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

(3.10)

$$\omega_{TY}^{n}=\frac{1}{2}i^{n^{2}}\Omega_{X}\wedge\overline{\Omega}_{X}.$$

Moreover, there is a constant $\delta_{0}=\delta_{0}(M,D)>0$ such that for all integers $k\geq 0$,

(3.11)

$$|\nabla_{g_{X}}^{k}\phi|_{g_{X}}=O(e^{-\delta_{0}r_{X}^{\frac{n}{n+1}}})\ \text{as}\ r_{X}\to\infty.$$