the restrictions on $\alpha$ and $\delta$ that we have imposed in . Hence we satisfy part (d) so long as $${\varepsilon}\leq\frac{\delta^2}{A^2}.$$

Finally, it is clear that part (e) is satisfied, because $gg_s\geq(1-{\varepsilon})f\geq0$.

Characterizing Kähler metrics {#SingularityModel}
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We prove in this paper that as they become singular, solutions originating from initial data satisfying our Closeness Assumptions asymptotically approach the blowdown soliton. For the reader’s convenience, we include here a brief review of metrics related to that singularity model.

The Calabi construction {#CalabiConstruction}
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We call a metric on ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$ or ${\mathcal}L_{-1}^2$ a *Calabi metric* if it is Kähler and ${\mathrm}U(2)$-invariant. As part of a much more general construction [@Calabi82], Calabi has observed that any ${\mathrm}U(2)$-invariant Kähler metric on ${\mathbb}C^2\backslash(0,0)$ has the form $$\label{U2-complex}
h_{\,{\mathbb}C^2\backslash(0,0)}
=\Big\{e^{-r}\phi\,\delta_{\alpha\beta}
+e^{-2r}(\phi_r-\phi)\,\bar z^\alpha z^\beta\Big\}\,
    {\mathrm}dz^\alpha{\otimes}{\mathrm}d\bar z^\beta.$$ Here $r:=\log(|z_1|^2+|z_2|^2)$ is Calabi’s coordinate, and $\phi(r)=P_r(r)$, where $P$ is the Kähler potential. The metric closes smoothly at the origin, hence induces a smooth metric on the total space of the bundle ${\mathcal}L_{-1}^2$ (or on a neighborhood of ${\mathcal}S^2_-$ in ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$) if and only if there are $a_0,a_1>0$ such that $$\label{ClosingCondition}
\phi(r)=a_0+a_1 e^r + a_2 e^{2r}+ {\mathcal}O(e^{3r})\quad\mbox{ as }\quad
r\rightarrow-\infty.$$ The metric closes smoothly at spatial infinity, hence induces a 