant -c（\ln（r+1））^{-1/4}$，其中$r$是到$M$中不动点的距离函数。作为应用，我们证明了如果标量曲率在无穷远处收敛到零，则孤子一定是渐近圆锥的。作为一个单独的问题，导出了任意维紧致收缩梯度Ricci孤子的直径上界。
The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.