Instability of graphical strips and a positive answer to the Bernstein problem in theHeisenberg group H^1

In the first Heisenberg group H-1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C-2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C-2 entire H-minimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H-1 the only stable C-2 H-minimal entire graphs, with empty characteristic locus, are the vertical planes.