A CR invariant sphere theorem

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.