Panoramas et synthèses - Parutions - 26 (2008) < Parutions

Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques
Zindine Djadli, Colin Guillarmou, Marc Herzlich
Panoramas et synthèses 26 (2008), vi+171 pages
Acheter l'ouvrage

Résumé :
En 1985, Fefferman et Graham ont introduit un programme ambitieux (dit de la métrique ambiante) d'étude des invariants locaux de la géométrie conforme. Celui-ci s'est considérablement développé ces dernières années, menant à la définition de nombreux objets nouveaux: opérateurs de Graham-Jenne-Mason-Sparling (GJMS) généralisant ceux de Yamabe et de Paneitz, -courbure de Branson... et à des applications parfois spectaculaires et inattendues: classification des variétés conformément plates de dimension  à caractéristique d'Euler positive, théorème de pincement conforme^Mde la sphère, etc. Absentes de la stratégie originelle, la géométrie et l'analyse sur les variétés asymptotiquement hyperboliques d'Einstein (ou Poincaré-Einstein) se sont révélées un élément essentiel du programme. L'objectif de ce livre est de présenter un panorama des développements récents et une synthèse des principaux résultats, accessible à des lecteurs ayant une connaissance de base de la géométrie riemannienne.

Abstract:
Geometric operators, conformal invariants and asymptotically hyperbolic manifolds
In 1985, Fefferman and Graham initiated an ambitious program of study of conformal geometry (known as the ambient metricmethod). This has known tremendous developments in the last few years, leading to the definition of a number of new invariants: Graham-Jenne-Mason-Sparling (GJMS) operators generalizing the Yamabe and Paneitz operators, Branson -curvatures... and to remarkable applications to conformally flat manifolds of dimension  and nonnegative Euler characteristic, or to conformally invariant pinching theorems. An essential role is played in the theory by asymptotically hyperbolic Einstein metrics (or Poincaré-Einstein metrics) associated to a conformal class. The book is devoted to a presentation of the theory together with a description of the latest developments. It should be accessible to all readers having a basic knowledge of Riemannian geometry.

Bibliographie:

1

Adams, David R.

A sharp inequality of J. Moser for higher order derivatives

Ann. of Math. 128 (1988) 385–398

Math Reviews MR960950

2

Albin, Pierre

A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem

Adv. Math. 213 (2007) 1–52

Math Reviews MR2331237

3

Albin, Pierre

Renormalizing curvature integrals on Poincaré-Einstein manifolds

Adv. Math. 221 (2009) 140–169

4

Alexakis, Spyros

On the decomposition of global conformal invariants, II

Adv. Math. 206 (2006) 466–502

Math Reviews MR2263712

5

Alexakis, Spyros

On global conformal invariants, I

6

Alexakis, Spyros

The decomposition of global conformal invariants: on a conjecture of Deser and Schwimmer

7

Anderson, Michael T.

curvature and volume renormalization of AHE metrics on 4-manifolds

Math. Res. Lett. 8 (2001) 171–188

Math Reviews MR1825268

8

Anderson, Michael T.

Geometric aspects of the AdS/CFT correspondence

in AdS/CFT correspondence: Einstein metrics and their conformal boundaries

IRMA Lect. Math. Theor. Phys. 8 (2005) 1–31

Math Reviews MR2160865

9

Anderson, Michael T.

Einstein metrics with prescribed conformal infinity on 4-manifolds

Geom. Funct. Anal. 18 (2008) 305–366

Math Reviews MR2421542

10

Anderson, Michael T. and Herzlich, Marc

Unique continuation results for Ricci curvature and applications

J. Geom. Phys. 58 (2008) 179–207

Math Reviews MR2384310

11

Andersson, Lars and Dahl, Mattias

Scalar curvature rigidity for asymptotically locally hyperbolic manifolds

Ann. Global Anal. Geom. 16 (1998) 1–27

Math Reviews MR1616570

12

Aubry, E. and Guillarmou, Colin

Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity

13

Bahri, A. and Coron, J.-M.

