5.3 Gromov-Hausdorff convergence [00TX]

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5.3 Gromov-Hausdorff convergence

On the regular locus $\mathcal{R}\subset\partial\Delta_{\lambda}^{\vee}$ we have a well defined real MA metric,

g_{\infty}=\begin{cases}\frac{1}{2}\sum_{i,j}\frac{\partial^{2}u_{\infty}}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},\quad\text{ on the face regions},\\ \frac{1}{2}\sum_{i,j}\frac{\partial^{2}u_{\infty,m}}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},\quad\text{ on the star regions}.\end{cases}

(34)

Notice the definitions are compatible on overlapping regions. Let $(\bar{\mathcal{R}},g_{\infty})$ be the metric completion. The metric asymptotes (32)(33) say that in some $C^{\infty}_{loc}$ sense the collapsing CY metrics $g_{CY,s}$ converge to the metric $g_{\infty}$ on $\mathcal{R}$ , and we know $\mathcal{R}$ is path connected because its complement has zero $\mathcal{H}^{n-1}$ -measure.

Remark 5.9.

We do not know if $\bar{\mathcal{R}}$ is homeomorphic to $\partial\Delta_{\lambda}^{\vee}\simeq S^{n}$ , as the regularity theory of the real MA equation on a singular affine manifold is not yet developed, and we know little about what can happen near singularities.

The goal of this section is to show

Theorem 5.10.

The subsequence of collapsing CY metrics $(X_{s},g_{CY,s})$ converges in the Gromov-Hausdorff sense to $(\bar{\mathcal{R}},g_{\infty})$ .

Proposition 5.11.

There is a uniform diameter bound

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

Proof.

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

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. ∎

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. ∎