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 CR-hypersurfaces 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 V2n. 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 3-manifold M3, its CR Paneitz operator P is a closed operator. There are examples of non-emendable strongly pseudoconvex CR-manifolds M3, for which the corresponding ∂b-operator and Paneitz operators are not closed operators.[[fileno]]2010205010016[[department]]數學