Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A minimal regularity for the area formula in the Engel group

Francesca Corni, Fares Essebei, Valentino Magnani

TL;DR

This work tackles the problem of representing the spherical measure on $2$-dimensional submanifolds in the Engel group, linking the area formula to an upper blow-up analysis in a non-Euclidean, step-3 Carnot group. It develops a general framework for upper blow-up in homogeneous groups via a special coordinate system and intrinsic Taylor-type expansion, then applies it to the Engel group. A key novelty is proving the vanishing of the structure coefficient $\xi_{13}$ at maximum-degree points for degree-3 surfaces using Stokes' theorem and Maurer–Cartan relations, which enables $C^1$ blow-up in the lowest regularity regime previously unresolved. Combined with the negligibility of the singular set, the authors derive an integral representation of the spherical measure for $C^{1,\alpha}$ surfaces with $d(\Sigma)\le 4$, and discuss implications for transversal and multiradial-distance settings, contributing a robust toolbox for area formulas 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 $C^{1,α}$ 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.

A minimal regularity for the area formula in the Engel group

TL;DR

This work tackles the problem of representing the spherical measure on -dimensional submanifolds in the Engel group, linking the area formula to an upper blow-up analysis in a non-Euclidean, step-3 Carnot group. It develops a general framework for upper blow-up in homogeneous groups via a special coordinate system and intrinsic Taylor-type expansion, then applies it to the Engel group. A key novelty is proving the vanishing of the structure coefficient at maximum-degree points for degree-3 surfaces using Stokes' theorem and Maurer–Cartan relations, which enables blow-up in the lowest regularity regime previously unresolved. Combined with the negligibility of the singular set, the authors derive an integral representation of the spherical measure for surfaces with , and discuss implications for transversal and multiradial-distance settings, contributing a robust toolbox for area formulas in stratified groups.

Abstract

We prove that the upper blow-up theorem in the Engel group holds for 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 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.
Paper Structure (10 sections, 19 theorems, 141 equations)

This paper contains 10 sections, 19 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic facts on homogeneous groups
  4. Degree of submanifolds and the homogeneous tangent space
  5. Intrinsic measure and measure-theoretic area formula
  6. Some general results for the upper blow-up in homogeneous groups
  7. The Engel group with respect to a general graded basis
  8. Non-existence of horizontal surfaces in the Engel group
  9. Upper Blow-up in the Engel group
  10. Appendix

Key Result

Theorem 1.1

Let $\Sigma \subset {\mathbb{G}}$ be a $2$-dimensional $C^{1,\alpha}$ regular submanifold with $d(\Sigma)\leqslant 4$. Then the following formula holds for every Borel set $B \subset \Sigma$ whenever $(d(\Sigma)-2)/d(\Sigma)<\alpha\leqslant 1$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2: Degree of $k$-vectors and of $k$-vector fields
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Homogeneous tangent space
  • Definition 2.6
  • Definition 2.7
  • ...and 29 more