CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of $\mathrm{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in ${\mathbb{C}}^2$ and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of $\mathrm{SU}(2)$ to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.