  We consider a functional related with phase transition models in the
Heisenberg group framework. We prove that level sets of local minimizers
satisfy some density estimates, that is, they behave as "codimension one" sets.
We thus deduce a uniform convergence property of these level sets to interfaces
with minimal area.
  These results are then applied in the construction of (quasi)periodic,
plane-like minimizers, i.e., minimizers of our functional whose level sets are
contained in a spacial slab of universal size in a prescribed direction. As a
limiting case, we obtain the existence of hypersurfaces contained in such a
slab which minimize the surface area with respect to a given periodic metric.
