Measure of submanifolds in the Engel group

We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.

💡 Research Summary

The paper investigates the geometric measure theory of submanifolds in the Engel group, a four‑dimensional step‑3 nilpotent Lie group that serves as a canonical example of a non‑commutative Carnot group. The authors focus on submanifolds that are $C^{1,1}$ smooth, meaning that they possess a Lipschitz continuous first derivative. For each such submanifold $M$ they introduce the notion of “degree” $d$, which is determined by how the tangent space $T_pM$ at a point $p\in M$ aligns with the stratification of the Engel Lie algebra. Roughly, $d$ counts the highest layer of the stratification that is non‑trivially intersected by $T_pM$: if only the first layer is involved, $d=1$; if the first and second layers appear, $d=2$; inclusion of the third layer yields $d=3$; and when the full four‑dimensional algebra is spanned, $d=4$.

The main theorem states that for every $C^{1,1}$ submanifold $M$ of degree $d$, the intrinsic measure $\mu_M$—the natural measure induced by the Haar measure of the Engel group restricted to $M$—is equivalent to the $d$‑dimensional spherical Hausdorff measure $\mathcal{S}^d$ restricted to $M$. More precisely, there exists a positive constant $c_d$, depending only on the degree, such that \