## 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.

Original language | English |
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Pages (from-to) | 197-227 |

Number of pages | 31 |

Journal | Analysis and PDE |

Volume | 1 |

Issue number | 2 |

DOIs | |

State | Published - 2008 |

### Bibliographical note

Funding Information:MSC2000: primary 58J50; secondary 32W20, 53C55. Keywords: CR geometry, Q curvature, geometric scattering theory. Hislop supported in part by NSF grant DMS-0503784. Perry supported in part by NSF grants DMS-0408419 and DMS-0710477. Tang supported in part by NSF grants DMS-0408419 and DMS-0503784.

Publisher Copyright:

© 2008, Analysis and PDE. All Rights Reserved.

## Keywords

- Cr geometry
- Geometric scattering theory
- Q curvature

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics