Given some constant $\delta>0$, a complete Riemannian manifold $(X^{4},g)$ is said to be $\delta$-asymptotically Calabi if there exist a compact subset $K\subset X$ and a Calabi model space $(\mathcal{C},g_{\mathcal{C}})$ as defined in Section 3, and a diffeomorphism

(4.1)

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

with $K^{\prime}=\{|\xi|_{h}\geq\frac{1}{2}\}\subset\mathcal{C}$ such that for all $k\geq 0$,

(4.2)

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

where $z=(-{\log|\xi|_{h}^{2}})^{1/2}$ denotes the natural moment map coordinate on $(\mathcal{C},g_{\mathcal{C}})$.