Tubular neighborhoods in the sub-Riemannian Heisenberg groups
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AuthorRitore Cortés, Manuel María
Walter De Gruyter Gmbh
Heisenberg groupCarnot–Carathéodory distancetubular neighborhoodsDistance functionSingular setSteiner’s formula
Ritoré, M. (2021). Tubular neighborhoods in the sub-Riemannian Heisenberg groups. Advances in Calculus of Variations, 14(1), 1-36. [https://doi.org/10.1515/acv-2017-0011]
SponsorshipEuropean Union (EU) Spanish Government MTM2007-61919 MTM2013-48371-C2-1-P; Junta de Andalucía FQM-325
In the present paper we consider the Carnot–Carathéodory distance δE to a closed set E in the sub-RiemannianHeisenberg groupsℍn, n ⩾ 1. Theℍ-regularity of δE is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean Ck submanifold, k ⩾ 2, we prove that δE is Ck out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of E is of class C2 are obtained, out of the singular set, in terms of the horizontal principal curvatures of ∂E and of the function ⟨N, T⟩/|Nh| and its tangent derivatives.