Let $(M_{j}^{n},g_{j},p_{j})$ be a sequence of Einstein manifolds with $|\Ric_{g_{j}}|\leq\epsilon_{j}\to 0$ such that

(7.41)

$$(M_{j}^{n},g_{j},p_{j})\xrightarrow{GH}\mathbb{R}^{n-1}$$

and $\Gamma_{2}(p_{j})\equiv\Image[\pi_{1}(B_{2}(p_{j}))\to\pi_{1}(M_{j}^{n})]$ is of infinite order. Then for any $R>0$,

(7.42)

$$\sup\limits_{B_{R}(p_{j})}|\Rm|\leq\frac{C_{0}(n)}{R^{2}}.$$