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 $\partial D$ or on the Lipschitz boundary datum $\varphi : \partial D \to \mathbb{R}$. The proof relies on a calibration argument. We then apply these techniques to establish a new regularity result for $H$-perimeter minimizers.