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 $({\text{He}}^n,d_{\epsilon })$, where $d_{\epsilon }$ is distance associated to either $\mathrm{\mid }\cdot {\mathrm{\mid }}_{\epsilon }$, $\epsilon >0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\epsilon =0$. For $\mathrm{\Omega }\subseteq {\text{He}}^n$ an open set with smooth boundary ${\mathrm{\Sigma }}_0=\mathrm{\partial }\mathrm{\Omega }$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{{\mathrm{\Sigma }}_s^{\epsilon }{\}}_{s\ge 0}\subseteq {\mathbb{H}}^n$ which are level sets of a proper globally Lipschitz function with logarithmic growth at infinity. Here, both in the Riemannian and in the sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to Moser and based on the the link between IMCF and $p$-harmonic functions.