There is a diffeomorphism

(6.18)

$\displaystyle\Phi^{N}_{-}:(-T_{-},-T_{-}+1,)\times\Nil^{3}_{b_{-}}(\epsilon,\tau)\rightarrow\mathcal{N}_{m_{0}}^{4}(-T_{-},-T_{-}+1),$

which preserves the $z$-coordinate, such that

(6.19)

$$(\Phi^{N}_{-})^{*}\theta=\theta_{b_{-}}+O(e^{-\delta_{N}T_{-}}),$$

as $T_{-}\rightarrow\infty$. Similarly, there is a diffeomorphism

(6.20)

$\displaystyle\Phi^{N}_{+}:(T_{+}-1,T_{+})\times\Nil^{3}_{-b_{+}}(\epsilon,\tau)\rightarrow\mathcal{N}_{m_{0}}^{4}(T_{+}-1,T_{+}),$

which preserves the $z$-coordinate, such that

(6.21)

$$(\Phi^{N}_{+})^{*}\theta=\theta_{-b_{+}}+O(e^{-\delta_{N}T_{+}}),$$

as $T_{+}\rightarrow\infty$. Furthermore, there exist triples of $1$-forms on the ends of the neck such that

(6.22)		$\displaystyle(\Phi^{N}_{-})^{*}\bm{\omega}^{N}-\bm{\omega}_{b_{-}}=d(\bm{a}^{N}_{-})$
(6.23)		$\displaystyle(\Phi^{N}_{+})^{*}\bm{\omega}^{N}-\bm{\omega}_{-b_{+}}=d(\bm{a}^{N}_{+}),$

with

(6.24)

$\displaystyle|\nabla^{k}\bm{a}^{N}_{\pm}|\leq Ce^{-\delta_{N}|z|}$

for any integer $k\geq 0$ and $\epsilon>0$, where $\delta_{N}>0$ is a uniform constant in Proposition 3.4, $\bm{\omega}_{b_{-}}$ and $\bm{\omega}_{-b_{+}}$ are the hyperkähler triples on the corresponding Calabi model spaces.