To start with, for two positive integers

(7.1)

$$b_{-},b_{+}\in\{1,2,\ldots,9\},$$

let $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ and $(X_{b_{{}_{+}}}^{4},g_{b_{+}},q_{+})$ be fixed hyperkähler Tian-Yau spaces with reference points $q_{-}\in X_{b_{-}}^{4}$ and $q_{+}\in X_{b_{+}}^{4}$ such that their degrees are $b_{-}$ and $b_{+}$ respectively. See Section 3 for the definition of a Tian-Yau space and the natural coordinate outside a large compact subset. On $(X_{b_{-}}^{4},g_{b_{-}},q_{-})$ and $(X_{b_{+}}^{4},g_{b_{+}},q_{+})$, there are diffeomorphisms

(7.2)

$$\Phi_{\pm}:[\zeta_{0}^{\pm},+\infty)\times\Nil_{b_{\pm}}^{3}\rightarrow X_{b_{\pm}}^{4}\setminus K_{\pm}$$

between the Gibbons-Hawking space which models over a flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$. We define the definite constants $D_{0}^{\pm}$ by

(7.3)

$\displaystyle D_{0}^{\pm}\equiv\text{The distance between}\ q_{\pm}\ \text{and the level set}\ \{\bm{x}\in X_{b_{\pm}}^{4}|z_{\pm}(\bm{x})=\zeta_{0}^{\pm}\}.$

Proposition 3.4 shows that there are some positive constants

(7.4)

$$\dl>0,\ \dr>0$$

such that for any $k\in\mathbb{N}$,

(7.5)

$$|\nabla_{g_{\mathcal{C}}^{\pm}}^{k}(\Phi^{*}\omega_{TY}^{\pm}-\omega_{\mathcal{C}}^{\pm})|_{g_{\mathcal{C}}^{\pm}}=O(e^{-\dl z_{\pm}}),$$

where $\omega_{TY}^{\pm}$, $\omega_{\mathcal{C}}^{\pm}$ are the Kähler forms on the Tian-Yau spaces $X_{b_{\pm}}^{4}$,and the Calabi model spaces respectively.

Now we fix some parameters in the neck region for the convenience of our discussions in the later sections.

Let $\mathcal{P}_{m_{0}}\equiv\{p_{1},\ldots,p_{m_{0}}\}$ be the set of monopoles on the flat cylinder $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ with coordinates $(x,y,z)$ such that

1. (1)

   $z(p_{1})=0$.

2. (2)

   There are definite constants

   (7.6)

   $$\iota_{0}>0,\ T_{0}>0$$

   such that for all $k\neq l$, we have

   (7.7)

   $$\iota_{0}\leq d_{g_{0}}(p_{k},p_{l})\leq T_{0}.$$

Around each monopole $p_{m}\in\mathcal{P}_{m_{0}}$, we define the associated distance function

(7.8)

$$d_{m}(\bm{x})\equiv d_{g}(p_{m},\bm{x}),\ p_{m}\in\mathcal{P}_{m_{0}},\ \bm{x}\in(\mathcal{M},g).$$

In our proof, the following notations will also be needed. We fix definite constants

(7.9)

$$\iota_{0}^{\prime}>0,\ T_{0}^{\prime}>0$$

such that for all $1\leq m<l\leq m_{0}$, then in terms of the Gibbons-Hawking metric of the neck region, we have

(7.10)

$$\iota_{0}^{\prime}\cdot(\beta)^{\frac{1}{2}}\leq d_{g}(p_{m},p_{l})\leq T_{0}^{\prime}\cdot(\beta)^{\frac{1}{2}}.$$

We have already defined in Section 6 the Gibbons-Hawking metric in the neck region $\mathcal{N}_{m_{0}}^{4}$. Given a gluing parameter $\beta>0$, by Theorem 2.6, the defining Green’s function $V_{\beta}$ satisfies the asymptotic property that there are constants

(7.11)

$$\el>0,\ \er>0$$

such that for any $k\in\mathbb{N}$ we have

(7.12)

$$\Big|\nabla^{k}\Big(V_{\beta}-\Big(\frac{2\pi b_{-}}{A}z+\beta\Big)\Big)\Big|=O(e^{\el z}),\qquad z\to-\infty$$

and

(7.13)

$$\Big|\nabla^{k}\Big(V_{\beta}-\Big(-\frac{2\pi b_{+}}{A}z+\beta\Big)\Big)\Big|=O(e^{-\er z}),\qquad z\to+\infty.$$

The following functions defined on $\mathcal{N}_{m_{0}}^{4}$ as well as on the Tian-Yau pieces are crucial in analyzing the rescaled limits and the definition of the weight function in the next section, which naturally comes from the construction of the model metric:

1. (1)

