+ 8 \frac{f^3}{g^5} g_s \\
&\leq\left(\frac{4}{\sqrt3}-2C\right)\frac{g_s^2}{g^2}
+\left(\frac{12}{\sqrt3}\frac{f^4}{g^4}-6C\right)\frac{f^2}{g^4}\\
&\leq 0\end{aligned}$$ if $C \ge 2/\sqrt3$, because $f/g\leq1$.

A similar argument shows that $\frac{d}{dt} (f_s)_{\min}\geq0$ if $(f_s)_{\min}=-C$ at some time.

These uniform bounds on the first derivatives of $f$ and $g$ lead to the following.

\[lem-curv-1-der\] For any solution originating from initial data satisfying our Closeness Assumptions, there exists a uniform constant $C$ such that $$|\kappa_{12}| + |\kappa_{31}| + |\kappa_{23}| \le \frac{C}{g^2}$$ for as long as the flow exists.

It follows from Lemma \[NearCalabi\] that the inequality $(gg_s/f)^2\leq1$ persists if it is true initially. This implies that $|g_s| \leq f/g$ for as long as the flow exists. Combining this estimate with the identities in , using Corollary \[cor-gs\], Lemma \[lem-fs\], and the fact that $f \leq g$, we obtain $$|\kappa_{12}| + |\kappa_{31}| + |\kappa_{23}| \le \frac{C}{g^2}.$$

Second derivative estimates {#SecondOrder}
---------------------------

Here we derive estimates for the remaining curvatures — those that depend on second-order derivatives of $(f,g)$.

\[lem-k02\] For any solution originating from initial data satisfying our Closeness Assumptions, there exists a uniform constant $C$ such that $$|\kappa_{02}| = \left|\frac{g_{ss}}{g}\right| \le \frac{C}{g^2}$$ for as long as the flow exists.

We define $Q = g g_{ss} - A g_s^2 - Bf_s^2$, where $A, 