The Bernstein problem for embedded surfaces in the Heisenberg group $mathbb{H}^1$

Abstract

In the paper [13] we proved that the only stable C 2 minimal surfaces in the first Heisenberg group ℍ 1 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 C 2 complete embedded minimal surfaces in ℍ 1 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.