On uniform measures in the Heisenberg group

Abstract

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 $\mathbb{H}$.

31 pages

Cited by in corpus (4)

• Lipschitz graphs and currents in Heisenberg groups
• Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups
• Geometry of $1$-codimensional measures in Heisenberg groups
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