A CR invariant sphere theorem

TL;DR

This paper establishes a classification of certain closed CR three-manifolds based on their Yamabe constant and total $Q'$-curvature, showing they are contact diffeomorphic or CR equivalent to standard models.

Contribution

It proves a CR invariant sphere theorem characterizing CR three-manifolds with specific curvature conditions as quotients of standard models.

Findings

CR three-manifolds with positive Yamabe constant are contact diffeomorphic to quotients of the sphere.

CR three-manifolds with zero Yamabe constant are CR equivalent to quotients of the Heisenberg group.

The results classify CR three-manifolds under curvature conditions.

Abstract

We prove that every closed, universally embeddable CR three-manifold with nonnegative Yamabe constant and positive total $Q^\prime$-curvature is contact diffeomorphic to a quotient of the standard contact three-sphere. We also prove that every closed, embeddable CR three-manifold with zero Yamabe constant and nonnegative total $Q^\prime$-curvature is CR equivalent to a compact quotient of the Heisenberg group with its flat CR structure.