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

Peter D. Hislop; Peter A. Perry; Siu Hung Tang

doi:10.1016/j.crma.2006.03.003

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

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Suppose that M is a CR manifold bounding a compact complex manifold X. 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 as CR-covariant differential operators and obtain the CR Q-curvature of M from the scattering operator as well. Our results are an analogue in CR-geometry of Graham and Zworski's result that certain residues of the scattering operator on a conformally compact manifold with a Poincaré-Einstein metric are natural, conformally covariant differential operators, and the Q-curvature of the conformal infinity can be recovered from the scattering operator. To cite this article: P.D. Hislop et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Original language	English
Pages (from-to)	651-654
Number of pages	4
Journal	Comptes Rendus Mathematique
Volume	342
Issue number	9
DOIs	https://doi.org/10.1016/j.crma.2006.03.003
State	Published - May 1 2006

Bibliographical note

Funding Information:
We are grateful to Charles Epstein, Robin Graham, John Lee, and Antônio Sá Barretto for helpful discussions and correspondence. P.A.P. thanks the National Center for Theoretical Science at the National Tsing Hua University, Hsinchu, Taiwan, for hospitality during part of the time that this work was done. The authors are partially supported by NSF grants DMS 05-03784 (Hislop) and DMS 04-08419 (Perry).

Funding

We are grateful to Charles Epstein, Robin Graham, John Lee, and Ant\u00F4nio S\u00E1 Barretto for helpful discussions and correspondence. P.A.P. thanks the National Center for Theoretical Science at the National Tsing Hua University, Hsinchu, Taiwan, for hospitality during part of the time that this work was done. The authors are partially supported by NSF grants DMS 05-03784 (Hislop) and DMS 04-08419 (Perry).

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS 04-08419, DMS 05-03784

ASJC Scopus subject areas

General Mathematics

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Cite this

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title = "CR-invariants and the scattering operator for complex manifolds with CR-boundary",

abstract = "Suppose that M is a CR manifold bounding a compact complex manifold X. The manifold X admits an approximate K{\"a}hler-Einstein metric g which makes the interior of X a complete Riemannian manifold. We identify certain residues of the scattering operator as CR-covariant differential operators and obtain the CR Q-curvature of M from the scattering operator as well. Our results are an analogue in CR-geometry of Graham and Zworski's result that certain residues of the scattering operator on a conformally compact manifold with a Poincar{\'e}-Einstein metric are natural, conformally covariant differential operators, and the Q-curvature of the conformal infinity can be recovered from the scattering operator. To cite this article: P.D. Hislop et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).",

author = "Hislop, \{Peter D.\} and Perry, \{Peter A.\} and Tang, \{Siu Hung\}",

note = "Funding Information: We are grateful to Charles Epstein, Robin Graham, John Lee, and Ant{\^o}nio S{\'a} Barretto for helpful discussions and correspondence. P.A.P. thanks the National Center for Theoretical Science at the National Tsing Hua University, Hsinchu, Taiwan, for hospitality during part of the time that this work was done. The authors are partially supported by NSF grants DMS 05-03784 (Hislop) and DMS 04-08419 (Perry).",

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AU - Hislop, Peter D.

AU - Perry, Peter A.

AU - Tang, Siu Hung

N1 - Funding Information: We are grateful to Charles Epstein, Robin Graham, John Lee, and Antônio Sá Barretto for helpful discussions and correspondence. P.A.P. thanks the National Center for Theoretical Science at the National Tsing Hua University, Hsinchu, Taiwan, for hospitality during part of the time that this work was done. The authors are partially supported by NSF grants DMS 05-03784 (Hislop) and DMS 04-08419 (Perry).

PY - 2006/5/1

Y1 - 2006/5/1

N2 - Suppose that M is a CR manifold bounding a compact complex manifold X. 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 as CR-covariant differential operators and obtain the CR Q-curvature of M from the scattering operator as well. Our results are an analogue in CR-geometry of Graham and Zworski's result that certain residues of the scattering operator on a conformally compact manifold with a Poincaré-Einstein metric are natural, conformally covariant differential operators, and the Q-curvature of the conformal infinity can be recovered from the scattering operator. To cite this article: P.D. Hislop et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

AB - Suppose that M is a CR manifold bounding a compact complex manifold X. 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 as CR-covariant differential operators and obtain the CR Q-curvature of M from the scattering operator as well. Our results are an analogue in CR-geometry of Graham and Zworski's result that certain residues of the scattering operator on a conformally compact manifold with a Poincaré-Einstein metric are natural, conformally covariant differential operators, and the Q-curvature of the conformal infinity can be recovered from the scattering operator. To cite this article: P.D. Hislop et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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JF - Comptes Rendus Mathematique

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ER -

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

Abstract

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