A minimal regularity for the area formula in the Engel group

We prove that the upper blow-up theorem in the Engel group holds for $C^1$ submanifolds. Combining this result with the known negligibility of the singular set, we obtain an integral representation of the spherical measure for all surfaces of class $C^{1,α}$ in the Engel group. A new and central aspect of our method is the suitable use of Stokes’ theorem to prove the upper blow-up, which relies on the special algebraic structure of left-invariant forms in the Engel group. Some general tools are also introduced to establish area formulas in arbitrary stratified group.

💡 Research Summary

The paper investigates area formulas for submanifolds in the Engel group, a step‑three Carnot (or stratified) group with Lie algebra relations (