Let $(M^{n},g)$ satisfy $|\Rm_{g}|\leq 1$ in $B_{2}(p)$, then the following holds:

1. (1)

   there exists a dimensional constant $r_{0}(n)>0$ such that $r_{1,\alpha}(x)\geq r_{0}(n)>0$ for all $x\in B_{1}(p)$ and $\alpha\in(0,1)$. Moreover, $r_{1,\alpha}(p)\geq r_{0}(n)\cdot r_{|\Rm|}(p)>0$, where

   (4.182)

   $$r_{|\Rm|}(p)\equiv\sup\Big\{r>0\Big||\Rm|_{C^{0}(B_{r}(p))}\leq r^{-2}\Big\}$$

   denotes the curvature scale at $p$.

2. (2)

   In particular, if $\Rm_{g}\equiv 0$ on a complete manifold $M^{n}$, then $r_{k,\alpha}(x)=+\infty$ for all $x\in M^{n}$, $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$.