# On uniform measures in the Heisenberg group

created by magnani on 08 Nov 2018
modified on 18 Sep 2020

[BibTeX]

Published Paper

Inserted: 8 nov 2018
Last Updated: 18 sep 2020

Volume: 363
Number: 106980
Pages: 42
Year: 2020
We study uniform measures in the first Heisenberg group $\mathbb H$ equipped with the Korányi metric $d_H$. 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. We also show that each $3$-uniform measure which is supported on a vertically ruled surface is proportional to the restriction of spherical $3$-Hausdorff measure to an affine vertical plane, and that no quadratic $x_3$-graph can be the support of a $3$-uniform measure. According to a result of Merlo, every $3$-uniform measure is supported on a quadratic variety; in conjunction with our results, this shows that all $3$-uniform measures are proportional to spherical $3$-Hausdorff measure restricted to an affine vertical plane. We establish our conclusions by deriving 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 $\mathbb H$.