Horizontal inverse mean curvature flow in the Heisenberg group

Huisken and Ilmanen in [37] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. The present paper investigates and constructs a sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of hypersurfaces in the first Heisenberg group ${\mathbb{H}}^1$, and provides a positive answer to an open problem: the Heintze-Karcher inequality in ${\mathbb{H}}^1$. Furthermore, we introduce a $\mathbb{H}$-perimeter preserving flow (1.8) in the first Heisenberg group ${\mathbb{H}}^1$, which is derived by applying the Heisenberg dilation to HIMCF. This rescaled flow is subsequently applied to establish a Minkowski-type formula in ${\mathbb{H}}^1$.