### Abstract

We study uniform measures in the first Heisenberg group 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 (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 H.

Original language | English (US) |
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Article number | 106980 |

Journal | Advances in Mathematics |

Volume | 363 |

DOIs | |

State | Published - Mar 25 2020 |

### Keywords

- Heisenberg group
- Uniform measures

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*363*, [106980]. https://doi.org/10.1016/j.aim.2020.106980