The bernstein problem for embedded surfaces in the heisenberg group ℍ¹

In the paper [13] we proved that the only stable C² minimal surfaces in the first Heisenberg group ℍ¹ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in [13] to C² complete embedded minimal surfaces in ℍ¹ with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian counterpart of the classical theorems of Fischer-Colbrie and Schoen, [16], and do Carmo and Peng, [15], and answers a question posed by Lei Ni.