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 $(\He^n, d_\varepsilon)$, where dε is distance associated to either |⋅|ε, ε>0, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterpart for ε=0. For $Ω\subseteq \He^n$ 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 {Σεs}s≥0⊆Hn 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, following the approach due to Moser and based on the link between IMCF and p-harmonic functions. The talk is based on a joint work with Eugenio Vecchi (University of Bologna).