The scalar-curvature problem on the standard three-dimensional sphere

J. Funct. Anal. 95 (1991) 106–172

Math Reviews MR1087949

14

Bailey, Toby N. and Eastwood, Michael G. and Gover, A. Rod

Thomas's structure bundle for conformal, projective and related structures

Rocky Mountain J. Math. 24 (1994) 1191–1217

Math Reviews MR1322223

15

Bailey, Toby N. and Eastwood, Michael G. and Graham, C. Robin

Invariant theory for conformal and CR geometry

Ann. of Math. 139 (1994) 491–552

Math Reviews MR1283869

16

Bakry, Dominique and Émery, Michel

Inégalités de Sobolev pour un semi-groupe symétrique

C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 411–413

Math Reviews MR808640

17

Berline, Nicole and Getzler, Ezra and Vergne, Michèle

Heat kernels and Dirac operators

Springer, 1992

Math Reviews MR1215720

18

Besse, Arthur L.

Einstein manifolds

Springer, 1987

Math Reviews MR867684

19

Biquard, Olivier

Métriques d'Einstein asymptotiquement symétriques

Soc. Math. France, 2000

Math Reviews MR1760319

20

Biquard, Olivier

Métriques autoduales sur la boule

Invent. Math. 148 (2002) 545–607

Math Reviews MR1908060

21

AdS/CFT correspondence: Einstein metrics and their conformal boundaries

European Mathematical Society (EMS), Zürich, 2005

Math Reviews MR2160864

22

Biquard, Olivier

Continuation unique à l'infini pour les métriques d'Einstein

Math. Res. Lett. 15 (2008) 1091–1099

23

Birman, M. Š. and Kren, M. G.

On the theory of wave operators and scattering operators

Dokl. Akad. Nauk SSSR 144 (1962) 475–478

Math Reviews MR0139007

24

Birman, M. Š. and Yafaev, Dmitri

The spectral shift function. The papers of M. G. Kreand their further development

Algebra i Analiz 4 (1992) 1–44

Math Reviews MR1202723

25

Borthwick, David

Spectral theory of infinite-area hyperbolic surfaces

Birkhäuser, 2007

Math Reviews MR2344504

26

Bourguignon, Jean-Pierre

Les variétés de dimension à signature non nulle dont la courbure est harmonique sont d'Einstein

Invent. Math. 63 (1981) 263–286

Math Reviews MR610539

27

Branson, Thomas P.

Differential operators canonically associated to a conformal structure

Math. Scand. 57 (1985) 293–345

Math Reviews MR832360

28

Branson, Thomas P.

Sharp inequalities, the functional determinant, and the complementary series

Trans. Amer. Math. Soc. 347 (1995) 3671–3742

Math Reviews MR1316845

29

Branson, Thomas P. and Chang, Sun-Yung A. and Yang, Paul C.

Estimates and extremals for zeta function determinants on four-manifolds

Comm. Math. Phys. 149 (1992) 241–262

Math Reviews MR1186028

30

Branson, Thomas P. and Gilkey, Peter and Pohjanpelto, Juha

Invariants of locally conformally flat manifolds

Trans. Amer. Math. Soc. 347 (1995) 939–953

Math Reviews MR1282884

31

Branson, Thomas P. and Gover, A. Rod

Conformally invariant operators, differential forms, cohomology and a generalisation of -curvature

Comm. Partial Differential Equations 30 (2005) 1611–1669

Math Reviews MR2182307

32

Branson, Thomas P. and rsted, Bent

Explicit functional determinants in four dimensions

Proc. Amer. Math. Soc. 113 (1991) 669–682

Math Reviews MR1050018

33

Bunke, Ulrich and Olbrich, Martin

Selberg zeta and theta functions

Akademie-Verlag, 1995 A differential operator approach

Math Reviews MR1353651

34

Buser, Peter

Geometry and spectra of compact Riemann surfaces

Birkhäuser, 1992

Math Reviews MR1183224

35

Caffarelli, Luis A. and Yang, Yi Song

Vortex condensation in the Chern-Simons Higgs model: an existence theorem

Comm. Math. Phys. 168 (1995) 321–336

Math Reviews MR1324400

36

Čap, Andreas and Gover, A. Rod

Standard tractors and the conformal ambient metric construction

Ann. Global Anal. Geom. 24 (2003) 231–259

Math Reviews MR1996768

37

Čap, Andreas and Schichl, Hermann

Parabolic geometries and canonical Cartan connections

Hokkaido Math. J. 29 (2000) 453–505

Math Reviews MR1795487

38

Carron, Gilles

Déterminant relatif et la fonction Xi

Amer. J. Math. 124 (2002) 307–352

Math Reviews MR1890995

39

Chang, Sun-Yung A. and Fang, H.

