Stable minimal graphs in Heisenberg group ${\mathbb{H}}^n$

We prove that a strictly stable minimal $C_h^2$ intrinsic graph G is locally area-minimizing, i.e. given any $C_h^1$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^{\mathrm{\infty }}$ minimal graphs with prescribed small boundary datum.