On a conjecture of J. M. Lee

Abstract

We deal with the Lee conjecture (compact strictly pseudoconvex CR manifolds whose CR structure has a vanishing first Chern class admit a global pseudo-Einstein structure). We solve in affirmative the Lee conjecture for compact strictly pseudoconvex CR manifolds with a regular (in the sense of R. Palais, [Pal]) contact vector. The regularity assumption leads (via the Boothby-Wang theorem ([BoO-Wan]) and B. O’Neill’s f- damental equations of a submersion ([Nei])) to zero pseudohermitian torsion (and we may apply a result of [Lee2]). Moreover we construct a family ${\mathbf H}_{n}(s)$, $0<s<1$, of compact strictly pseudoconvex CR manifolds, so that each ${\mathbf H}_{n}(s)$ satisfies the Lee conjecture. We endow ${\mathbf H}_{n}(s)$ with the contact form (4); our construction is reminiscent of W. C. Boothby’s Hermitian metric (cf. [Boo]) on a complex Hopf manifold.