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 $\mathbb H$.

中文翻译：

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海森堡群中的统一测度

我们在配备 Kor\'anyi 度量 $d_H$ 的第一个海森堡群 $\mathbb H$ 中启动了统一测度的分类，它代表了配备齐次距离的非交换分层群的第一个例子。我们证明$1$-uniform 测度与限制在仿射水平线上的球形$1$-Hausdorff 测度成正比，而$2$-uniform 测度与限制在仿射垂直线上的球形$2$-Hausdorff 测度成正比。$3$-均匀测度是否与球面$3$-Hausdorff测度对仿射垂直平面的限制成正比仍然是一个悬而未决的问题。如果测量的支撑是垂直规则的表面，我们建立这个结论。在此过程中，我们推导出了 $({\mathbb H}, d_H)$ 与光滑的子流形相交。我们的幂级数展开中的系数涉及与 $\mathbb H$ 中的平滑曲线和曲面相关的内在曲率概念。