Instability of graphical strips and a positive answer to the bernstein problem in the heisenberg group ℍ¹

In the first Heisenberg group ℍ¹ with its sub-Riemannian struc-ture generated by the horizontal subbundle, we single out a class of C² 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 deforma-tions 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 C² 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 ℍ¹ the only stable C2 H- minimal entire graphs, with empty characteristic locus, are the vertical planes.