Global weak solutions for the inverse mean curvature flow in the Heisenberg group

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group (ⁿ, d_ε), where d_ε is distance associated to either | · |_ε, ε>0, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for ε=0. For Ω ⊆ ⁿ an open set with smooth boundary Σ₀=∂ Ω satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces \Σ^εₛ ≥ 0 ⊆ Hⁿ which is proper. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in {HuiskenIlmanen}, following the approach in {Moser} due to Moser and based on the the link between IMCF and p-harmonic functions.