A class of variational functionals in conformal geometry

Int. Math. Res. Not. IMRN 7 (2008) 16 pp

40

Chang, Sun-Yung A. and Gursky, Matthew J. and Yang, Paul C.

An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature

Ann. of Math. 155 (2002) 709–787

Math Reviews MR1923964

41

Chang, Sun-Yung A. and Gursky, Matthew J. and Yang, Paul C.

A conformally invariant sphere theorem in four dimensions

Publ. Math. Inst. Hautes Études Sci. 98 (2003) 105–143

Math Reviews MR2031200

42

Chang, Sun-Yung A. and Gursky, Matthew J. and Yang, Paul C.

Conformal invariants associated to a measure

Proc. Natl. Acad. Sci. USA 103 (2006) 2535–2540 (electronic)

Math Reviews MR2203156

43

Chang, Sun-Yung A. and Yang, Paul C.

A perturbation result in prescribing scalar curvature on

Duke Math. J. 64 (1991) 27–69

Math Reviews MR1131392

44

Chang, Sun-Yung A. and Yang, Paul C.

Extremal metrics of zeta function determinants on -manifolds

Ann. of Math. 142 (1995) 171–212

Math Reviews MR1338677

45

Chang, Sun-Yung Alice and Qing, J. and Yang, Paul C.

Renormalized volumes for conformally compact Einstein manifolds

Sovrem. Mat. Fundam. Napravl. 17 (2006) 129–142

Math Reviews MR2336463

46

Chang, Sun-Yung Alice and Yang, Paul C.

Prescribing Gaussian curvature on

Acta Math. 159 (1987) 215–259

Math Reviews MR908146

47

Chen, Chiun-Chuan and Lin, Chang-Shou

Topological degree for a mean field equation on Riemann surfaces

Comm. Pure Appl. Math. 56 (2003) 1667–1727

Math Reviews MR2001443

48

Chruściel, Piotr T. and Delay, Erwann and Lee, John M. and Skinner, Dale N.

Boundary regularity of conformally compact Einstein metrics

J. Differential Geom. 69 (2005) 111–136

Math Reviews MR2169584

49

Cycon, H. L. and Froese, R. G. and Kirsch, W. and Simon, Barry

Schrödinger operators with application to quantum mechanics and global geometry

Springer, 1987

Math Reviews MR883643

50

Demailly, J.-P.

Complex analytic and differential geometry

51

Ding, Weiyue and Jost, Jürgen and Li, Jiayu and Wang, Guofang

Existence results for mean field equations

Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 653–666

Math Reviews MR1712560

52

Djadli, Zindine

Existence result for the mean field problem on Riemann surfaces of all genuses

Commun. Contemp. Math. 10 (2008) 205–220

Math Reviews MR2409366

53

Djadli, Zindine and Malchiodi, Andrea

Existence of conformal metrics with constant -curvature

Ann. of Math. 168 (2008) 813–858

Math Reviews MR2456884

54

Eastwood, Michael G. and Rice, John W.

Conformally invariant differential operators on Minkowski space and their curved analogues

Comm. Math. Phys. 109 (1987) 207–228

Math Reviews MR880414

55

Falk, C. and Juhl, Andreas

Universal recursive formulas for -curvature

56

Fefferman, Charles

Parabolic invariant theory in complex analysis

Adv. in Math. 31 (1979) 131–262

Math Reviews MR526424

57

Fefferman, Charles and Graham, C. Robin

Conformal invariants

in The mathematical heritage of Élie Cartan (Lyon, 1984)

Astérisque hors série (1985) 95–116

Math Reviews MR837196

58

Fefferman, Charles and Graham, C. Robin

-curvature and Poincaré metrics

Math. Res. Lett. 9 (2002) 139–151

Math Reviews MR1909634

59

Fefferman, Charles and Graham, C. Robin

The ambient metric

60

Fefferman, Charles and Hirachi, Kengo

Ambient metric construction of -curvature in conformal and CR geometries

Math. Res. Lett. 10 (2003) 819–831

Math Reviews MR2025058

61

Fried, David

The zeta functions of Ruelle and Selberg. I

Ann. Sci. École Norm. Sup. 19 (1986) 491–517

Math Reviews MR875085

62

Gauduchon, P.

Connexion canonique en géométrie conforme

(1990)

63

Gover, A. Rod

Aspects of parabolic invariant theory

Rend. Circ. Mat. Palermo (2) Suppl. 59 (1999) 25–47 The 18th Winter School ``Geometry and Physics'' (Srní, 1998)

Math Reviews MR1692257

64

Gover, A. Rod and Leitner, F.

A sub-product construction of Poincare-Einstein metrics

65

Gover, A. Rod and Peterson, Lawrence J.

Conformally invariant powers of the Laplacian, -curvature, and tractor calculus

Comm. Math. Phys. 235 (2003) 339–378

Math Reviews MR1969732

66

Graham, C. Robin

Volume and area renormalizations for conformally compact Einstein metrics

in The Proceedings of the 19th Winter School ``Geometry and Physics'' (Srn, 1999)

Rend. Circ. Mat. Palermo (2) Suppl. 63 (2000) 31–42

Math Reviews MR1758076

67

Graham, C. Robin

Dirichlet-to-Neumann map for Poincaré-Einstein metrics

Oberwolfach Rep. 2 (2005)

68

Graham, C. Robin

Jet isomorphism for conformal geometry

Arch. Math. (Brno) 43 (2007) 389–415

Math Reviews MR2381784

69

Graham, C. Robin and Hirachi, Kengo

The ambient obstruction tensor and -curvature

in AdS/CFT correspondence: Einstein metrics and their conformal boundaries

IRMA Lect. Math. Theor. Phys. 8 (2005) 59–71

Math Reviews MR2160867

70

Graham, C. Robin and Hirachi, Kengo

Inhomogeneous ambient metrics

in Symmetries and overdetermined systems of partial differential equations

IMA Vol. Math. Appl. 144 (2008) 403–420

Math Reviews MR2384722

71

Graham, C. Robin and Jenne, Ralph and Mason, Lionel J. and Sparling, George A. J.

Conformally invariant powers of the Laplacian. I. Existence

J. London Math. Soc. 46 (1992) 557–565

Math Reviews MR1190438

72

Graham, C. Robin and Juhl, Andreas

Holographic formula for -curvature

Adv. Math. 216 (2007) 841–853

Math Reviews MR2351380

73

Graham, C. Robin and Lee, John M.

Einstein metrics with prescribed conformal infinity on the ball

Adv. Math. 87 (1991) 186–225

Math Reviews MR1112625

74

Graham, C. Robin and Zworski, Maciej

Scattering matrix in conformal geometry

Invent. Math. 152 (2003) 89–118

Math Reviews MR1965361

75

Guillarmou, Colin

Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Duke Math. J. 129 (2005) 1–37

Math Reviews MR2153454

76

Guillarmou, Colin

Generalized Krein formula, determinants and Selberg zeta function in even dimension

77

Guillopé, Laurent and Zworski, Maciej

Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity

Asymptotic Anal. 11 (1995) 1–22

Math Reviews MR1344252

78

Guillopé, Laurent and Zworski, Maciej

Scattering asymptotics for Riemann surfaces

Ann. of Math. 145 (1997) 597–660

Math Reviews MR1454705

79

Gursky, Matthew J.

Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic

Indiana Univ. Math. J. 43 (1994) 747–774

Math Reviews MR1305946

80

Gursky, Matthew J.

The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics

Ann. of Math. 148 (1998) 315–337

Math Reviews MR1652920

81

Gursky, Matthew J.

The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE

Comm. Math. Phys. 207 (1999) 131–143

Math Reviews MR1724863

82

Gursky, Matthew J.

Four-manifolds with and Einstein constants of the sphere

Math. Ann. 318 (2000) 417–431

Math Reviews MR1800764

83

Gursky, Matthew J. and Viaclovsky, Jeff A.

A fully nonlinear equation on four-manifolds with positive scalar curvature

J. Differential Geom. 63 (2003) 131–154

Math Reviews MR2015262

84

Hejhal, Dennis A.

