## V. Chousionis - V. Magnani - J. T. Tyson

# On uniform measures in the Heisenberg group

created by magnani on 08 Nov 2018

modified on 18 Sep 2020

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BibTeX]

*Published Paper*

**Inserted:** 8 nov 2018

**Last Updated:** 18 sep 2020

**Journal:** Adv. Math.

**Volume:** 363

**Number:** 106980

**Pages:** 42

**Year:** 2020

**Links:**
postprint

**Abstract:**

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 $\mathbb H$.

**Keywords:**
Heisenberg group, uniform measure

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