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Euclidean rectifiability of sub-Finsler spheres in free-Carnot groups of step 2

Enrico Le Donne, Luca Nalon

Abstract

We consider 2-step free-Carnot groups equipped with sub-Finsler distances. We prove that the metric spheres are codimension-one rectifiable from the Euclidean viewpoint. The result is obtained by studying how the Lipschitz constant for the distance function behaves near abnormal geodesics.

Euclidean rectifiability of sub-Finsler spheres in free-Carnot groups of step 2

Abstract

We consider 2-step free-Carnot groups equipped with sub-Finsler distances. We prove that the metric spheres are codimension-one rectifiable from the Euclidean viewpoint. The result is obtained by studying how the Lipschitz constant for the distance function behaves near abnormal geodesics.
Paper Structure (8 sections, 14 theorems, 118 equations)

This paper contains 8 sections, 14 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Sub-Finsler free-Carnot groups of step 2
  3. The sphere as graph
  4. Abnormal set and Minkowski content
  5. Minimal height and rectifiability of spheres
  6. Some remarks on minimal height
  7. Final proofs
  8. A sharp example

Key Result

Theorem 1.2

Let $\mathbb G$ be a sub-Finsler free-Carnot group of step $2$. Denote by $d_{\mathrm{cc}}$ the corresponding Carnot-Carathéodory distance. Then the unit sphere, defined as is Euclidean $(d-1)$-rectifiable, where $d$ is the topological dimension of $\mathbb G$ and $0_\mathbb G$ is its identity element.

Theorems & Definitions (40)

  • Definition 1.1: Euclidean rectifiable subset
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 30 more