The proof only requires straightforward calculations, so we only sketch the calculations. We use the following formula for the pointwise norm squared of the curvature of a Gibbons-Hawking metric

see [GW00]. We just need to consider the case of $1$ monopole point located at the origin, the case of several monopole points follows easily from this case. Let $r_{0}\equiv\frac{1}{2}\InjRad_{g_{0}}(\mathbb{T}^{2})$, then we have the expansion

where $h$ is a bounded harmonic function.

First, we estimate the curvature in the case $r(\bm{x})<\frac{1}{\beta}$. By (7.28),

so it follows that

for $r<\beta^{-1}$, then

and the first claimed estimate follows from this.

Before showing the curvature estimates in other regions, we relate the intrinsic distance function $d_{m}(\bm{x})$ and the Euclidean radial function $r(\bm{x})$. By directly estimating the integral of $\sqrt{V_{\beta}}$, we have that

where $C^{\prime}>0$ is some universal constant. So the first part of the curvature estimate in (7.23) immediately follows.

Next, let $\bm{x}\in B_{r_{0}}(0^{3})$ satisfy $r(\bm{x})\geq\beta^{-1}$. Substituting (7.28) into (7.27), then similar expansion formula shows that for some uniform constant $C>0$,

Correspondingly in terms of the intrinsic distance function, the curvature estimate turns out to be

The above in fact covers the curvature estimates in Region I and Region II.

From now on, we consider the case that $\bm{x}$ is in the neck region satisfying $\frac{r_{0}}{10}\leq|z(\bm{x})|\leq\beta$. In this case, the harmonic function $V_{\beta}$ has the expansion,

We apply the above expansion to the curvature formula (7.27), then we obtain the following curvature estimate

where $C>0$ is a uniform curvature estimate. Similarly, one can calculate that in the damage zones,

for some uniform constant $C>0$. Note that the cutoff function and its derivatives up to third order are uniformly bounded, the curvature of the glued metric is therefore also of order $\beta^{-3}$ in the damage zone region.

Next, we recall from Section 2.2 that for the model spaces, the defining harmonic functions are $V_{-}(\bm{x})=\frac{2\pi b_{-}z(\bm{x})}{A}$ and $V_{+}(\bm{x})=\frac{2\pi b_{+}z(\bm{x})}{A}$, so (7.27) implies that

for some uniform constant $C>0$, so the complete end of the model space has exactly inverse quadratic curvature decay. It follows from Proposition 3.4 that the Tian-Yau metric does also.

∎

Let $(\widetilde{M_{j}^{n}},\tilde{g}_{j},\Gamma_{j},\tilde{p}_{j})$ be the Riemannian universal covers of $(M_{j}^{n},g_{j})$ which converge to the limit product space $(\mathbb{R}^{n-1}\times Y,\tilde{d}_{\infty},\Gamma_{\infty},\tilde{p}_{\infty})$ in the equivariant Gromov-Hausdorff topology, where $\Gamma_{j}\equiv\pi_{1}(M_{j}^{n})$ and $\Gamma_{j}\to\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$. See Section 3 of [FY92] for the precise definition of the equivariant Gromov-Hausdorff convergence. In summary, we have the following diagram

where the covering maps $\pr_{j}:\widetilde{M_{j}^{n}}\to M_{j}^{n}$ converge to a natural projection map $\pr_{\infty}:\mathbb{R}^{n-1}\times Y\to\mathbb{R}^{n-1}$.

The main part is to prove the claim that $Y$ is isometric to $\mathbb{R}$.

Applying Cheeger-Colding’s quantitative splitting theorem (see [CC96]), the convergence assumption (7.41) implies that for any fixed $R>0$, there are harmonic splitting maps $\Phi_{j}\equiv(u_{j}^{(1)},\ldots,u_{j}^{(n-1)}):B_{10R}(p_{j})\to\mathbb{R}^{n-1}$ which realize the Gromov-Hausdorff maps such that

Let $\widetilde{\Phi}_{j}\equiv(\tilde{u}_{j}^{(1)},\ldots,\tilde{u}_{j}^{(n-1)})$ be the lifted harmonic functions on the universal covers, then the volume comparison theorem implies that

By the definition of the splitting maps, $Y$ is the Gromov-Hausdorff limit of the level sets of the lifted splitting maps $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$. Since $\Gamma_{2}(p_{j})\leq\pi_{1}(M_{j}^{n})$ is of infinite order which acts on $\widetilde{M}_{j}^{n}$ isometrically and discretely, the limit space $Y$ must be non-compact.

On other hand hand, notice that $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$ is invariant under the deck transformation group $\Gamma_{j}$ and $\Gamma_{j}$ converge to some limiting group $\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$ such that $\pr_{\infty}$ is given by the quotient $(\mathbb{R}^{n-1}\times Y)/\Gamma_{\infty}=\mathbb{R}^{n-1}$. Hence $\Gamma_{\infty}\leq\Isom(Y)$ and $\Gamma_{\infty}$ acts homogeneously on $Y$.

Therefore, by standard arguments, the noncompact homogeneous space $Y$ admits a line (see [CG72] or lemma 2.4 in [NZ16]). The Ricci curvature assumption implies that $Y$ is isometric to $\mathbb{R}$. This completes the proof of the claim.

The curvature estimate (7.42) immediately follows from the $\epsilon$-regularity theorem for noncollapsed Einstein manifolds (for example see Section 7 in [CC97]).

∎