Preprint

Inserted: 19 oct 2026
Last Updated: 19 oct 2026

Pages: 67
Year: 2026

Abstract:

We establish a new Steiner-type formula for smooth surfaces in the sub-Riemannian Heisenberg group. The main novelty is that we admit characteristic points, whose presence produces nontrivial singularities both of the distance from the surface and of its tubular neighborhood.

Such singularities are reflected in the Steiner formula: the expansion up to a given order $N\geq 4$ of the volume of the $\varepsilon$-tubular neighborhood $\mathcal T_S(\varepsilon)$ of a surface $S$ is no longer polynomial, in general, taking the form \[ vol(T_S(\varepsilon)) = \sum_{j=1}^3\alpha_j\epsilon^j+\sum_{n=4}^N \left(\alpha_n+\beta_n\log\varepsilon\right)\varepsilon^n+o(\epsilon^N),\qquad\text{as }\varepsilon\to0^+ . \] More precisely, under a geometric stability condition on characteristic points, we prove that the first three coefficients are integrals over $S$ of locally integrable curvature densities, coinciding with those known in the non-characteristic case.

A new phenomenon appears at order $4$, equal to the homogeneous dimension of the Heisenberg group, where logarithmic terms may occur. We show that the higher-order coefficients are naturally described by a distributional analogue of Federer’s curvature measures, whose logarithmic components are supported on the characteristic set. We compute the fourth-order coefficients explicitly.

The proof is based on different ingredients of independent interest: (i) an explicit formula for the cut time of homogeneous surfaces, (ii) a fine analysis of the normal exponential map through blow-ups, (iii) the study of the stability of the surface cut locus under perturbations.

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