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 $\mathbb H^2$.