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

Abstract

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C_{\mathbb{G}}^1$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups ${\mathbb{H}}^n$ are $C_{{\mathbb{H}}^n}^1$-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.