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

TL;DR

This paper establishes a rectifiability criterion for 1-codimensional measures in Carnot groups, extending classical results to non-Euclidean settings and providing new tools for analyzing geometric measure theory in these groups.

Contribution

It introduces a Marstrand-Mattila type rectifiability criterion for measures in Carnot groups, including the first non-Euclidean extension of Preiss's theorem.

Findings

Measures with (2n+1)-density in Heisenberg groups are rectifiable.

Provides a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets.

Shows flatness of tangents implies $C^1_ ext{G}$-rectifiability in Carnot groups.

Abstract

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