Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Abstract

We consider the Carnot-Carathéodory distance ${\delta }_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups ${\mathbb{H}}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of ${\delta }_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that ${\delta }_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\partial E$ and of the function $⟨N,T⟩/|N_h|$ and its tangent derivatives.

44 pages. Accepted version to appear in Adv. Calc. Var

References in corpus (4)

• The Hessian of the distance from a surface in the Heisenberg group
• Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n
• An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group
• The kinematic formula in the 3D-Heisenberg group

Cited by in corpus (4)

• On the essential self-adjointness of singular sub-Laplacians
• Heat content asymptotics for sub-Riemannian manifolds
• Heat content and horizontal mean curvature on the Heisenberg group
• Steiner and tube formulae in 3D contact sub-Riemannian geometry