Families of Conformally Covariant Differential Operators, Q-Curvature and Holography (Progress in Mathematics)

Progress in Mathematics Volume 275 Series Editors H. Bass J. Oesterlé A. Weinstein Andreas Juhl Families of Conform...

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Progress in Mathematics Volume 275

Series Editors H. Bass J. Oesterlé A. Weinstein

Andreas Juhl

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

Birkhäuser Basel · Boston · Berlin

Author: Andreas Juhl Humboldt-Universität Institut für Mathematik 10099 Berlin Germany e-mail: [email protected] and Universitet Uppsala Matematiska Institutionen Box-480 75106 Uppsala Sweden e-mail: [email protected]

2000 Mathematics Subject Classiﬁcation 53BXX (several: 20, 25, 50), 53C25, 53C80, 58 J50, 81T50 Library of Congress Control Number: 2009924478 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-9899-6 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microﬁlms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN978-3-7643-9899-6

e-ISBN 978-3-7643-9900-9

987654321

www.birkhauser.ch

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 contents . . . . . . . . . . . . . . . . . . 1.12 Some further perspectives . . . . . . . . . . . . . . . .

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9 11 17 20 32 42 46

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50 55 58

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63 67 71 81 82

3 Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory 3.1 GJMS-operators and Q-curvature . . . . . . . . . . . . . . . . . . 3.2 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 91

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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 . . . . . .

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vi

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4 Paneitz Operator and Paneitz Curvature 4.1 P4 , Q4 and their transformation properties . . . . . . . . . . . . . 106 4.2 The fundamental identity for the Paneitz curvature . . . . . . . . 108 4.3 Q4 and v4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5 Intertwining Families 5.1 The algebraic theory . . . . . . . . . . . . . . . . . . . 5.1.1 Even order families D2N (λ) . . . . . . . . . . . 5.1.2 Odd order families D2N +1 (λ) . . . . . . . . . . 5.1.3 DN (λ) as homomorphism of Verma modules . . 5.2 Induced families . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Induction . . . . . . . . . . . . . . . . . . . . . nc c (λ) and D2N (λ) . . . 5.2.2 Even order families: D2N nc c 5.2.3 Odd order families: D2N +1 (λ) and D2N +1 (λ) . nc n 5.2.4 Eigenfunctions of ΔH and the families DN (λ) 5.3 Some low order examples . . . . . . . . . . . . . . . . 5.4 Families for (Rn , S n−1 ) . . . . . . . . . . . . . . . . . . b (λ) . . . . . . . . . . . . . . . 5.4.1 The families DN b b 5.4.2 D1 (λ), D2 (λ) and D3b (λ) . . . . . . . . . . . . . 5.4.3 D3b (0) for n = 4 and (P3 , T ) for (B4 , S 3 ) . . . . 5.5 Automorphic distributions . . . . . . . . . . . . . . . .

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6 Conformally Covariant Families 6.1 Fundamental pairs and critical families . . . . . . . . . . . 6.2 The family D1 (g; λ) . . . . . . . . . . . . . . . . . . . . . . 6.3 D2 (g; λ) 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 D2 (g; λ) as a residue family . . . . . . . . . . . . . . . . . 6.8 D3res (h; λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 The holographic coeﬃcients v2 , v4 and v6 . . . . . . . . . 6.10 The holographic formula for Q6 . . . . . . . . . . . . . . . 6.11 Factorization identities for residue families. Recursive relations . . . . . . . . . . . . . . . . . . . . . . 6.12 A recursive formula for P6 . Universality . . . . . . . . . . 6.13 Recursive formulas for Q8 and P8 . . . . . . . . . . . . . . 6.14 Holographic formula for conformally ﬂat metrics . . . . . 6.15 v4 as a conformal index density . . . . . . . . . . . . . . . 6.16 The holographic formula for Einstein metrics . . . . . . . 6.17 Semi-holonomic Verma modules and their role . . . . . . .

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117 117 127 129 131 131 139 148 154 161 165 165 172 176 178

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190 194 195 201 208 214 235 236 239 254

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264 318 325 329 339 343 356

Contents

6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26

vii

Zuckerman translation and DN (λ) . From Verma modules to tractors . . Some elements of tractor calculus . . T The tractor families DN (M, Σ; g; λ) . Some results on tractor families . . . J and Fialkow’s fundamental forms D2 (g; λ) as a tractor family . . . . . The family D3T (M, Σ; g; λ) . . . . . . The pair (P3 , Q3 ) . . . . . . . . . . .

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360 381 388 403 418 445 450 455 463

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Preface A basic problem in geometry is to ﬁnd canonical metrics on smooth manifolds. Such metrics can be speciﬁed, for instance, by curvature conditions or extremality properties, and are expected to contain basic information on the topology of the underlying manifold. Constant curvature metrics on surfaces are such canonical metrics. Their distinguished role is emphasized by classical uniformization theory. A more recent characterization of these metrics describes them as critical points of the determinant functional for the Laplacian. The key tool here is Polyakov’s variational formula for the determinant. In higher dimensions, however, it is necessary to further restrict the problem, for instance, to the search for canonical metrics in conformal classes. Here two metrics are considered to belong to the same conformal class if they diﬀer by a nowhere vanishing factor. A typical question in that direction is the Yamabe problem ([165]), which asks for constant scalar curvature metrics in conformal classes. In connection with the problem of understanding the structure of Polyakov type formulas for the determinants of conformally covariant diﬀerential operators in higher dimensions, Branson ([31]) discovered a remarkable curvature quantity which now is called Branson’s Q-curvature. It is one of the main objects in this book. Q-curvature is a scalar local Riemannian curvature invariant on manifolds of even dimension. On surfaces it coincides with Gauß curvature. On four-manifolds it ﬁrst appeared in connection with the conformally covariant Paneitz operator. In this case, it is a certain linear combination of squared scalar curvature, the squared norm of Ricci curvature and the Laplacian of scalar curvature. On a manifold of dimension n, Q-curvature is an nth -order curvature invariant. One of its remarkable properties is that its behaviour under conformal changes of the metric is governed by an nth -order linear conformally covariant diﬀerential operator. In dimensions two and four, the respective operators are the Laplacian and the Paneitz operator. In higher dimensions, these operators are replaced by certain conformally covariant powers of the Laplacian (GJMS-operators) ([124]). Besides their signiﬁcance in conformal geometry, GJMS-operators also play an important role in physics. This is due to the fact that their deﬁnition extends to Lorentzian manifolds. They are common generalizations of the Yamabe operator and the conformally covariant powers of the wave operator on Minkowski space.

x

Preface

In recent years, new connections of Q-curvature with other parts of mathematics and theoretical physics have been discovered. Probably the most remarkable one is the relation to geometric scattering on asymptotically hyperbolic Einstein spaces. This is a relation in the spirit of the conjectural AdS/CFT-duality ([171]), which connects gravitation with gauge ﬁeld theory, and which led to an outburst of activities in theoretical physics (for reviews see [1], [82]). In geometric analysis, recent eﬀorts are being directed towards an understanding of the geometric significance of Q-curvature in dimension four, for instance, by studying Yamabe type problems and Q-curvature ﬂows (for a review see [170]). Here one of the problems is to characterize the conformal classes which contain a metric with constant Q-curvature. Much less is known in higher dimensions. Although Q-curvature is an intrinsic Riemannian curvature invariant, all known conceptual deﬁnitions in general dimension take one or another extrinsic point of view. The situation somewhat resembles Weyl’s formula for the volume of a tube ([130]). This formula shows that the volume of a tube of a closed submanifold of Euclidean space is a polynomial in its radius, and the coeﬃcients depend only on the intrinsic curvature of the submanifold. In particular, the Euler characteristic of the submanifold appears in the leading coeﬃcient. In the present book, we develop a new extrinsic point of view towards Qcurvature with the emphasis on general structural results. The guiding idea is to associate to a hypersurface i : Σ → M and a general background metric g on M certain one-parameter families of conformally covariant local operators which map functions on M to functions on Σ. Q-curvature and the GJMS-operator of the submanifold (Σ, i∗ (g)) appear in the respective linear and constant coeﬃcients of these families, and the fundamental transformation law of Q-curvature is a direct consequence of the covariance of the family. In particular, we introduce two speciﬁc constructions of conformally covariant families with such properties: the residue families and the tractor families. The setting of residue families is more restricted, however. Here Σ is the boundary of M , and the background metric on M is a canonical extension of a given metric on Σ. Such situations arise in connection with conformally compact Einstein metrics and the Feﬀerman-Graham construction of an ambient metric. The closely related Poincar´e-Einstein metrics associate to any conformal class on Σ a diﬀeomorphism class of conformally compact Einstein metrics on M with the given class as conformal inﬁnity ([199]). The method of the ambient metric was introduced in [99] as a fundamental systematic construction of conformal invariants. During the last two decades the ambient metric had a major inﬂuence on the subject of conformal geometry. For full details see [96]. Poincar´e-Einstein metrics are used in theoretical physics in connection with the speculative holographic principle in quantum gravity ([29], [227]). The bulk space/boundary duality between superstring theory on AdS-space and supersymmetric Yang-Mills theory on Minkowski space is regarded as a concrete manifestation of the principle.

