Stable complete embedded minimal surfaces in $\mathbb H^1$ with empty characteristic locus are vertical planes

Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh; Pauls, Scott

Stable complete embedded minimal surfaces in $\mathbb H^1$ with empty characteristic locus are vertical planes

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In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ 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 \cite{DGNP} 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.