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

Abstract

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 .

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

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Cited by in corpus (5)

• Variational formulas for submanifolds of fixed degree
• Tubular neighborhoods in the sub-Riemannian Heisenberg groups
• On the classification of complete area-stationary and stable surfaces in the sub-Riemannian Sol manifold
• Area-stationary and stable surfaces of class $C^1$ in the sub-Riemannian Heisenberg group ${\mathbb{H}}^1$
• Regularity of $C^1$ surfaces with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds