Instability of Graphical Strips and a Positive Answer to the Bernstein Problem in the Heisenberg Group H1

One of the most celebrated problems in geometry and calculus of variations is the Bernstein problem, which asserts that a C2 minimal graph in R3 must necessarily be an affine plane. Following an old tradition, here minimal means of vanishing mean curvature. Bernstein [Be] established this property in 1915. Almost fifty years later a new insight of Fleming [Fle] sparked a major development in the geometric measure theory which, through the celebrated works [DG3], [Al], [Sim], [BDG] culminated in the following solution of the Bernstein problem. Theorem 1.1. Let S = {(x, u(x)) ∈ Rn+1|x ∈ Rn, xn+1 = u(x)} be a C2 minimal graph in Rn+1, i.e., let u ∈ C2(Rn) be a solution of the minimal surface equation