CR embeddability of quotients of the Rossi sphere via spectral theory

TL;DR

This paper investigates the CR embeddability of quotient manifolds derived from the Rossi sphere and standard sphere under finite subgroup actions of SU(2), linking subgroup order to spectral properties of the Kohn Laplacian.

Contribution

It establishes the relationship between subgroup order and CR embeddability of quotient manifolds, and demonstrates how spectral data can determine subgroup size in the unperturbed case.

Findings

Quotient manifolds are CR manifolds under subgroup actions.

Subgroup order influences the spectral distribution of the Kohn Laplacian.

Spectral data can determine subgroup size in the unperturbed case.

Abstract

We look at the action of finite subgroups of $\operatorname{SU}(2)$ 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 $\operatorname{SU}(2)$ 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.