First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

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 + σ = 0, where H is analogous to the mean curvature and σ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in \(\mathbb {H}^{2}\).