ScalingStacks

Proof. (Thm. 5.10) By Thm 5.6 we already know the metric convergence over any properly contained open subset of $\mathcal{R}$, which corresponds to a region $U_{s}\subset X_{s}$, with nearly the full measure:

$$\text{Vol}(U_{s})>(1-\epsilon)\text{Vol}(X_{s}),$$

where $\epsilon$ can be chosen arbitrarily small. It now suffices to show any point $p\in X_{s}\setminus U_{s}$ is close to $U_{s}$. For any $r>0$ such that the geodesic ball $B_{g_{CY,s}}(p,r)\subset X_{s}\setminus U_{s}$, the Bishop-Gromov inequality implies

$$\left(\frac{r}{\text{diam}(X_{s})}\right)^{2n}\leq\frac{\text{Vol}(B_{g_{CY,s}}(p,r))}{\text{Vol}(X_{s})}\leq\frac{\text{Vol}(X_{s}\setminus U_{s})}{\text{Vol}(X_{s})}<\epsilon.$$

Taking the sup of all such $r$,

$$\text{dist}_{g_{CY,s}}(p,U_{s})\leq\epsilon^{1/2n}\text{diam}(X_{s})\leq C\epsilon^{1/2n},$$

which can be made arbitrarily small. ∎