  In the Heisenberg group framework, we study rigidity properties for stable
solutions of $(-\Delta_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a
Poincar\'e type inequality in connection with a degenerate elliptic equation in
$\R^4_+$; through an extension (or "lifting") procedure, this inequality will
be then used for giving a criterion under which the level sets of the above
solutions are minimal surfaces in $H$, i.e. they have vanishing mean curvature.
