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arXiv · 1808.01130

On uniform measures in the Heisenberg group

Vasilis Chousionis·Valentino Magnani·Jeremy T. Tyson
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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 $\mathbb H$.

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Vasilis Chousionis, Valentino Magnani, Jeremy T. Tyson. 2018-08-03. On uniform measures in the Heisenberg group. https://doi.org/10.1016/j.aim.2020.106980

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