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 .
Keywords
Cite
@article{arxiv.1109.6213,
title = {First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds},
author = {Matteo Galli},
journal= {arXiv preprint arXiv:1109.6213},
year = {2014}
}
Comments
36 pages. Misprints corrected. Statement of Proposition 9.8 slightly changed and Remark 9.9 added