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Families of Conformally Covariant Differential Operators, Q-Curvature and Holography / by Andreas Juhl
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Progress in Mathematics ; 275 | SpringerLink BücherPublisher: Basel ; Boston, Mass. ; Berlin : Birkhäuser, 2009Publisher: [Heidelberg] : [Springer], 2009Description: Online-Ressource (XIII, 488 p, digital)ISBN:- 9783764399009
- 516.36
- 516.373
- 510
- QA641-670
- QA649
Contents:
Summary: Spaces, Actions, Representations and Curvature -- Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory -- Paneitz Operator and Paneitz Curvature -- Intertwining Families -- Conformally Covariant Families.Summary: The central object of the book is a subtle scalar Riemannian curvature quantity in even dimensions which is called Branson’s Q-curvature. It was introduced by Thomas Branson about 15 years ago in connection with an attempt to systematise the structure of conformal anomalies of determinants of conformally covariant differential operators on Riemannian manifolds. Since then, numerous relations of Q-curvature to other subjects have been discovered, and the comprehension of its geometric significance in four dimensions was substantially enhanced through the studies of higher analogues of the Yamabe problem. The book attempts to reveal some of the structural properties of Q-curvature in general dimensions. This is achieved by the development of a new framework for such studies. One of the main properties of Q-curvature is that its transformation law under conformal changes of the metric is governed by a remarkable linear differential operator: a conformally covariant higher order generalization of the conformal Laplacian. In the new approach, these operators and the associated Q-curvatures are regarded as derived quantities of certain conformally covariant families of differential operators which are naturally associated to hypersurfaces in Riemannian manifolds. This method is at the cutting edge of several central developments in conformal differential geometry in the last two decades such as Fefferman-Graham ambient metrics, spectral theory on Poincaré-Einstein spaces, tractor calculus, and Cartan geometry. In addition, the present theory is strongly inspired by the realization of the idea of holography in the AdS/CFT-duality. This motivates the term holographic descriptions of Q-curvature.PPN: PPN: 1648314163Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
CONTENTS; Preface; 1 Introduction; 1.1 Hyperbolic geometry and conformal dynamics; 1.2 Automorphic distributions and intertwining families; 1.3 Asymptotically hyperbolic Einstein metrics.Conformally covariant powers of the Laplacian; 1.4 Intertwining families; 1.5 The residue method for the hemisphere; 1.6 Q-curvature, holography and residue families; 1.7 Factorization of residue families. Recursive relations; 1.8 Families of conformally covariant differential operators; 1.9 Curved translation and tractor families; 1.10 Holographic duality. Extrinsic Q-curvature.Odd order Q-curvature
1.11 Review of the contents1.12 Some further perspectives; 2 Spaces, Actions, Representations and Curvature; 2.1 Lie groups, Lie algebras, spaces and actions; 2.2 Stereographic projection; 2.3 Poisson transformations and spherical principal series; 2.4 The Nayatani metric; 2.5 Riemannian curvature and conformal change; 3 Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory; 3.1 GJMS-operators and Q-curvature; 3.2 Scattering theory; 4 Paneitz Operator and Paneitz Curvature; 4.1 P 4 , Q 4 and their transformation properties
4.2 The fundamental identity for the Paneitz curvature4.3 Q 4 and v 4; 5 Intertwining Families; 5.1 The algebraic theory; 5.2 Induced families; 5.3 Some low order examples; 5.4 Families for (R n ,S n-1 ); 5.5 Automorphic distributions; 6 Conformally Covariant Families; 6.1 Fundamental pairs and critical families; 6.2 The family D 1 (g; ?); 6.3 D 2 (g; ?) for a surface in a 3-manifold; 6.4 Second-order families. General case; 6.5 Families and the asymptotics of eigenfunctions; 6.6 Residue families and holographic formulas for Q-curvature; 6.7 D 2 (g; ?) as a residue family; 6.8 D3 res (h
?)6.9 The holographic coefficients v 2 , v 4 and v 6; 6.10 The holographic formula for Q 6; 6.11 Factorization identities for residue families.Recursive relations; 6.12 A recursive formula for P 6 . Universality; 6.13 Recursive formulas for Q 8 and P 8; 6.14 Holographic formula for conformally flat metrics; 6.15 v 4 as a conformal index density; 6.16 The holographic formula for Einstein metrics; 6.17 Semi-holonomic Verma modules and their role; 6.18 Zuckerman translation and D N (?); 6.19 From Verma modules to tractors; 6.20 Some elements of tractor calculus
6.21 The tractor families DN T (M,S g; ?); 6.22 Some results on tractor families; 6.23 J and Fialkow's fundamental forms; 6.24 D 2 (g; ?) as a tractor family; 6.25 The family D3 T (M,S; g; ?); 6.26 The pair (P 3 ,Q 3 ); Bibliography; Index;
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