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

SISSA/ISAS, via Beirut 2-4, 34100, Trieste, Italy. chtch@sissa.it

Received: 29 August 2007
Revised: 20 April 2008

Abstract

We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical 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. The singularities of minimal surfaces turn out to be 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.