Steiner’s formula in the Heisenberg group

Balogh, Zoltan; Ferrari, Fausto; Franchi, Bruno; Vecchi, Eugenio; Wildrick, Kevin Michael (2015). Steiner’s formula in the Heisenberg group. Nonlinear analysis: theory, methods & applications, 126, pp. 201-217. Elsevier 10.1016/j.na.2015.05.006

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Steiner’s tube formula states that the volume of an ϵ-neighborhood of a smooth regular domain in Rn is a polynomial of degree n in the variable ϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.

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