There is a smooth function $\phi$ on $Z$ such that $\omega_{TY}\equiv\omega_{Z}+\sqrt{-1}\partial\bar{\partial}\phi$ is a complete Ricci-flat Kähler metric on $Z$ solving the Monge-Ampère equation

Moreover, there is a diffeomorphism $\Phi:\mathcal{C}\setminus K^{\prime}\rightarrow Y\setminus K$, where $K\subset Z$ is compact and $K^{\prime}=\{|\xi|\geq\frac{1}{2}\}$ and constant $\delta_{Z}>0$, such that the following asymptotics hold uniformly for all $z$ large

1. (1)

   (7.46)

   $$|\nabla_{g_{Z}}^{k}\phi|_{g_{Z}}=O(e^{-\delta_{Z}(-\log|S|^{2})^{1/2}})\ \text{for all}\ k\geq 0.$$

2. (2)

   (7.47)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}J_{Z}-J_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.$$

3. (3)

   (7.48)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\Omega_{Z}-\Omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.$$

4. (4)

   (7.49)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\omega_{TY}-\omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-{\delta_{Z}}z^{n/2}}).$$

5. (5)

   There is a constant $C>0$ such that

   (7.50)

   $$C^{-1}z\leq\Phi^{*}((-\log|S|^{2})^{\frac{1}{n}})\leq Cz.$$