by the evolution equations for $f$ and $g$, which (as shown in (14) of [@IKS14]) take the following form:

\[RF\] $$\begin{aligned}
    \label{f-evolution}
f_t&=f_{ss}+2\frac{g_s}{g}f_s-2\frac{f^3}{g^4},\\
    \label{g-evolution}
g_t&=g_{ss}+\left(\frac{f_s}{f}+\frac{g_s}{g}\right)g_s
+2\frac{f^2-2g^2}{g^3}.\end{aligned}$$

The variable $s=s(x,t)$, representing arclength from the ${\mathcal}S^3$ at $x=0$, is a choice of gauge that results in this system being manifestly strictly parabolic. The cost one pays for this is the non-vanishing commutator, $$\label{Commutator}
    \left[\frac{\partial}{\partial t},\frac{\partial}{\partial s}\right]
    =-(\log\rho)_t\frac{\partial}{\partial s}
    =-\left(\frac{f_{ss}}{f}+2\frac{g_{ss}}{g}\right)\frac{\partial}{\partial s}.$$

Closeness Assumptions {#sec-assume}
---------------------

The Riemannian Ricci flow solutions we study here originate from an open set of cohomogeneity-one metrics that is defined by certain mild hypotheses, which effectively guarantee that at least initially, the metrics are “somewhat close” to the subspace of Kähler metrics.

\[assumption-main\] At time $t=0$, the metric $G$ of the form  determined by the pair $(f,g)$ satisfies the following:

1.  $f\leq g$;

2.  $g|g_s|\leq f$;

3.  $|f_s| \le 2/\sqrt3$;

4.  $g^2(s_+)-3g^2(s_-)\geq\delta^2$ for some $\delta>0$; and

5.  $g_s\geq0$, with strict inequality off ${\mathcal}S^2_\pm$.

It follows from Lemma 26 of [@IKS14] that condition (a) is preserved under the flow. We prove in Section \[FirstOrder\] that condition (b) — which, as we show there, may be regarded as a “Kähler pinching condition” — and condition (c) are preserved by the flow. 