Proposition 5.11 (00SZ) · ScalingStacks

Proposition 5.11. There is a uniform diameter bound

\text{diam}(X_{s},g_{CY,s})\leq C.

00T0

Proof. This argument is essentially the same as [36, Thm 3.1]. We quote [36, Lem 3.2]:

00T1

Lemma 5.12. Let $(M^{2n},g)$ be a closed Riemannian manifold with $Ric(g)\geq 0$ , let $p\in M$ and $1<R\leq diam(X,g)$ . Then $\frac{R-1}{4n}\leq\frac{\text{Vol}(B(p,2(R+1)))}{\text{Vol}(B(p,1))}$ .

Using Thm. 5.6, we can find inside the regular region of $X_{s}$ some geodesic ball $B_{g_{CY,s}}(p,r)$ of radius $r<1$ , occupying a nontrivial portion of the total volume:

\frac{\text{Vol}(B_{g_{CY,s}}(p,r)))}{\text{Vol}(X_{s})}\geq\epsilon>0,

with $\epsilon$ independent of $s$ . Now applying the Lemma to the rescaled CY metric $r^{-2}g_{CY,s}$ ,

\frac{\text{diam}(X_{s})-r}{4nr}\leq\frac{\text{Vol}(B_{g_{CY,s}}(p,2(\text{diam}(X_{s})+r)))}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\frac{\text{Vol}(X_{s})}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\epsilon^{-1},

so $\text{diam}(X_{s})\leq Cr\leq C$ as required. ∎