A characterization of gauge balls in $\mathbb{H}^n$ by horizontal curvature

TL;DR

This paper characterizes gauge balls in the Heisenberg group by their horizontal mean curvature, providing uniqueness results for certain hypersurfaces and advancing understanding of geometric structures in sub-Riemannian geometry.

Contribution

It offers a novel characterization of gauge balls in $H^n$ through horizontal curvature and establishes uniqueness under specific conditions.

Findings

Uniqueness of gauge balls in $H^1$ under singular set assumptions

Characterization of gauge balls in $H^n$ for $n extgreater 1$ among horizontally umbilical hypersurfaces

Advancement in understanding the geometry of hypersurfaces in the Heisenberg group

Abstract

In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces