Pseudo-Einstein and Q-flat Metrics with Eigenvalue Estimates on CR-hypersurfaces

ABSTRACT. In this paper, we will use the Kohn’s ¯ ∂b-theory on CRhypersurfaces to derive some new results in CR-geometry. Main Theorem. Let M2n−1 be the smooth boundary of a bounded strongly pseudo-convex domain Ω in a complete Stein manifold V 2n.Then: (1) For n ≥ 3, M2n−1 admits a pseudo-Einstein metric. (2) For n ≥ 2, M2n−1 admits a Fefferman metric of zero CR Q-curvature. (3) In addition, for a compact strictly pseudoconvex CR emendable 3manifold M3, its CR Paneitz operator P is a closed operator. There are examples of non-emendable strongly pseudoconvex CRmanifolds M 3, for which the corresponding ¯ ∂b-operator and Paneitz operators are not closed operators. In this paper, we study several questions, including the existence of Q-flat metrics, pseudo-Einstein metrics, and the closedness of the CR Paneitz operators. First, we will use an approach proposed by Fefferman and his school to prov