Minimal surfaces in contact Sub-Riemannian manifolds

Details Minimal surfaces in contact Sub-Riemannian manifolds Minimal surfaces in contact Sub-Riemannian manifolds

2006-04-23

arxiv.org/abs/math/0604494v1

Mathematics

Differential Geometry

24 pages, 2 figures

Scientific paper

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 Sub-Riemannian 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 Sub-Riemannian manifold and can be easily generalized to the case of non-holonomic 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 Sub-Riemannian 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 classification of the characteristic points of the Sub-Riemannian minimal surfaces in the Heisenberg group $H^1$ 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.

Affiliated with

Also associated with

No associations

Rating

Minimal surfaces in contact Sub-Riemannian manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Minimal surfaces in contact Sub-Riemannian manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimal surfaces in contact Sub-Riemannian manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-120780