Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group H to be area-minimizing in the slab { - 1 < x< 1 }. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.