The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg group

We prove that, in the first Heisenberg group H, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of H. Moreover two examples are given for stressing result’s sharpness.