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.

References in corpus (5)

• Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups
• Geometry of $1$-codimensional measures in Heisenberg groups
• On uniform measures in the Heisenberg group
• Area of intrinsic graphs and coarea formula in Carnot Groups
• Marstrand's density theorem in the Heisenberg group

Cited by in corpus (5)

• Lipschitz graphs and currents in Heisenberg groups
• Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups
• Geometry of $1$-codimensional measures in Heisenberg groups
• Rickman rugs and intrinsic bilipschitz graphs
• Unique continuation for locally uniformly distributed measures