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

Abstract

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_{\epsilon }$ is distance associated to either $|\cdot {|}_{\epsilon }$, $\epsilon >0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\epsilon =0$. For $Ω\subseteq \He^n$ an open set with smooth boundary ${\Sigma }_0=\partial \Omega$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{{\Sigma }_s^{\epsilon }{\}}_{s\ge 0}\subseteq {\mathbb{H}}^n$ 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.

A new section on the relevant Bochner inequality has been added. A relevant error in the computation of the Ricci curvature has been fixed, together with the proofs of Theorem 1.2 and 1.4