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

Adriano Pisante, Eugenio Vecchi

TL;DR

The paper develops a rigorous weak formulation for inverse mean curvature flow in the Heisenberg group, using Riemannian approximations $d_ε$ and a $p$-Laplace regularization that connects IMCF to $p$-harmonic potentials. It establishes uniform gradient bounds and two-sided asymptotics for $p$-capacitary potentials, proves a Bochner-type inequality for the energy density, and constructs barrier-based boundary estimates that extend to the exterior domain, culminating in the existence of global weak IMCF level-sets that grow at an explicit rate dictated by Korányi geometry. The core contributions include a Cheng–Yau-type interior gradient bound, boundary gradient control, and a limiting framework that yields proper, Lipschitz weak solutions $u^ε$ whose level sets $\{u^ε=s\}$ behave like expanding Korányi spheres as time progresses. This work lays foundational steps toward a full understanding of IMCF in sub-Riemannian settings, with implications for geometric analysis on Carnot groups and asymptotic geometric behavior at infinity.

Abstract

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group $(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for $\varepsilon=0$. For $Ω\subseteq \He^n$ an open set with smooth boundary $Σ_0=\partial Ω$ satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces $\{Σ^\varepsilon_s\}_{s \geq 0} \subseteq \mathbb{H}^n$ 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.

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

TL;DR

The paper develops a rigorous weak formulation for inverse mean curvature flow in the Heisenberg group, using Riemannian approximations and a -Laplace regularization that connects IMCF to -harmonic potentials. It establishes uniform gradient bounds and two-sided asymptotics for -capacitary potentials, proves a Bochner-type inequality for the energy density, and constructs barrier-based boundary estimates that extend to the exterior domain, culminating in the existence of global weak IMCF level-sets that grow at an explicit rate dictated by Korányi geometry. The core contributions include a Cheng–Yau-type interior gradient bound, boundary gradient control, and a limiting framework that yields proper, Lipschitz weak solutions whose level sets behave like expanding Korányi spheres as time progresses. This work lays foundational steps toward a full understanding of IMCF in sub-Riemannian settings, with implications for geometric analysis on Carnot groups and asymptotic geometric behavior at infinity.

Abstract

We consider the inverse mean curvature flow (IMCF) in the Heisenberg group , where is distance associated to either , , the natural family of left-invariant Riemannian metrics, or with their sub-Riemannian counterparts for . For an open set with smooth boundary satisfying a uniform exterior gauge-ball condition and bounded complement we show existence of a global weak IMCF of generalized hypersurfaces 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 -harmonic functions.
Paper Structure (8 sections, 32 theorems, 261 equations)

This paper contains 8 sections, 32 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Basic notations and preliminary results
  3. Construction of barriers
  4. Two uniform estimates in multiscale geometries.
  5. Two-sided control at infinity for $p$-capacitary potentials.
  6. Pointwise Bochner's inequality for the energy density.
  7. Uniform gradient estimates for $p$-capacitary potentials.
  8. Proof of the main results

Key Result

Theorem 1.2

Let $\Omega \subset \mathbb{H}^n$ be an open set with $C^2$-smooth boundary and bounded complement satisfying the exterior uniform gauge-ball condition ($\mathbf{HP_\Omega}$) with parameter $R_0$. For any $1<p\leq 2$ and $\varepsilon \in [0,1]$ let $v^\varepsilon_p \in \dot{W}^{1,p}_{1,\varepsilon}( Moreover, for $1<p\leq 2$ and $\varepsilon=0$ the corresponding finite energy solution $v^0_p \in

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 53 more