The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds

Abstract

We study the eigenvalues and eigenfunctions of the Folland-Stein operator $\mathscr{L}_\alpha$ on some examples of 3-dimensional Heisenberg Bieberbach manifolds, that is, compact quotients $\Gamma\backslash\mathbb{H}$ of the Heisenberg group $\mathbb{H}$ by a discrete torsion-free subgroup $\Gamma$ of $\mathbb{H}\rtimes U(1)$.