Given some constant $\delta>0$, a complete Riemannian manifold $(X^{2n},g)$ of dimension $2n$ is said to be $\delta$-asymptotically Calabi if there exist a compact subset $K\subset X^{2n}$, a Calabi model space $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$ (as in Section 2.2) with $\dim_{\mathbb{C}}(\mathcal{C}^{n})=n$, and a diffeomorphism

(5.1)

$$\Phi:\mathcal{C}^{n}\setminus K^{\prime}\rightarrow X^{2n}\setminus K$$

with $K^{\prime}=\{|\xi|\geq C\}\subset\mathcal{C}^{n}$ (for some $C>0$) such that for all $k\geq 0$,

(5.2)

$$|\nabla_{g_{\mathcal{C}^{n}}}^{k}(\Phi^{*}g-g_{\mathcal{C}^{n}})|_{g_{\mathcal{C}^{n}}}=O(e^{-\delta z^{\frac{n}{2}}})\ \text{as}\ z\to+\infty,$$

where $z\equiv(-{\log|\xi|^{2}})^{\frac{1}{n}}$ denotes the natural moment map coordinate on $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$.