initial data, one may therefore assume without loss of generality that $\rho$ is identical to Calabi’s coordinate $r=\log(|z_1|^2+|z_2|^2)$.

It follows from equation (56) in [@IKS14] and from   above that the gauge quantity $\frac{\partial s}{\partial x}$ evolves according to the equation $$\Big(\frac{\partial s}{\partial x}\Big)_t=
\left\{\frac{f_{ss}}{f}+2\frac{g_{ss}}{g}\right\}\frac{\partial s}{\partial x}.$$ Hence, using equations  and , we determine that the time derivative of $\rho$ at fixed $x$ is given by $$\begin{aligned}
\frac12\rho_t&=\frac{\partial}{\partial t}
\left(\int_0^x f^{-1}\big(s(\bar x,t),t\big)\,\frac{\partial s}{\partial\bar x}\,{\mathrm}d\bar x\right)\\
&=\int_0^x\left\{f^{-1}\Big(\frac{\partial s}{\partial\bar x}\Big)_t-f^{-2}f_t\Big(\frac{\partial s}{\partial\bar x}\Big)
\right\}{\mathrm}d\bar x\\
&=2\int_0^s\left\{\frac{g_{\bar s\bar s}}{fg}-\frac{f_{\bar s}g_{\bar s}}{f^2g}+\frac{f}{g^4}\right\}{\mathrm}d\bar s.\end{aligned}$$ This proves the first claim. The second follows by direct computation.

Ricci flow of Calabi metrics
----------------------------

Lemma \[CalabiCondition\] states that the initial metric is Calabi if and only if $f=gg_s$. Because Ricci flow preserves the Kähler condition with respect to the original complex structure (here, the unique complex structure on ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$) and also preserves initial symmetries, a solution originating from Calabi initial data remains Calabi for as long as it exists. This can be seen directly, as we now observe.

In this section, which is not needed for the rest of the paper, we find it convenient to work with $u:=f^2$ and $v:=g^2$. The Ricci flow evolution equations for these quantities are 