Vol. 1, No. 2, 2008

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 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
Milestones
Received: 20 February 2008
Revised: 30 July 2008
Accepted: 28 September 2008
Published: 29 June 2009
Authors
Peter D. Hislop
Department of Mathematics
University of Kentucky
Lexington KY 40506-0027
United States
http://www.ms.uky.edu/~hislop
Peter A. Perry
Department of Mathematics
University of Kentucky
Lexington KY 40506-0027
United States
http://www.ms.uky.edu/~perry
Siu-Hung Tang
Department of Mathematics
University of Kentucky
Lexington KY 40506-0027
United States