closing construction at either “pole” $s_\pm$. We note that while ${\mathcal}S^2\tilde\times{\mathcal}S^2$ is diffeomorphic to ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$, the Ricci flow evolutions we study are not Kähler.

The metrics $G = {{\mathrm}{d}s}{\otimes}{{\mathrm}{d}s}+f^2\,\omega^1{\otimes}\omega^1+g^2\omega^2{\otimes}\omega^2+h^2\omega^3{\otimes}\omega^3$ described in Appendix A of [@IKS14] are clearly ${\mathrm}{SU}(2)$-invariant. The simplifying assumption $h\equiv g$ made here enlarges their symmetry group to ${\mathrm}U(2)$. However, although ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$ admits Kähler metrics, including the ${\mathrm}U(2)$-invariant Kähler–Ricci soliton mentioned above, we observe in Lemma \[CalabiCondition\] that metrics of the form  cannot be Kähler unless they satisfy the closed condition $f=gg_s$.

Ricci flow equations
--------------------

We study in this work solutions $\big({\mathcal}S^2\tilde\times{\mathcal}S^2,G(t)\big)$ of Ricci flow that originate from smooth initial data $G(0)$ satisfying the closing conditions and for the \[${\mathcal}S^2\tilde\times{\mathcal}S^2$\]-warped Berger geometries, as outlined above. For as long as such solutions remain smooth, the functions $f$ and $g$ continue to satisfy conditions and , and hence remain \[${\mathcal}S^2\tilde\times{\mathcal}S^2$\]-warped Berger geometries.

Since the metrics studied in [@IKS14] and those studied here are the same apart from boundary conditions, we may use formulas (10)–(13) of [@IKS14] to obtain the sectional curvatures of the metric[^3] $G$:

\[curvatures\] $$\begin{aligned}
\kappa_{12}=\kappa_{31}&=\frac{f^2}{g^4}-\frac{f_s g_s}{fg},\\
\kappa_{23}&=\frac{4g^2-3f^2}{g^4}-\frac{g_s^2}{g^2},\\
\kappa_{01}&=-\frac{f_{ss}}{f},\\
\kappa_{02}=\kappa_{03}&=-\frac{g_{ss}}{g}.\end{aligned}$$

Writing the metric in coordinate form , we note that its evolution under Ricci flow is governed 