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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 Hminimal 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 Hminimal entire graphs, with empty characteristic locus, are the vertical planes.

Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1