2006/04/23 by Nataliya Shcherbakova, Shcherbakova, Nataliya
Computer Science · Mathematics · #49Q05 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Topological and Geometric Data Analysis #math.DG #msc:49Q05 #msc:53C17
paper · pdf · doi:10.48550/arxiv.math/0604494
24 pages, 2 figures
arxiv created 2006/04/23 · openalex publication_date 2006/04/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H1 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.