CR-invariants and the scattering operator for complex manifolds with boundary

Abstract

Suppose that $M$ is a strictly pseudoconvex CR manifold bounding a compact complex manifold $X$ of complex dimension $m$. Under appropriate geometric conditions on $M$, the manifold $X$ admits an approximate Kähler–Einstein metric $g$ which makes the interior of $X$ a complete Riemannian manifold. We identify certain residues of the scattering operator on $X$ as conformally covariant differential operators on $M$ and obtain the CR $Q$-curvature of $M$ from the scattering operator as well. In order to construct the Kähler–Einstein metric on $X$, we construct a global approximate solution of the complex Monge–Ampère equation on $X$, using Fefferman’s local construction for pseudoconvex domains in ${ℂ}^m$. Our results for the scattering operator on a CR-manifold are the analogue in CR-geometry of Graham and Zworski’s result on the scattering operator on a real conformal manifold.