A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups

In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group ${\mathbb{H}}^n$, with $n\ge 2$, we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces.