Recall the distance to the origin is $O(|x|^{\frac{n+2}{2n}})$. Thus the $O(x_{1}^{-\frac{n}{n-1}})$ decay is slower than quadratic, and we need further correction terms to improve the ansatz. The linearization of the complex Monge-Ampère equation will naturally lead to a Poisson equation.

Using section 4.3, we can solve (a rescaled version of) the Poisson equation on the Tian-Yau space $D_{1}\setminus D_{2}$:

$$\frac{(n+2)(n-1)}{2\pi nd_{2}^{2}}v_{0}a^{n-2}(\frac{d_{1}}{d_{2}})^{\frac{n(n-2)}{n-1}}(dd^{c}U)\wedge\omega_{TY}^{n-2}=-\mathcal{I}_{1},$$

(34)

where we recall from (33) that $\mathcal{I}_{1}$ is a top degree form on $D_{1}\setminus D_{2}$, with exponential decay $O(e^{-cx_{2}^{1/2}})$ for some $c>0$ to all orders of derivatives. The caveat is that $\int_{D_{1}\setminus D_{2}}\mathcal{I}_{1}$ is not guaranteed to be zero, so $U$ may not decay at infinity. Instead,

$$U=a^{\prime}u_{0}+\tilde{U},$$

where $|\nabla^{k}_{TY}\tilde{U}|=O(e^{-cx_{2}^{1/2}})$ for some possibly shrinked $c>0$, and $a^{\prime}$ is a constant.

We regard $U$ as a function in the region $x_{2}\ll x_{1}$, namely the tubular neighbourhood around $D_{1}\setminus D_{2}$, and let $\phi_{D_{1}}^{(2)}=\phi_{D_{1}}+\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}U$. From the leading term $u_{0}\sim x_{2}^{\frac{1}{n-1}}$ in $U$, we can compute using Lemma 4.7 that the correction term $\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}U$ has local $C^{k,\alpha}$-norm $O(x_{1}^{-\frac{n}{n-1}}x_{2}^{-1})$. This small correction term leads to better volume form decay: