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

Partial Differential Equations

CR-invariants and the scattering operator for complex manifolds with CR-boundary
[Des CR-invariants et la matrice de diffusion pour des variétés complexes avec CR-frontière]

Présenté par : Peter Lax

Peter D. Hislop ¹ ; Peter A. Perry ¹ ; Siu-Hung Tang ²

¹ Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA
² Departamento de Matemática, Universidade Federal da Pernambuco, Racife, Brazil

Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 651-654.

Résumé
Abstract

Soit M une variété CR qui est aussi la frontière d'une variété complexe et compacte X. Il y a une métrique g de type Kähler–Einstein sur X telle que $Int (X)$ est une variété riemannienne complète. Nous étudions la matrice de diffusion sur $(X, g)$ et nous montrons que les résidus à certains points sont des opérateurs différentiels CR-covariants. Nous montrons aussi qu'on peut recuperer la courbure CR Q en utilisant la matrice de diffusion. Nos résultats sont les analogues des résultats de Graham–Zworski pour le cas réel et asymptotiquement hyperbolique.

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.

Reçu le : 2006-01-31
Accepté le : 2006-02-28
Publié le : 2006-04-11

DOI : 10.1016/j.crma.2006.03.003

Affiliations des auteurs :

Peter D. Hislop ¹ ; Peter A. Perry ¹ ; Siu-Hung Tang ²

¹ Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA
² Departamento de Matemática, Universidade Federal da Pernambuco, Racife, Brazil

@article{CRMATH_2006__342_9_651_0,
     author = {Peter D. Hislop and Peter A. Perry and Siu-Hung Tang},
     title = {CR-invariants and the scattering operator for complex manifolds with {CR-boundary}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {651--654},
     publisher = {Elsevier},
     volume = {342},
     number = {9},
     year = {2006},
     doi = {10.1016/j.crma.2006.03.003},
     language = {en},
}

TY  - JOUR
AU  - Peter D. Hislop
AU  - Peter A. Perry
AU  - Siu-Hung Tang
TI  - CR-invariants and the scattering operator for complex manifolds with CR-boundary
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 651
EP  - 654
VL  - 342
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2006.03.003
LA  - en
ID  - CRMATH_2006__342_9_651_0
ER  -

%0 Journal Article
%A Peter D. Hislop
%A Peter A. Perry
%A Siu-Hung Tang
%T CR-invariants and the scattering operator for complex manifolds with CR-boundary
%J Comptes Rendus. Mathématique
%D 2006
%P 651-654
%V 342
%N 9
%I Elsevier
%R 10.1016/j.crma.2006.03.003
%G en
%F CRMATH_2006__342_9_651_0

Peter D. Hislop; Peter A. Perry; Siu-Hung Tang. CR-invariants and the scattering operator for complex manifolds with CR-boundary. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 651-654. doi : 10.1016/j.crma.2006.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.003/

Bibliographie
Cité par

[1] T.P. Branson Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc., Volume 347 (1995), pp. 3671-3742

[2] C. Epstein; R.M. Melrose; G.A. Mendoza Resolvent of the Laplacian on pseudoconvex domains, Acta Math., Volume 167 (1991), pp. 1-106

[3] C.L. Fefferman Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2), Volume 103 (1976) no. 2, pp. 395-416 Correction Ann. of Math. (2), 104, 2, 1976, pp. 393-394

[4] C. Fefferman; K. Hirachi Ambient metric construction of Q-curvature in conformal and CR-geometries, Math. Res. Lett., Volume 10 (2003), pp. 819-832

[5] C.R. Graham, Private communication

[6] C.R. Graham; A.R. Gover CR-invariant powers of the sub-Laplacian, J. Reine Angew. Math., Volume 583 (2005), pp. 1-27

[7] C.R. Graham; C. Fefferman Conformal invariants, Astérisque (1985) no. Numéro Hors Série, pp. 95-116

[8] C.R. Graham; R. Jenne; L.J. Mason; G.A.J. Sparling Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2), Volume 46 (1992) no. 3, pp. 557-565

[9] C.R. Graham; J.M. Lee Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J., Volume 57 (1988) no. 3, pp. 697-720

[10] C.R. Graham; M. Zworski Scattering matrix in conformal geometry, Invent. Math., Volume 152 (2003) no. 1, pp. 89-118

[11] F.R. Harvey; H.B. Lawson On boundaries of complex analytic varieties. I, Ann. of Math. (2), Volume 102 (1975) no. 2, pp. 223-290

[12] P.D. Hislop, P.A. Perry, S.-H. Tang, The scattering operator for complex manifolds with boundary, in preparation

[13] D. Jerison; J.M. Lee A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR-manifolds, Contemp. Math., Volume 27 (1984), pp. 57-63

[14] J. Lee; R. Melrose Boundary behavior of the complex Monge–Ampère equation, Acta Math., Volume 148 (1982), pp. 160-192

[15] R. Melrose Scattering theory for strictly pseudoconvex domains, Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 161-168

[16] N. Seshadri Volume renormalization for complete Einstein–Kähler metrics, April 2004 (Preprint, arXiv:) | arXiv

[17] N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, in: Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975

[18] A. Vasy; J. Wunsch Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature, 2004 (Preprint, arXiv:) | arXiv

[19] S.M. Webster Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., Volume 13 (1978) no. 1, pp. 25-41

Cité par Sources :

Commentaires - Politique