The Plateau's Problem for intrinsic graphs in the Heisenberg Group

Using a geometric construction, we solve Plateau's Problem in the Heisenberg group H¹ for intrinsic graphs defined on a convex domain D, under a smallness condition either on the boundary ∂ D or on the Lipschitz boundary datum φ: ∂ D → R. The proof relies on a calibration argument. We then apply these techniques to establish a new regularity result for H-perimeter minimizers.