Horizontal inverse mean curvature flow in the Heisenberg group

Huisken and Ilmanen [J. Differential Geom., 2001] 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 surfaces in the first Heisenberg group ${\mathbb{H}}^1$. The level set formulation of the IMCF in ${\mathbb{H}}^1$ is given by (0.1), where $\Omega \subset {\mathbb{H}}^1$ is an open set with smooth boundary, and ${\Omega }^c={\mathbb{H}}^1\setminus \Omega =\{u\le 0\}$ is bounded. Let $w_p=\exp \left(\frac{u_p}{1-p}\right)$ and $w_p$ satisfies (0.2). Following the argument by Moser, the key ingredient in proving the existence of weak solutions to (0.1) is to establish a uniform interior estimate for $|{\nabla }_0{u}_p|$. However, due to the lack of boundary continuity of $|{\nabla }_0{u}_p|=(p-1)\frac{|{\nabla }_0{w}_p|}{w_p}\in C^{0,\beta }(\Omega )$ ($0<\beta <1,p>1$) by Zhong and Mukherjee [Anal. PDE, 2021], the standard method in [R. Moser, J. Eur. Math. Soc., 2007] cannot be applied to obtain a uniform interior estimate for $|{\nabla }_0{u}_p|$. Fortunately, the present paper discovers two refined inequalities: Harnack inequality and Lipschitz estimate for $w_p$, which allow one to obtain interior estimates for $|{\nabla }_0{u}_p|$ independent of $p$. By further combining them with Arzel$\grave{\mathrm{a}}$-Ascoli theorem, the weak solution of (0.1) can then be generated as the limit of $u_p$ as $p\to 1$, where $w_p=\exp \left(\frac{u_p}{1-p}\right)$ and $w_p$ is of solutions to (0.2). As an important application of the IMCF in ${\mathbb{H}}^1$, a positive answer to an open problem posed in [F. Montefalcon, Ann. Mat. Pura Appl. (4), 2014]:Heintze-Karcher inequality in ${\mathbb{H}}^1$ is provided.