#### Vol. 1, No. 2, 2008

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

### Peter D. Hislop, Peter A. Perry and Siu-Hung Tang

Vol. 1 (2008), No. 2, 197–227
##### 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.

##### Keywords
CR geometry, Q curvature, geometric scattering theory
##### Mathematical Subject Classification 2000
Primary: 58J50
Secondary: 32W20, 53C55