The so-called Bernstein problem consists in characterizing global minimizers of suitable
perimeter functionals. Roughly speaking, is it true that (smooth) boundaries of global
perimeter minimizers are flat in a certain sense? While this topic is well understood in the
Euclidean framework, the Bernstein problem in sub-Riemannian Heisenberg groups leaves
many interesting questions still unanswered, especially with regard to its high-dimensional
formulation. A crucial step in the study of minimal surfaces in H^1, where the Bernstein
problem is answered affirmatively, is to show that they are ruled by horizontal lines.
In this talk, after a survey of the known results, we introduce a generalization of the notion
of ruled surface to higher dimensional Heisenberg groups. We link this differential
notion with the vanishing of a suitable horizontal second fundamental form, and we provide
a characterization of those hypersurfaces which share the above-mentioned properties. To
conclude, we discuss some consequences and we present some possible developments.