First variation of the Hausdorff measure of non-horizontal submanifolds in stratified Lie groups

Abstract

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation $H+\sigma =0$, where $H$ is analogous to the mean curvature and $\sigma$ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in ${\mathbb{H}}^2$.

arXiv comment25 pages