Horizontal inverse mean curvature flow in the Heisenberg group

Abstract

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 $Ω\subset \mathbb{H}^1$ is an open set with smooth boundary, and $Ω^{c} = \mathbb{H}^1\setminus Ω= \{ u \leq 0\}$ 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 $|\nabla_{0} u_p| = (p-1)\frac{|\nabla_{0} w_p|}{w_p} \in C^{0,β}(Ω)$ ($0< β<1, p>1$) 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{\rm 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.