Curvature distributions and the Steiner formula for characteristic surfaces in the Heisenberg group
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≥ 4 of the volume of the ε-tubular neighborhood 𝒯S(ε) of a surface S is no longer polynomial, in general, taking the form vol(𝒯S(ε)) = ∑j=13αjεj+∑n=4N (αn+βnlogε)εn+o(εN), as ε→0+ . 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.
Related
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- K3 surfaces with finite automorphism group in positive characteristic
- Bilinear maximal functions associated with some degenerate surfaces on the Heisenberg group
- Foliations on abelian surfaces in positive characteristic
- Finite Symplectic Automorphism Groups of K3 Surfaces in Characteristic Seven
BibTeX
@misc{barilari2026curvature,
title = {{Curvature distributions and the Steiner formula for characteristic surfaces in the Heisenberg group}},
author = {Davide Barilari and Eugenio Bellini and Tommaso Rossi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26706},
eprint = {2610.26706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26706},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature distributions and the Steiner formula for characteristic surfaces in the Heisenberg group
AU - Davide Barilari
AU - Eugenio Bellini
AU - Tommaso Rossi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26706
UR - https://arxiv.org/abs/2610.26706
AB - 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≥ 4 of the volume of the ε-tubular neighborhood 𝒯_S(ε) of a surface S is no longer polynomial, in general, taking the form vol(𝒯_S(ε)) = ∑_j=1^3α_jε^j+∑_n=4^N (α_n+β_nlogε)ε^n+o(ε^N), as ε→0^+ . 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.
ER -