B > 0$ are to be suitably chosen below. We first show that there exists a uniform constant $C$ so that $Q \ge -C$ for as long as the flow exists. A straightforward computation shows that the evolution of $Q$ is given by $$\begin{gathered}
\label{eq-Q}
\frac{\partial Q}{\partial t} = \Delta Q + \frac{12 B f^2 f_s^2}{g^4}
+ \frac{4 f_s^2}{g^2} + \frac{24 f^2 g_s^2}{g^4} + \frac{12A f^2 g_s^2}{g^4} + \frac{2Af_s^2 g_s^2}{f^2} + \frac{4Bf_s^2g_s^2}{g^2} \\
+ \frac{2g_s^4}{g^2} + \frac{2A g_s^4}{g^2} + 2B f_{ss}^2 + 2(A - 1) g_{ss}^2 
- g g_{ss}\left(\frac{4f^2}{g^4} + \frac{2f_s^2}{f^2} + \frac{4Ag_s^2}{g^2}\right) \\
- \frac{16 B f^3 f_s g_s}{g^5} - \frac{24 f f_s g_s}{g^3} - \frac{8A f f_s g_s}{g^3} + \frac{2g f_s^3 g_s}{f^3} - \frac{8 g_s^2}{g^2} - \frac{8A g_s^2}{g^2}  \\
+ \frac{4 f f_{ss}}{g^2} + \frac{4B f_s^2 f_{ss}}{f} - \frac{8B f_s g_s f_{ss}}{g} - \frac{2g f_s g_s f_{ss}}{f^2},\end{gathered}$$ where as noted above, $\Delta Q=Q_{ss}+(f_s/f+2g_s/g)Q_s$. We observe that l’Hôpital’s rule implies that the terms $$\frac{Q_s f_s}{f}, \qquad \frac{2Af_s^2 g_s^2}{f^2}, \qquad 2g f_s^2\, \frac{(f_s g_s - f g_{ss})}{f^3},
\qquad \frac{4B f_s^2 f_{ss}}{f},$$ appearing in equation  are well defined and smooth at ${\mathcal}S^2_{\pm}$. We now distinguish between two cases.

\[case-1\] A minimum of $Q$ occurs away from ${\mathcal}S^2_\pm$.

We assume that at 