We mentioned in the beginning that the generalized Calabi ansatz geometrically describes an iterated fibration, which is supposedly the model for the generic region near infinity on the noncompact Calabi-Yau $X=\bar{X}\setminus D$. Figuring out the order of magnitude of various length scales is essentially a matter of dimensional analysis. In the generic region $t$ is of order $\sim 1$, $v$ is smooth and of order $\sim 1$. Our homogeneous ansatz prescribes

$$u\sim O(|x|^{(n+2)/n}),\quad|du|\sim O(|x|^{2/n}),\quad|D^{2}u|\sim O(|x|^{(2-n)/n}).$$

We have $|D^{2}u|\ll 1\ll|du|$. From the descriptions in section 2.1, the Hessian term is responsible for the base and torus direction of the metric, while $du$ is responsible for the effective Kähler class, and therefore the size of the fibres diffeomorphic to $Y\simeq D_{1}\cap D_{2}$. Thus

$$\text{diam}(T^{2})=O(|D^{2}u|^{1/2})=O(|x|^{(2-n)/2n}),\quad\text{diam}(Y)=O(|du|^{1/2})=O(|x|^{1/n}),$$

the distance to the origin is of order

$$O(\int|D^{2}u|^{1/2})=O(|x|^{(2+n)/2n}),$$

and the volume within $|x|\leq r$ is $O(r^{2})$ from the two log directions of the base. In terms of geodesic distance to the origin, the volume grows with power $4n/(n+2)$.

For $|x|\gg 1$, this reaffirms the intuition that the $T^{2}$ length scale is far smaller than $Y\simeq D_{1}\cap D_{2}$, which is far smaller than the real 2-dimensional base.