There exists $\epsilon(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq\epsilon$, ${\rm Vol}(B_{1}(p))>{\rm v}>0$. and

$\displaystyle d_{GH}\big(B_{2}(p),B_{2}(0)\big)<\epsilon\,,$

(6.1)

where $0$ is a vertex of the cone $\mathds{R}^{n-3}\times C(Y)$, for some metric space $Y$, then we have

$\displaystyle r_{h}(p)\geq 1\,.$

(6.2)

Consequently, if $M^{n}$ is Einstein, we have the bound

$\displaystyle\sup_{B_{1}(p)}|{\rm Rm}|\leq 1\,.$

(6.3)