Uniformization of spherical CR manifolds

TL;DR

This paper proves that under certain conditions, spherical CR manifolds can be globally uniformized via their developing maps, extending the understanding of their geometric structure.

Contribution

The paper establishes injectivity of the CR developing map for spherical CR manifolds with positive Yamabe invariant, confirming their uniformizability in higher dimensions.

Findings

Injectivity of the CR developing map for n ≥ 3.

Injectivity under the condition s(M) < 1 when n=2.

M is shown to be uniformizable under these conditions.

Abstract

Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $% \Phi $ denote a $CR$ developing map {equation*} \Phi :\widetilde{M}\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$% . Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\Phi $ is injective for $n\geq 3$. In the case $n=2$, we also show that $\Phi $ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable.