Plateau's Problem for intrinsic graphs in the Heisenberg Group

Using a geometric construction, we solve Plateau’s Problem in the Heisenberg group ${\mathbb{H}}^1$ for intrinsic graphs defined on a convex domain $D$, under a smallness condition either on the boundary $\mathrm{\partial }D$ or on the Lipschitz boundary datum $\phi :\mathrm{\partial }D\to \mathbb{R}$. The proof relies on a calibration argument. We then apply these techniques to establish a new regularity result for $H$-perimeter minimizers.