Marstrand–Mattila rectifiability criterion for 1-codimensional measures in Carnot groups

Merlo, Andrea

doi:10.2140/apde.2023.16.927

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Marstrand–Mattila rectifiability criterion for $1$-codimensional measures in Carnot groups

Andrea Merlo

Vol. 16 (2023), No. 4, 927–996

DOI: 10.2140/apde.2023.16.927

Abstract

In this paper, we show that the flatness of tangents of $1$ -codimensional measures in Carnot groups implies $C_{𝔾}^{1}$ -rectifiability. As applications we prove a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups and we show that measures with $(2 n + 1)$ -density in the Heisenberg groups $ℍ^{n}$ are $C_{ℍ^{n}}^{1}$ -rectifiable, providing the first non-Euclidean extension of Preiss’s rectifiability theorem.

Keywords

Marstrand–Mattila rectifiability criterion, Preiss's rectifiability theorem, Carnot groups

Mathematical Subject Classification

Primary: 28A75, 53C17

Secondary: 22E25, 49Q15

Milestones

Received: 16 December 2020

Revised: 14 September 2021

Accepted: 28 October 2021

Published: 15 June 2023

Authors

Andrea Merlo
Departamento de Matemáticas Universidad del País Vasco Leioa Spain

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Vol. 16, No. 4, 2023