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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 |
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Vol. 1 (2008), No. 2, 197–227
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Abstract |
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Suppose that is a strictly pseudoconvex CR manifold bounding a compact complex manifold of complex dimension . Under appropriate geometric conditions on , the manifold admits an approximate Kähler–Einstein metric which makes the interior of a complete Riemannian manifold. We identify certain residues of the scattering operator on as conformally covariant differential operators on and obtain the CR -curvature of from the scattering operator as well. In order to construct the Kähler–Einstein metric on , we construct a global approximate solution of the complex Monge–Ampère equation on , using Fefferman’s local construction for pseudoconvex domains in . 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
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Mathematical Subject Classification 2000
Primary: 58J50
Secondary: 32W20, 53C55
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Milestones
Received: 20 February 2008
Revised: 30 July 2008
Accepted: 28 September 2008
Published: 29 June 2009
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Authors |
| Peter D. Hislop | |
| Department of Mathematics University of Kentucky Lexington KY 40506-0027 United States |
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| http://www.ms.uky.edu/~hislop | |
| Peter A. Perry | |
| Department of Mathematics University of Kentucky Lexington KY 40506-0027 United States |
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| http://www.ms.uky.edu/~perry | |
| Siu-Hung Tang | |
| Department of Mathematics University of Kentucky Lexington KY 40506-0027 United States |
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