Given $m_{0}\in\mathbb{Z}_{+}$, the neck region $(\mathcal{N}_{m_{0}}^{4},g_{GH})$ is obtained from a Gibbons-Hawking space based on $\mathbb{T}^{2}\times\mathbb{R}$ with $m_{0}$ monopole points. The Gibbons-Hawking metric is determined by a harmonic function on $\mathbb{T}^{2}\times\mathbb{R}$. Notice that, the topology of $\K 3$ requires

Consider the flat cylinder $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ and denote by $\mathcal{P}_{m_{0}}\equiv\{p_{1},\ldots,p_{m_{0}}\}\subset\mathbb{T}^{2}\times\mathbb{R}$ a finite set of monopoles, let $V_{\infty}$ be the Green’s function from Corollary 2.7, which satisfies

In the gluing procedure, we need to modify the above Green’s function $V_{\infty}$ such that the metric on the neck matches up with the metric of the Tian-Yau parts. For this purpose, we analyze the asymptotic behavior of $V_{\infty}$ at the two ends of the neck region. By Corollary 2.7, there are bounded harmonic functions $h_{+}$ and $h_{-}$ on $\mathbb{T}^{2}\times\mathbb{R}$ such that

and $h_{\pm}$ satisfy the asymptotic behavior $h_{\pm}(\bm{x})=O(e^{-\epsilon_{0}|z(\bm{x})|})$.

For fixed $\beta>0$, we define a new harmonic function on $\mathbb{T}^{2}\times\mathbb{R}$,

such that the metric on the neck can be glued with the metric on the Tian-Yau space. First, we need to match the slopes, that is, the slope parameter $k$ should be chosen as

Then near to the two ends of the neck region, the harmonic function $V_{\beta}$ can be written as

Using this potential, we next define the neck metric through the Gibbons-Hawking ansatz. Letting $\mathcal{P}_{m_{0}}$ denote the set of monopole points, we note that $H^{2}((\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}},\mathbb{Z})$ has dimension $m_{0}+1$ with generators being small spheres around the monopole points, and any torus of the form $\mathbb{T}^{2}\times\{z^{\prime}\}$ where $z^{\prime}$ is any value of $z$ for which there are no monopole points. It is easy to see that the $2$-form $\frac{1}{2\pi}*dV_{\beta}$ attains integer values on these cycles, which implies that the cohomology class $[\frac{1}{2\pi}*dV_{\beta}]$ lies in the image of the natural inclusion

Therefore, we let $\mathcal{N}_{m_{0}}^{4}$ be the total space of the $S^{1}$-bundle over $(\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}}$ corresponding to the class $[\frac{1}{2\pi}*dV_{\beta}]$, completed by adding finitely many points corresponding to $\mathcal{P}_{m_{0}}$. Choose a connection $1$-form $\theta$ on $\mathcal{N}_{m_{0}}$ so that

Then applying the Gibbons-Hawking construction to $V_{\beta}$, we obtain a smooth hyperkähler triple $\bm{\omega}^{N}$ over $\mathcal{N}_{m_{0}}^{4}$, which induces an incomplete hyperkähler metric over the part in $\mathbb{T}^{2}\times\mathbb{R}$ where $V_{\beta}$ is strictly positive.

For parameters $T_{-}>0$ and $T_{+}>0$, we define,

where $\pi:\mathcal{N}_{m_{0}}^{4}\rightarrow U\equiv(\mathbb{T}^{2}\times\mathbb{R})\setminus\mathcal{P}_{m_{0}}$ is the bundle projection.