  The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$,
$\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric is
constructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametric
form related to a Schwarzian equation and prove new relations for $N=2$ Super
$SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz's
lemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem related
to the Renormalization Group. Furthermore, we obtain a closed form for the
prepotential as function of $a$. Finally, we show that $\partial_{\hat\tau}
\langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle
\phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the beta
function.
