that $\rho$ and its time evolution depend only on $s(x,t)$ and $f(s(x,t),t)$, neither of which depend on $\rho$.

We observe that equation  has the form  of a Calabi metric $h$ if and only if $g^2=\phi$ and $f^2=(g^2)_\rho$, in which case one has $$f=gg_s\qquad\mbox{ and }\qquad f_s=gg_{ss}+g_s^2.$$

We summarize this simple observation, which is crucial to our work here, as follows:

\[CalabiCondition\] A $[{\mathcal}S^2\tilde\times{\mathcal}S^2]$-warped Berger metric  is Kähler if and only if $f=gg_s$.

If $G$ is Kähler, then its sectional curvatures (which generally take the form ) take the following special form: $$\begin{aligned}
\kappa_{12}=\kappa_{31}=\kappa_{02}=\kappa_{03}&=-\frac{g_{ss}}{g},\\
\kappa_{23}&=4\frac{1-g_s^2}{g^2},\\
\kappa_{01}&=-\frac{g_{sss}}{g_s}-3\frac{g_{ss}}{g}.\end{aligned}$$ As must be true for a Kähler metric on a complex surface, the Ricci endomorphism then has only two eigenvalues, $$R_0^0=R_1^1=-\frac{g_{sss}}{g_s}-5\frac{g_{ss}}{g}\qquad\mbox{ and }\qquad
R_2^2=R_3^3=-2\frac{g_{ss}}{g}+4\frac{1-g_s^2}{g^2}.$$

Because Kähler–Ricci flow is strictly parabolic, no time-dependent choice of gauge $s(x,t)$ is needed to ensure parabolicity. Rather, one can write the Kähler–Ricci <span style="font-variant:small-caps;">pde</span> with respect to a time-independent coordinate. The following observation is a particular instance of this general fact.

\[rho-evolution-proof\] The evolution equation for the coordinate $\rho$ under Ricci flow takes the form $$\label{rho-evolution}
\rho_t=2\int_0^\rho\left\{\frac{g_{ss}}{g}-\frac{f_sg_s}{fg}+\frac{f^2}{g^4}\right\}{\mathrm}d\bar\rho.$$ For a Kähler geometry, the integrand in  vanishes pointwise; hence, for Kähler initial data, the coordinate $\rho$ is independent of $t$.

For Kähler 