Rigidity results for non local phase transitions in the Heisenberg group $H$

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-{\mathrm{\Delta }}_H{)}^{s}v=f(v)$ in $H$, $s\in (0,1)$. We obtain a Poincar'e type inequality in connection with a degenerate elliptic equation in ${\mathbb{R}}_{+}^4$; 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.