A pr 2 00 6 Minimal surfaces in contact Sub - Riemannian manifolds

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ”horizontal” area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the ’‘horizontal” area functional. We derive an intrinsic equation for minimal surfaces associated to a generic SubRiemannian structure of co-rank 1 in terms of the canonical volume form and the “horizontal” normal. The presented construction permits to describe the Sub-Riemannian minimal surfaces in a generic SubRiemannian manifold and can be easily generalized to the case of nonholonomic vector distributions of greater co-rank. The case of contact vector distributions, in particular the (2, 3)-case, is studied more in detail. In the latter case the geometry of the SubRiemannian minimal surfaces is determined by the structure of their characteristic points (i.e., the points where the hyper-surface touches the horizontal distribution) and characteristic curves. It turns out that the known (see [4]) classification of the characteristic points of the SubRiemannian minimal surfaces in the Heisenberg group H holds true for the minimal surfaces associated to a generic contact (2, 3) distribution. Moreover, we show that in the (2, 3) case the Sub-Riemannian minimal surfaces are the integral surfaces of a certain system of ODE in the extended state space. In some particular cases the Cauchy problem for this system can be solved explicitly. We illustrate our results considering Sub-Riemannian minimal surfaces in the Heisenberg group and the group of roto-translations.