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

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We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C² surface and non-singular variation for Cₕ² surfaces. These formulas enable us to construct a stability operator for non-singular C² surfaces and another one for C2 (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 term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we classify complete stable surfaces in the roto-traslation group RT.