The Selberg trace formula for . Vol. 2

Springer, 1983

Math Reviews MR711197

85

Henningson, M. and Skenderis, Kostas

The holographic Weyl anomaly

J. High Energy Phys. 7 (1998) Paper 23, 12 pp.(electronic)

Math Reviews MR1644988

86

Hitchin, Nigel J.

Compact four-dimensional Einstein manifolds

J. Differential Geometry 9 (1974) 435–441

Math Reviews MR0350657

87

Hitchin, Nigel J.

Twistor spaces, Einstein metrics and isomonodromic deformations

J. Differential Geom. 42 (1995) 30–112

Math Reviews MR1350695

88

Hitchin, Nigel J.

Einstein metrics and the eta-invariant

Boll. Un. Mat. Ital. B 11 (1997) 95–105

Math Reviews MR1456253

89

Iwaniec, Henryk

Spectral methods of automorphic forms

Amer. Math. Soc., 2002

Math Reviews MR1942691

90

Joshi, Mark S. and Sá Barreto, Antônio

Inverse scattering on asymptotically hyperbolic manifolds

Acta Math. 184 (2000) 41–86

Math Reviews MR1756569

91

Kato, Tosio

Wave operators and unitary equivalence

Pacific J. Math. 15 (1965) 171–180

Math Reviews MR0180138

92

Kazdan, Jerry L. and Warner, F. W.

Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures

Ann. of Math. 101 (1975) 317–331

Math Reviews MR0375153

93

Kontsevich, Maxim and Vishik, Simeon

Geometry of determinants of elliptic operators

in Functional analysis on the eve of the 21st century, Vol.1 (New Brunswick, NJ, 1993)

Progr. Math. 131 (1995) 173–197

Math Reviews MR1373003

94

Kren, M. G.

On the trace formula in perturbation theory

Mat. Sbornik N.S. 33(75) (1953) 597–626

Math Reviews MR0060742

95

Kren, M. G.

On perturbation determinants and a trace formula for unitary and self-adjoint operators

Dokl. Akad. Nauk SSSR 144 (1962) 268–271

Math Reviews MR0139006

96

Kuiper, N. H.

On conformally-flat spaces in the large

Ann. of Math. 50 (1949) 916–924

Math Reviews MR0031310

97

Kulkarni, Ravindra S.

On the Bianchi Identities

Math. Ann. 199 (1972) 175–204

Math Reviews MR0339004

98

Kulkarni, Ravindra S.

Index theorems of Atiyah-Bott-Patodi and curvature invariants

Les Presses de l'Université de Montréal, Montreal, Que., 1975

Math Reviews MR0440626

99

Labbi, M.-L.

Double forms, curvature structures and the -curvatures

Trans. Amer. Math. Soc. 357 (2005) 3971–3992 (electronic)

Math Reviews MR2159696

100

Labbi, M.-L.

Variational properties of the Gauss-Bonnet curvatures

Calc. Var. Partial Differential Equations 32 (2008) 175–189

Math Reviews MR2389988

101

Lax, Peter D. and Phillips, Ralph S.

Scattering theory for automorphic functions

Princeton Univ. Press, 1976 Annals of Mathematics Studies, No. 87

Math Reviews MR0562288

102

LeBrun, Claude R.

-space with a cosmological constant

Proc. Roy. Soc. London Ser. A 380 (1982) 171–185

Math Reviews MR652038

103

LeBrun, Claude R.

On the topology of self-dual -manifolds

Proc. Amer. Math. Soc. 98 (1986) 637–640

Math Reviews MR861766

104

Lee, John M.

The spectrum of an asymptotically hyperbolic Einstein manifold

Comm. Anal. Geom. 3 (1995) 253–271

Math Reviews MR1362652

105

Lee, John M.

Fredholm operators and Einstein metrics on conformally compact manifolds

Mem. Amer. Math. Soc. 183 (2006) 83

Math Reviews MR2252687

106

Lee, John M. and Parker, Thomas H.

The Yamabe problem

Bull. Amer. Math. Soc. (N.S.) 17 (1987) 37–91

Math Reviews MR888880

107

Lesch, Matthias

On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

Ann. Global Anal. Geom. 17 (1999) 151–187

Math Reviews MR1675408

108

Li, Yan Yan

Prescribing scalar curvature on and related problems. I

J. Differential Equations 120 (1995) 319–410

Math Reviews MR1347349

109

Lin, Chang-Shou

Topological degree for mean field equations on

Duke Math. J. 104 (2000) 501–536

Math Reviews MR1781481

110

Lott, John

Optimal transport and Ricci curvature for metric-measure spaces

in Surveys in differential geometry. Vol. XI

Surv. Differ. Geom. 11 (2007) 229–257

Math Reviews MR2408268

111

Malchiodi, Andrea

Compactness of solutions to some geometric fourth-order equations

J. reine angew. Math. 594 (2006) 137–174

Math Reviews MR2248155

112

Maldacena, Juan

The large limit of superconformal field theories and supergravity

Adv. Theor. Math. Phys. 2 (1998) 231–252

Math Reviews MR1633016

113

Marchioro, Carlo and Pulvirenti, Mario

Mathematical theory of incompressible nonviscous fluids

Springer, 1994

Math Reviews MR1245492

114

Margerin, Christophe

A sharp characterization of the smooth -sphere in curvature terms

Comm. Anal. Geom. 6 (1998) 21–65

Math Reviews MR1619838

115

Margerin, Christophe

Géométrie conforme en dimension 4: ce que l'analyse nous apprend

in Séminaire Bourbaki 2004/2005

Astérisque 307 (2006) Exp. No. 950, ix–415–468

Math Reviews MR2296426

116

Mazzeo, Rafe

The Hodge cohomology of a conformally compact metric

J. Differential Geom. 28 (1988) 309–339

Math Reviews MR961517

117

Mazzeo, Rafe

Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds

Amer. J. Math. 113 (1991) 25–45

Math Reviews MR1087800

118

Mazzeo, Rafe R. and Melrose, Richard B.

Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature

J. Funct. Anal. 75 (1987) 260–310

Math Reviews MR916753

119

Melrose, Richard B.

Weyl asymptotics for the phase in obstacle scattering

Comm. Partial Differential Equations 13 (1988) 1431–1439

Math Reviews MR956828

120

Melrose, Richard B.

Geometric scattering theory

Cambridge Univ. Press, 1995

Math Reviews MR1350074

121

Moller, N. M.

Extremals of Dirac operators, spectral determinants and zeta functions, and Branson's conjecture

122

Moser, J.

A sharp form of an inequality by N. Trudinger

Indiana Univ. Math. J. 20 (1970/71) 1077–1092

Math Reviews MR0301504

123

Okikiolu, K.

Critical metrics for the determinant of the Laplacian in odd dimensions

Ann. of Math. 153 (2001) 471–531

Math Reviews MR1829756

124

Onofri, E.

On the positivity of the effective action in a theory of random surfaces

Comm. Math. Phys. 86 (1982) 321–326

Math Reviews MR677001

125

Osgood, B. and Phillips, Ralph S. and Sarnak, P.

Compact isospectral sets of surfaces

J. Funct. Anal. 80 (1988) 212–234

Math Reviews MR960229

126

Osgood, B. and Phillips, Ralph S. and Sarnak, P.

Extremals of determinants of Laplacians

J. Funct. Anal. 80 (1988) 148–211

Math Reviews MR960228

127

Paneitz, S.

A quadratic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds

(1983)

128

Patterson, S. J.

The limit set of a Fuchsian group

Acta Math. 136 (1976) 241–273

Math Reviews MR0450547

129

Patterson, S. J.

The Selberg zeta-function of a Kleinian group

in Number theory, trace formulas and discrete groups (Oslo, 1987)

(1989) 409–441

Math Reviews MR993330

130

Patterson, S. J. and Perry, Peter A.

The divisor of Selberg's zeta function for Kleinian groups

Duke Math. J. 106 (2001) 321–390 Appendix A by Charles Epstein

Math Reviews MR1813434

131

Paycha, Sylvie and Rosenberg, Steven

Conformal anomalies via canonical traces

in Analysis, geometry and topology of elliptic operators

(2006) 263–294

Math Reviews MR2246772

132

Pedersen, H.

