Minimal surfaces in contact Sub-Riemannian manifolds
| dc.creator | Shcherbakova, Nataliya | |
| dc.date | 2006-04-23 | |
| dc.date.accessioned | 2026-07-07T07:11:11Z | |
| dc.date.available | 2026-07-07T07:11:11Z | |
| dc.description | 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. | |
| dc.description | 24 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/math/0604494 | |
| dc.identifier | http://arxiv.org/abs/math/0604494 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/111660 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53C17; 49Q05 | |
| dc.title | Minimal surfaces in contact Sub-Riemannian manifolds | |
| dc.type | text |