   On the negative part of the neck region, we define the function

   (7.14)

   $$L_{-}(\bm{x})\equiv\Big(\frac{2\pi b_{-}}{A}\cdot z(\bm{x})+\beta\Big)^{\frac{1}{2}},\ -T_{-}\leq z(\bm{x})<0$$

2. (2)

   On the positive part of the neck region, we define the function

   (7.15)

   $$L_{+}(\bm{x})\equiv\Big(-\frac{2\pi b_{+}}{A}\cdot z(\bm{x})+\beta\Big)^{\frac{1}{2}},\ 0\leq z(\bm{x})<T_{+}$$

3. (3)

   For $\bm{x}\in\mathcal{M}$ located in the end region of $X_{b_{-}}^{4}$ and satisfy $\zeta_{0}^{-}\leq z_{-}(\bm{x})\leq T_{-}$, we define

   (7.16)

   $$\underline{L}_{-}(\bm{x})\equiv\Big(\frac{2\pi b_{-}}{A}\cdot z_{-}(\bm{x})\Big)^{\frac{1}{2}}$$

4. (4)

   For $\bm{x}\in\mathcal{M}$ located in the end region of $X_{b_{+}}^{4}$ and satisfy $\zeta_{0}^{+}\leq z_{+}(\bm{x})\leq T_{+}$, we define

   (7.17)

   $$\underline{L}_{+}(\bm{x})\equiv\Big(\frac{2\pi b_{+}}{A}\cdot z_{+}(\bm{x})\Big)^{\frac{1}{2}}.$$

Fix a gluing parameter $\beta>1$, the manifold $(\mathcal{M},g_{\beta})$ will be divided into the following $9$ regions depending on the different collapsing behaviors of metric $g$:

		$\displaystyle\I:\ d_{m}(\bm{x})\leq\beta^{-\frac{1}{2}}\ \text{for some}\ 1\leq m\leq m_{0}$
		(in the neck, very close to a monopole point)
		$\displaystyle\II:\ 2\beta^{-\frac{1}{2}}\leq d_{m}(\bm{x})\leq\frac{\iota_{0}^{\prime}}{4}\cdot(\beta)^{\frac{1}{2}}\text{ for some}\ 1\leq m\leq m_{0}$
		(in the neck, not close, but not too far from any monopole point)
		$\displaystyle\III:\ z(\bm{x})\in[-m_{0}T_{0},m_{0}T_{0}]\ \text{and }d_{m}(\bm{x})\geq\frac{\iota_{0}^{\prime}}{2}\cdot(\beta)^{\frac{1}{2}}\text{ for all}\ 1\leq m\leq m_{0}$
		(in a bounded region of the neck, but far from any monopole point)
		$\displaystyle\IV_{-}:\ z(\bm{x})\in[-T_{-}/2,-2m_{0}T_{0}]$
		(in the negative end region of the neck)
		$\displaystyle\IV_{+}:\ z(\bm{x})\in[2m_{0}T_{0},T_{+}/2]$
		(in the positive end region of the neck)
		$\displaystyle\V_{-}:\ \bm{x}\in X_{b_{-}}^{4}\ \text{and}\ 2\zeta_{0}^{-}\leq z_{-}(\bm{x})\leq T_{-}$
		(in the end region of $X_{b_{-}}^{4}$)
		$\displaystyle\V_{+}:\ \bm{x}\in X_{b_{+}}^{4}\ \text{and}\ 2\zeta_{0}^{+}\leq z_{+}(\bm{x})\leq T_{+}$
		(in the end region of $X_{b_{+}}$)
		$\displaystyle\VI_{-}:\ \bm{x}\in\overline{B_{D_{0}^{-}}(q_{-})}\subset X_{b_{-}}^{4}$
		(in the bounded part of $X_{b_{-}}^{4}$)
		$\displaystyle\VI_{+}:\ \bm{x}\in\overline{B_{D_{0}^{+}}(q_{+})}\subset X_{b_{+}}^{4}$
		$\displaystyle\hskip 28.45274pt\mbox{(in the bounded part of $X_{b_{+}}^{4}$)}.$

We note that for $\bm{x}\in\IV_{\pm}$, we have

(7.18)

$\displaystyle T_{0}^{\prime}(\beta)^{\frac{1}{2}}\leq d_{m}(\bm{x})\leq R_{\pm}\text{ for all}\ 1\leq m\leq m_{0},$

where

(7.19)		$\displaystyle R_{-}$	$\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big|-T_{-}\leq z(\bm{x})\leq 0\Big\},$
(7.20)		$\displaystyle R_{+}$	$\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big|0\leq z(\bm{x})\leq T_{+}\Big\}.$

Immediately, there is some constant $C>0$ (independent of $\beta$) such that

(7.21)

$$C^{-1}\beta^{\frac{3}{2}}\leq R_{\pm}\leq C\beta^{\frac{3}{2}}.$$