On uniform measures in the Heisenberg group

TL;DR

This paper classifies uniform measures in the Heisenberg group with the Korányi metric, revealing their geometric structure and establishing results for measures supported on lines and surfaces, with open questions for higher-dimensional cases.

Contribution

It provides the first classification of uniform measures in the noncommutative stratified Heisenberg group, linking measures to geometric structures like lines and surfaces.

Findings

1-uniform measures are proportional to spherical 1-Hausdorff measure on affine horizontal lines

2-uniform measures are proportional to spherical 2-Hausdorff measure on affine vertical lines

Established asymptotic formulas for measures of small extrinsic balls intersected with smooth submanifolds

Abstract

We initiate a classification of uniform measures in the first Heisenberg group $\mathbb H$ equipped with the Kor\'anyi metric $d_H$, that represents the first example of a noncommutative stratified group equipped with a homogeneous distance. We prove that $1$-uniform measures are proportional to the spherical $1$-Hausdorff measure restricted to an affine horizontal line, while $2$-uniform measures are proportional to spherical $2$-Hausdorff measure restricted to an affine vertical line. It remains an open question whether $3$-uniform measures are proportional to the restriction of spherical $3$-Hausdorff measure to an affine vertical plane. We establish this conclusion in case the support of the measure is a vertically ruled surface. Along the way, we derive asymptotic formulas for the measures of small extrinsic balls in $({\mathbb H},d_H)$ intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in $\mathbb H$.