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 $(-Δ_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincaré 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.

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