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arXiv · 1109.6213

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

Matteo Galli
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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 .

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Matteo Galli. 2011-09-28. First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds. https://doi.org/10.1007/s00526-012-0513-4

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