We can translate the boundary condition near $t=0$ into the asymptotic behaviour of the Calabi-Yau metric for $1\ll x_{2}\ll x_{1}$. Using the second derivative computation (8), we get

$$\begin{cases}\frac{\partial^{2}u}{\partial x_{1}^{2}}\sim\frac{2(n+2)}{n^{2}}v_{0}x_{1}^{\frac{2}{n}-1},\\ \frac{\partial^{2}u}{\partial x_{1}\partial x_{2}}\sim-\frac{d_{1}}{d_{2}}\frac{2(n+2)}{n^{2}}v_{0}x_{1}^{\frac{2}{n}-1}\\ \frac{\partial^{2}u}{\partial x_{2}^{2}}\sim(\frac{d_{1}}{d_{2}})^{2}\frac{a}{n-1}t^{-\frac{n-2}{n-1}}x_{1}^{\frac{2}{n}-1}.\end{cases}$$

Recall that in the generalized Calabi-Yau ansatz, this Hessian matrix controls the base metric and the torus fibres. In the $x_{1}$-direction of the base (i.e. the direction transverse to $D_{1}$ at infinity), and the $\log\xi_{1}$ circle direction, the behaviour is similar to what happens in the generic region where $t$ is of order one. However, in the $x_{2}$ direction, the base metric has a different scaling law:

$$\frac{\partial^{2}u}{\partial x_{2}^{2}}\sim(\frac{d_{1}}{d_{2}})^{2}\frac{a}{n-1}t^{-\frac{n-2}{n-1}}x_{1}^{\frac{2}{n}-1}=O(x_{1}^{\frac{2}{n}-\frac{1}{n-1}}x_{2}^{-\frac{n-2}{n-1}}).$$

The distance to $x_{2}\sim O(1)$ is $O(x_{1}^{\frac{1}{n}-\frac{1}{2(n-1)}}x_{2}^{\frac{n}{2(n-1)}})$. The $\log\xi_{2}$ circle direction now has diameter of order

$$O(x_{2}^{-\frac{n-2}{2(n-1)}}x_{1}^{\frac{1}{n}-\frac{1}{2(n-1)}}),$$

which is much larger compared to the $\log\xi_{1}$ circle. In other words, the metric exhibits inhomogeneous collapsing phenomenon.

The potential is roughly

$$\begin{split}&u=x_{1}^{\frac{n+2}{n}}v(t)\approx x_{1}^{\frac{n+2}{n}}(v_{0}-\frac{n+2}{n}v_{0}t+\frac{n-1}{n}at^{\frac{n}{n-1}})\\ &\approx v_{0}(x_{1}-\frac{d_{1}}{d_{2}}x_{2})^{\frac{n+2}{n}}+\frac{n-1}{n}ax_{1}^{\frac{n+2}{n}}(\frac{d_{1}}{d_{2}}\frac{x_{2}}{x_{1}})^{\frac{n}{n-1}}.\end{split}$$

Here $x_{2}\ll x_{1}$. For fixed values of $x_{1}-\frac{d_{1}}{d_{2}}x_{2}$, the second term dominates the metric contribution on the slices. Up to scaling factors by powers of $x_{1}$, the key dependence on $x_{2}$ is $dd^{c}x_{2}^{n/(n-1)}$, which is the Calabi ansatz on an open subset of the $(n-1)$-dimensional total space of the line bundle $\mathcal{O}(D_{2})\to D_{1}\cap D_{2}$ (cf. section 2.2). When $\xi_{1}$ is allowed to vary, the scales of the $S^{1}$, the Calabi ansatz, and the base metric, all depend on $x_{1}$ in some power law fashion.

The significance to the compactification question, is that $\mathcal{O}(D_{2})\to D_{1}\cap D_{2}$ describes the infinity of the non-compact Calabi-Yau manifold $D_{1}\setminus D_{2}$, and the Calabi ansatz is the asymptote of the Tian-Yau metric on $D_{1}\setminus D_{2}$. We can thus glue the Calabi ansatz to the Tian-Yau metric, in a parametrized fashion over the $x_{1}$ variable, in order to achieve the partial completion of the generalized Calabi ansatz. After this, we would have an approximate Calabi-Yau metric outside a compact set in $X=\bar{X}\setminus D_{1}\cup D_{2}$, from which one can hope to use a non-compact version of Yau’s proof to the Calabi conjecture to obtain an actual Calabi-Yau metric on $X$.