in any sub-riemannian group one can introduce a levi-civita connection adapted to the so-called horizontal subbundle. this can be used to introduce a horizontal connection on a codimension one submanifold and obtain, among other things, a notion of mean curvature. a smooth hypersurface is called h-minimal if its horizontal mean curvature vanishes everywhere. it is natural to pose an analogue of the classical bernstein problem, starting with the basic prototype of the heisenberg group. i will discuss a conjecture of bernstein type in this setting, and show that, despite the fact that h-minimal surfaces are critical points of an appropriate area functional (the h-perimeter), a new phenomenon occurs: there exists smooth critical points of the h-perimeter which are not local minimizers. this is in striking discrepancy with the classical case. i will also discuss in detail how this pathological minimal surfaces suggest an attack to the bernstein problem, and some interesting open questions.