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

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 $n\ge 2$ in the proper class of horizontally umbilical hypersurfaces