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.