Some remarks on contact variations in the first Heisenberg group

Abstract

We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f:{\mathbb{R}}^2\to \mathbb{R}$ is a $C^1$-intrinsic function, and ${\nabla }^f{\nabla }^{f}f=0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.

arXiv comment27 pages; Revised argument in section 3, results unchanged