A CR invariant sphere theorem

We prove that every closed, universally embeddable CR three-manifold with nonnegative Yamabe constant and positive total <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q prime"> <mml:semantics> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Q^\prime</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q prime"> <mml:semantics> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Q^\prime</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -curvature is CR equivalent to a compact quotient of the Heisenberg group with its flat CR structure.