Calculus of Variations and Geometric Measure Theory

S. Nicolussi Golo

Some remarks on contact variations in the first Heisenberg group

created by nicolussigolo on 08 Feb 2016
modified on 23 Nov 2016


Submitted Paper

Inserted: 8 feb 2016
Last Updated: 23 nov 2016

Year: 2016


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.

We also prove that the only smooth diffeomorphisms that keep the intrinsic perimeter finite are contact diffeomorphisms.