Area-stationary surfaces in contact sub-riemannian manifolds

Resumen

In this thesis we study topics related to area-stationary surfaces in contact sub- Riemannian manifolds. In Chapter 1 we prove an existence result for isoperimetric regions in contact sub- Riemannian manifolds such that the quotient by the group of contact isometries, the diffeomorphisms that preserve the contact structure and the sub-Riemannian metric, is compact. This is the analogous result to of Morgan's Riemannian one. In Chapter 2, we prove a necessary condition for C^2 stable minimal surfaces with empty singular set in a large class of pseudo-hermitian manifolds, that includes the uni-modular Lie groups. The condition we found involves the Webster scalar curvature W and the pseudo-hermitian torsion of the manifold, that are pseudo- hermitian invariants. This characterization is obtained by the study of a stability operator, which is constructed from the second variation formula of the sub-Riemannian area. We construct another stability operator that takes account the singular set. With these two tools, we give a classification of complete stable surfaces in the group of the rigid motions of the Euclidean plane RT. The main purpose of Chapter 3 is to generalize variation formulas for the sub- Riemannian area in contact sub-Riemannian manifolds of arbitrary dimensions.