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

Andrea Merlo

Vol. 16 (2023), No. 4, 927–996
Abstract

In this paper, we show that the flatness of tangents of 1-codimensional measures in Carnot groups implies C𝔾1-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 (2n+1)-density in the Heisenberg groups n are Cn1-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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