Indiana University Mathematics Journal
In the paper  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  to C 2 complete em-bedded 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 coun-terpart of the classical theorems of Fischer-Colbrie and Schoen, , and do Carmo and Peng, , and answers a question posed by Lei Ni.
Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh; and Pauls, Scott D., "The Bernstein Problem for Embedded Surfaces in the Heisenberg Group H" (2010). Open Dartmouth: Faculty Open Access Scholarship. 3846.