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

TL;DR

This paper derives first and second variation formulas for sub-Riemannian area in three-dimensional pseudo-hermitian manifolds, enabling stability analysis and classification of stable surfaces, including in the roto-translation group RT.

Contribution

It introduces new variation formulas for both singular and non-singular surfaces, leading to stability criteria and classification results in pseudo-hermitian geometry.

Findings

Derived stability operators for different surface classes

Established necessary conditions for stability involving torsion and curvature

Classified complete stable surfaces in the roto-translation group RT

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 .