Improved Lipschitz approximation of $H$-perimeter minimizing boundaries

TL;DR

This paper advances the approximation of $H$-perimeter minimizing boundaries in the Heisenberg group using intrinsic Lipschitz functions, improving previous results and adapting classical methods to this non-Euclidean setting.

Contribution

It introduces two new approximation results for $H$-perimeter minimizing boundaries in $ H^n$, extending classical Lipschitz approximation techniques to the Heisenberg group.

Findings

Improved Lipschitz approximation results for $H$-perimeter minimizing boundaries.

Reformulation of classical Lipschitz approximation in the Heisenberg group setting.

Adaptation of maximal function approximation methods to $ H^n$.

Abstract

We prove two new approximation results of $H$-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group $\mathbb{H}^n$ with $n\ge2$. The first one is an improvement of a result of Monti and is the natural reformulation in $\mathbb{H}^n$ of the classical Lipschitz approximation in $\mathbb{R}^n$. The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro.