Lemma 8.9 . [01ZM]

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Lemma 8.9.

For each $\delta<\delta({\rm v})$ , there exists $\bar{\alpha}(\delta,{\rm v})$ with the following property. If $|{\rm Ric}_{M^{4}}|\leq 3\delta$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , and $\mathcal{V}^{\delta}_{r_{1}}(x)<\ln N-\delta$ for some $N\in\mathds{N}$ and $x\in B_{1}(p)$ , we have

\mathcal{V}^{\delta}_{r_{\bar{\alpha}}}(x)<\ln\Big(N-1\Big)+\delta\,.