Paper

On uniform measures in the Heisenberg group

Tyson, Jeremy T, Chousionis, Vasilis, Magnani, Valentino

2018

Overview

We initiate a classification of uniform measures in the first Heisenberg group \\(\\mathbb H\\) equipped with the Korányi 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\\).

• Publisher: Cornell University Library, arXiv.org
• Subject: Asymptotic methods / Curvature / Manifolds (mathematics) / Power series / Thermal expansion