First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and second variation formulae for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that move the singular set of a ${mathcal{C}^2}$ surface and non-singular variations for ${mathcal{C}^2_mathcal{H}}$ surfaces. These formulae enable us to construct a stability operator for non-singular ${mathcal{C}^2}$ surfaces and another one for ${mathcal{C}^2}$ (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in terms of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we give a classification of the complete stable surfaces in the roto-translation group ${mathcal{R}mathcal{T}}$ .