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.

∎