The Ginzburg-Landau equation in the Heisenberg group

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.