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

TL;DR

This paper extends a Bernstein-type theorem in the Heisenberg group, proving that complete, boundaryless, stable minimal surfaces with empty characteristic locus are necessarily vertical planes.

Contribution

It generalizes previous results to all complete embedded minimal surfaces without boundary in , showing they must be vertical planes.

Findings

Complete embedded minimal surfaces with empty characteristic locus are vertical planes.

Extension of Bernstein-type theorem to broader class of surfaces.

No such surfaces exist other than vertical planes.

Abstract

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.