Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case

Details Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case

2007-09-19

arxiv.org/abs/0709.2977v1

Mathematics

Analysis of PDEs

Scientific paper

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. Generic singularities of minimal surfaces turn out the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations

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