Some remarks on contact variations in the first Heisenberg group

TL;DR

This paper investigates the stability of certain smooth intrinsic graphs in the first Heisenberg group, showing they are critical and stable for the sub-Riemannian perimeter despite not being area-minimizing.

Contribution

It demonstrates that specific intrinsic graphs are critical and stable under contact diffeomorphisms, expanding understanding of geometric variational properties in the Heisenberg group.

Findings

Intrinsic graphs with $ abla^f abla^f=0$ are critical points of sub-Riemannian area.

Such graphs are stable under compactly supported contact variations.

These graphs are not necessarily area-minimizing surfaces.

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 $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff=0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.