Geometry of $1$-codimensional measures in Heisenberg groups

TL;DR

This paper investigates the geometric properties of measures with density in Heisenberg groups, establishing conditions for flat tangents and classifying uniform measures in the simplest case.

Contribution

It provides new insights into the tangential structure of measures in Heisenberg groups and classifies uniform measures in $ ext{Heisenberg}^1$, advancing geometric measure theory in sub-Riemannian contexts.

Findings

Measures with $(2n+1)$-density have only flat tangent measures.

Classification of uniform measures in $ ext{Heisenberg}^1$.

Enhanced understanding of measure geometry in sub-Riemannian spaces.

Abstract

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups $\mathbb{H}^n$. Among other results we prove that measures with $(2n+1)$-density have only flat tangents and conclude the classification of uniform measures in $\mathbb{H}^1$.