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

Authors: D. Danielli, N. Garofalo, D.M. Nhieu and S. D. Pauls

Issue: Volume 59 (2010), Issue 2, 563-594

Abstract: In the paper [D. Danielli, N. Garofalo, D.M. Nhieu, S.D. Paulsen, \textit{Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group $\mathbb{H}^{1}$}, J. Differential Geom. \textbf{81} (2009), 251--295}, we proved that the only stable $C^{2}$ minimal surfaces in the first Heisenberg group $\mathbb{H}^{1}$ 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.\par In this paper we extend the result in [ibid.] to $C^{2}$ complete embedded minimal surfaces in $\mathbb{H}^{1}$ 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, [D. Fischer-Colbrie and R. Schoen, \textit{The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature}, Comm. Pure Appl. Math. \textbf{33} (1980), 199--211], and do Carmo and Peng, [M. do Carmo and C.K. Peng, \textit{Stable complete minimal surfaces in $\mathbb{R}^{3}$ are planes}, Bull. Amer. Math. Soc. (N.S.) \textbf{1} (1979), 903--906], and answers a question posed by Lei Ni.