translation and scaling.

Basic estimates
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In this section, we prove a number of estimates that are needed for the proof of our Main Theorem.

A weak one-sided Kähler stability result {#Weak}
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We begin by introducing the useful quantity $$\label{KahlerQuantity}
\psi:=\left(\frac{gg_s}{f}\right)^2-1.$$ This quantity $\psi$ is well defined at ${\mathcal}S^2_\pm$, because it follows from l’Hôpital’s rule that $\frac{gg_s}{f}{\big|_{{\mathcal}S^2_\pm}}=gg_{ss}$.

Lemma \[CalabiCondition\] tells us that $\psi\equiv0$ if and only if the metric $G$ from is Kähler. Therefore, we use $\psi$ to measure, in a precise sense, how far away a solution is from being Kähler. The following result is thus a statement of weak (one-sided) stability for the Kähler condition. Note that part (b) of our Closeness Assumptions ensures that $\psi\leq0$ at $t=0$.

\[NearCalabi\] If $-1\leq\psi\leq0$ initially, then $-1\leq\psi\leq0$ as long as the flow exists.

It is only necessary to prove the upper bound. The quantity $\psi$ evolves under Ricci flow by $$\label{psi-evolution}
\psi_t=\psi_{ss}+\left\{3\frac{f_s}{f}-2\frac{g_s}{g}\right\}\psi_s
-\frac{\psi_s^2}{2(\psi+1)}+\left\{4\frac{g_s^2}{g^2}-8\frac{f_sg_s}{fg}\right\}\psi.$$ From this equation, it is clear that the condition $\psi\leq0$ is preserved if all maxima of $\psi$ occur away from ${\mathcal}S^2_\pm$.

If a maximum occurs instead at ${\mathcal}S^2_\pm$, then we apply l’Hôpital to determine that $$\frac{f_s\psi_s}{f}{\big|_{{\mathcal}S^2_\pm}}=\psi_{ss}\qquad\mbox{and}\qquad
\frac{f_sg_s}{fg}{\big|_{{\mathcal}S^2_\pm}}=\frac{g_{ss}}{g}.$$ Hence $$\psi_t{\big|_{{\mathcal}S^2_\pm}}=4\psi_{ss}-8\frac{g_{ss}}{g}\psi.$$ However, smoothness of either function $\psi_\pm(s,\cdot):=\psi(s-s_\pm,\cdot)$ at a 