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_), where d_ is distance associated to either | |_, >0, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for =0. For ⁿ an open set with smooth boundary ₀= satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces \^ₛ\ₒ 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.