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

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

2011-09-28

arxiv.org/abs/1109.6213v2

Mathematics

Differential Geometry

36 pages. Misprints corrected. Statement of Proposition 9.8 slightly changed and Remark 9.9 added

Scientific paper

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 C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 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 .

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