Area-stationary and stable surfaces of class C1 in the sub-Riemannian Heisenberg group H1

We consider surfaces of class C1 in the 3-dimensional sub-Riemannian Heisenberg group H1. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a C1 function in H1 is a vertical plane.