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Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Part of the Progress in Mathematics Book Series (volume 275)

DOI: 10.1007/978-3-7643-9900-9

ISBN: 978-3-7643-9899-6

Publisher: Birkhäuser Basel - Springer Journals

Jul 25, 2009

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Chapter 1Introduction

Pages: 1 –61

[In the present book we develop a new approach towards Branson’s Q-curvature. The central idea is to relate it to certain one-parameter families of conformally covariant differential operators which are associated to hypersurfaces in Riemannian manifolds.]
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Chapter 2Spaces, Actions, Representations and Curvature

Pages: 63 –86

[The present chapter contains background material. We fix conventions on the structure of the Lie group SO(1, n) and its Lie-algebra. Starting from the light-cone model, we derive formulas for its actions on Sn−1 and ℝn−1 ∪ {∞} using stereographic projection. These actions induce two models of...
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Chapter 3Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory

Pages: 87 –104

[In the present chapter, we review the GJMS-construction of conformally covariant powers of the Laplacian, describe Branson’s definition of Q-curvature, and discuss the relation of both to geometric scattering theory on conformally compact Einstein manifolds.]
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Chapter 4Paneitz Operator and Paneitz Curvature

Pages: 105 –116

[In the present chapter we derive basic facts concerning the Q-curvature Q4 in dimension 4. In particular, we prove its conformal transformation law, i.e., the fundamental identity, which involves the Paneitz operator P4.]
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Chapter 5Intertwining Families

Pages: 117 –181

[In the present chapter, we construct two sequences of polynomial families DNc(λ): C∞(Sn) → C∞(Sn−1) and DNnc(λ) : C∞(ℝn) → C∞(ℝn−1) of differential intertwining operators for spherical principal series representations. These families are induced by families \documentclass[12pt]{minimal}...
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Chapter 6Conformally Covariant Families

Pages: 183 –468

[Let M be a smooth oriented manifold and i: Σ ↪ M an oriented codimension one submanifold (hypersurface). In this chapter, we shall discuss conformally covariant families of differential operators C∞(M) → C∞(Σ) in the presence of general background metrics on M. The model cases (ℝn+1, ℝn) with...

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