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

Content Provider	SpringerLink
Author	Galli, Matteo
Copyright Year	2012
Abstract	We calculate the first and second variation formulae for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that move the singular set of a $${\mathcal{C}^2}$$ surface and non-singular variations for $${\mathcal{C}^2_\mathcal{H}}$$ surfaces. These formulae enable us to construct a stability operator for non-singular $${\mathcal{C}^2}$$ surfaces and another one for $${\mathcal{C}^2}$$ (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 terms of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we give a classification of the complete stable surfaces in the roto-translation group $${\mathcal{R}\mathcal{T}}$$ .
Ending Page	157
Page Count	41
Starting Page	117
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	1-2
Volume Number	47
Language	English
Publisher	Springer-Verlag
Publisher Date	2012-03-28
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Variational problems in a geometric measure-theoretic setting Sub-Riemannian geometry Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Analysis Applied Mathematics

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