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

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