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)

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