well-defined function $\theta$ such that $$f=\theta\,gg_s.$$ We note that $\theta\equiv1$ for a Kähler solution, and that the evolution equation for $\theta$ is $$\theta_t=\theta_{ss}+2\frac{fg_s-2f_sg}{g^2}(\theta^2-1),$$ which yields an easy direct proof that the Kähler condition is preserved for these geometries.

The blowdown soliton {#ModelSoliton}
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Under Kähler–Ricci flow, the evolution of an arbitrary Calabi metric $$\label{Calabih}
h =\Big\{e^{-r}\phi\,\delta_{\alpha\beta}
+e^{-2r}(\phi_r-\phi)\,\bar z^\alpha z^\beta\Big\}\,
    {\mathrm}dz^\alpha{\otimes}{\mathrm}d\bar z^\beta,$$ written in terms of Calabi’s fixed $r$-coordinate on ${\mathcal}L_{-1}^2$ or on ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$, is determined by the <span style="font-variant:small-caps;">pde</span> $$\label{KRF}
\phi_t=\frac{\phi_{rr}}{\phi_r}+\frac{\phi_r}{\phi}-2.$$

The blowdown soliton is specified by setting $\phi$ in equal to a function ${\varphi}$ which (following Lemma 6.1 and equation (27) of [@FIK03] with $\lambda=-1$, $\mu=\sqrt2$, and $\nu=0$) satisfies the separable first-order <span style="font-variant:small-caps;">ode</span> $$\label{phi-ode1}
{\varphi}_r=\frac{1}{\sqrt2}{\varphi}-(\sqrt2-1)-\left(1-\frac{1}{\sqrt{2}}\right){\varphi}^{-1}.$$ Rewriting this <span style="font-variant:small-caps;">ode</span> in the form $${\mathrm}dr = \frac{{\varphi}\,{\mathrm}d{\varphi}}{{\varphi}-1}-\frac{{\varphi}\,{\mathrm}d{\varphi}}{{\varphi}+\sqrt2-1},$$ one can solve it implicitly, obtaining $$\label{phi-implicit}
e^{r+\chi}=\frac{{\varphi}-1}{\big({\varphi}+\sqrt2-1\big)^{\sqrt2-1}}.$$ The arbitrary constant $\chi$ above reflects the fact that the soliton is unique only modulo translations in $r$. Examination of formula  also shows that the soliton is cone-like at spatial infinity and hence complete.

Equation (24) of [@FIK03] implies that the blowdown soliton function ${\varphi}$ also satisfies the second-order <span style="font-variant:small-caps;">ode</span> $$\label{phi-ode2}
\frac{{\varphi}_{rr}}{{\varphi}_r}+\frac{{\varphi}_r}{{\varphi}}-\sqrt2{\varphi}_r+{\varphi}-2=0.$$ It follows from that ${\varphi}$ satisfies ${\varphi}_t=\sqrt2{\varphi}_r-{\varphi}$; hence the soliton evolves by 