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

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

(2.9)

where $0\in\mathds{R}^{n-1}\times C(Z)$, then the harmonic radius $r_{h}(x)$ satisfies

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

(2.10)

If $M^{n}$ is further assumed to be Einstein, then the regularity scale $r_{x}$ satisfies $r_{x}\geq 1$.