Elsevier

Advances in Mathematics

Volume 363, 25 March 2020, 106980
Advances in Mathematics

On uniform measures in the Heisenberg group

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Abstract

We study uniform measures in the first Heisenberg group H equipped with the Korányi metric dH. 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 x3-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,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.

Keywords

Uniform measures
Heisenberg group

Cited by (0)

V.C. was funded by the Academy of Finland Grant SA 267047 and Simons Collaboration Grant 521845. J.T.T. supported by NSF grants DMS-1201875 and DMS-1600650. V.M. supported by the University of Pisa, Project PRA 2018 49.