given by $$\begin{aligned}
u_t &= u_{ss}-\frac{u_s^2}{2u}+\frac{u_s v_s}{v}-4\frac{u^2}{v^2},\\
v_t &= v_{ss}+\frac{u_s v_s}{2u}+4\frac{u-2v}{v}.\end{aligned}$$ On a Calabi solution, one can use the relation $u=v_s^2/4$ (equivalent to $f=gg_s$) to simplify the evolution equation above for $v$, thereby obtaining $$\label{Calabi-v-evolution}
v_t=2v_{ss}+\frac{v_s^2}{v}-8.$$ One now has two ways of computing the evolution of $u$. Evaluating the equation above for $u_t$ by using the Kähler condition $u=v_s^2/4$ to convert the <span style="font-variant:small-caps;">rhs</span> into terms involving only $v$ and its derivatives, one obtains $$\label{Calabi-u-evolution}
u_t=\frac12v_sv_{sss}+\frac12\frac{v_{ss}v_s^2}{v}-\frac14\frac{v_s^4}{v^2}.$$ On the other hand, one can differentiate the <span style="font-variant:small-caps;">rhs</span> of $u=v_s^2/4$ directly, use the commutator $[\partial_t,\partial_s]$ given in equation , and then apply , obtaining $$\begin{aligned}
\left(\frac{v_s^2}{4}\right)_t
    &=\frac12v_s(v_s)_t\\
    &=\frac12v_s\left\{(v_t)_s-\left(\frac{f_{ss}}{f}+2\frac{g_{ss}}{g}\right)v_s\right\}\\
    &=\frac12v_sv_{sss}+\frac12\frac{v_{ss}v_s^2}{v}-\frac14\frac{v_s^4}{v^2},\end{aligned}$$ as above. This calculation verifies directly what one knows from general principles: that the Calabi condition is preserved by the flow. We note in particular that for a Calabi solution, the Ricci flow system reduces to a scalar <span style="font-variant:small-caps;">pde</span>, in the sense that the evolution of $u$ is completely determined by the evolution of $v$.

For solutions with initial data satisfying our Closeness Assumptions, the fact that $g_s>0$ everywhere except at ${\mathcal}S^2_\pm$ holds initially. For as long as this remains true (presumably only a short time for non-Kähler solutions), there is a 