Einstein metrics, spinning top motions and monopoles

Math. Ann. 274 (1986) 35–59

Math Reviews MR834105

133

Perelman, G.

The entropy formula for the Ricci flow and its geometric applications

134

Petkov, V. and Popov, G.

Asymptotic behaviour of the scattering phase for nontrapping obstacles

Ann. Inst. Fourier (Grenoble) 32 (1982) 111–149

Math Reviews MR688023

135

Polyakov, A. M.

Quantum geometry of bosonic strings

Phys. Lett. B 103 (1981) 207–210

Math Reviews MR623209

136

Ratcliffe, John G.

Foundations of hyperbolic manifolds

Springer, 2006

Math Reviews MR2249478

137

Ray, D. B. and Singer, I. M.

-torsion and the Laplacian on Riemannian manifolds

Advances in Math. 7 (1971) 145–210

Math Reviews MR0295381

138

Reed, Michael and Simon, Barry

Methods of modern mathematical physics. I

Academic Press Inc., 1980 Functional analysis

Math Reviews MR751959

139

Richardson, Ken

Critical points of the determinant of the Laplace operator

J. Funct. Anal. 122 (1994) 52–83

Math Reviews MR1274583

140

Rosenberg, Steven

The Laplacian on a Riemannian manifold

Cambridge Univ. Press, 1997 An introduction to analysis on manifolds

Math Reviews MR1462892

141

Sá Barreto, Antônio

Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds

Duke Math. J. 129 (2005) 407–480

Math Reviews MR2169870

142

Selberg, Atle

Collected papers. Vol. I

Springer, 1989

Math Reviews MR1117906

143

Sharpe, R. W.

Differential geometry

Springer, 1997

Math Reviews MR1453120

144

Shubin, M. A.

Pseudodifferential operators and spectral theory

Springer, 1987

Math Reviews MR883081

145

Skenderis, Kostas and Solodukhin, Sergey N.

Quantum effective action from the AdS/CFT correspondence

Phys. Lett. B 472 (2000) 316–322

Math Reviews MR1742763

146

Struwe, Michael

The existence of surfaces of constant mean curvature with free boundaries

Acta Math. 160 (1988) 19–64

Math Reviews MR926524

147

Struwe, Michael and Tarantello, Gabriella

On multivortex solutions in Chern-Simons gauge theory

Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 1 (1998) 109–121

Math Reviews MR1619043

148

Sullivan, Dennis

The density at infinity of a discrete group of hyperbolic motions

Publ. Math. I.H.É.S. 50 (1979) 171–202

Math Reviews MR556586

149

Tang, S. H. and Zworski, Maciej

Potential scattering on the line

150

Tarantello, Gabriella

Multiple condensate solutions for the Chern-Simons-Higgs theory

J. Math. Phys. 37 (1996) 3769–3796

Math Reviews MR1400816

151

Thomas, T. Y.

On conformal geometry

Proc. Natl. Acad. Sci. USA 12 (1926) 352–359

152

Trudinger, Neil S.

On imbeddings into Orlicz spaces and some applications

J. Math. Mech. 17 (1967) 473–483

Math Reviews MR0216286

153

Uhlenbeck, Karen K. and Viaclovsky, Jeff A.

Regularity of weak solutions to critical exponent variational equations

Math. Res. Lett. 7 (2000) 651–656

Math Reviews MR1809291

154

Viaclovsky, Jeff A.

Conformal geometry, contact geometry, and the calculus of variations

Duke Math. J. 101 (2000) 283–316

Math Reviews MR1738176

155

Weyl, Hermann

The classical groups

Princeton Univ. Press, 1997 Their invariants and representations, Fifteenth printing, Princeton Paperbacks

Math Reviews MR1488158

156

Yafaev, Dmitri

Mathematical scattering theory: General Theory

Amer. Math. Soc., 1992

157

Yafaev, Dmitri

Lectures on scattering theory

in Surveys in analysis and operator theory (Canberra, 2001)

Proc. Centre Math. Appl. Austral. Nat. Univ. 40 (2002) 61–88

Math Reviews MR1953480

158

Yang, Yisong

On a system of nonlinear elliptic equations arising in theoretical physics

J. Funct. Anal. 170 (2000) 1–36

Math Reviews MR1736194

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