A minimal regularity for the area formula in the Engel group

TL;DR

This paper establishes an area formula for $C^{1,eta}$ surfaces in the Engel group by proving an upper blow-up theorem using Stokes' theorem and algebraic structures, extending geometric measure theory in stratified groups.

Contribution

It proves the upper blow-up theorem for $C^1$ submanifolds in the Engel group and introduces tools for area formulas in stratified groups, utilizing Stokes' theorem and algebraic structures.

Findings

Upper blow-up theorem holds for $C^1$ submanifolds in the Engel group.

Integral representation of spherical measure for $C^{1,eta}$ surfaces.

Method relies on algebraic structure and Stokes' theorem in stratified groups.

Abstract

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 $C^{1,\alpha}$ 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.