Preface

xi

In a pure gravity setting with homogeneous metrics, a related bulk/boundary duality is Helgason’s well-known theory of Poisson transformations in harmonic analysis on symmetric spaces ([140]). Versions of that transform for conformally compact Einstein spaces play an important role in attempts to establish rigorous statements in the AdS/CFT-duality. The residue families are deﬁned by a certain residue construction, which has its origin in the spectral theory of Kleinian manifolds. This explains the name. These families can be regarded as local counterparts of the global scattering operator. They naturally lead to an understanding of Q-curvature of a metric on the boundary at inﬁnity as part of a hologram of the associated Poincar´e-Einstein metric in one higher dimension. More precisely, the holographic formulas describe Q-curvature in terms of holographic coeﬃcients of the Poincar´e-Einstein metric and its harmonic functions. Combining that relation between Q-curvature and residue families with structural properties of residue families (factorization relations), uncovers recursive structures among Q-curvatures and GJMS-operators. It is here where the lower order relatives Q2N (2N < n) of Branson’s Q-curvature Qn become important. All in all, the residue families are an eﬀective tool for the systematic study of the interplay between the asymptotic geometry of Poincar´eEinstein metrics on bulk space and GJMS-operators (and Q-curvatures) of their conformal inﬁnities. The theory of the tractor families is an attempt to take a wider perspective. Here the conformal compactiﬁcations of Poincar´e-Einstein metrics are replaced by arbitrary background metrics, and we extract the intrinsic Q-curvature of the submanifold using an appropriate extrinsic construction near Σ. That perspective leads to the notions of extrinsic and odd order Q-curvatures, which relate the subject of Q-curvature with conformal submanifold theory. The tractor families are deﬁned in terms of the conformally invariant tractor calculus ([17]). A closely related construction was used in [40] in a diﬀerent connection. For certain classes of metrics, residue families and tractor families coincide. Such relations imply tractor formulas for GJMS-operators and Q-curvature, and will be termed holographic duality. The new approach to Q-curvature grew out of results which relate the divisors of Selberg zeta functions to automorphic distributions. Such results are related to the dream of an interpretation of the Riemann-Weil explicit formula in analytic number theory as a version of a Lefschetz ﬁxed point formula. The hope is that a cohomological interpretation may also contain a key to the Riemann hypothesis ([78]). In the same spirit, it was shown in [151] that, using Osborne’s character formula, the Selberg trace formula can be regarded as a Lefschetz formula for the geodesic ﬂow. This leads to characterizations of the divisors of Selberg zeta functions in terms of cohomology of Anosov foliations and representation theory. The basic principle is that the complex numbers which appear as zeros or poles of a zeta function are characterized by the non-vanishing of the Euler characteristics of associated complexes. Moreover, the values of the corresponding Euler char-

xii

Preface

acteristics yield the multiplicities. Equivariant Poisson transformations translate these results into characterizations in terms of group cohomology with values in distributions on the geodesic boundary of rank one symmetric spaces. The latter result can be regarded as a version of holography: the divisor, i.e., zeros and poles with multiplicities, of a zeta function, which is deﬁned by the lengths of closed geodesics of a hyperbolic manifold, is completely characterized in terms of a theory which is formulated on a manifold of one dimension less. The cohomological objects which correspond to the zeros of the Riemann zeta function remain to be found, however. The fascination of Q-curvature stems from its central role in the complex web of ideas outlined above. In this framework, we observe how classical and modern diﬀerential geometry, geometric analysis, harmonic analysis and theoretical physics meet each other. Although in four dimensions the geometric meaning of Q-curvature has been studied intensively in recent years, there are only few results in higher dimensions. It would be pleasing if the perspectives and the structural insights presented here help to enter this unexplored ﬁeld. Presently, the future role of Q-curvature is hard to predict, and it seems that we are now taking only the ﬁrst steps towards its comprehension. The reader will easily notice that the theory in this book has open ends on diﬀerent levels. In addition to a number of explicitly formulated conjectures, there are results that are derived under conditions which certainly can be relaxed, and the full consequences of some arguments and constructions are not yet predictable. Moreover, the basic ideas should apply also in diﬀerent contexts. We hope that this will motivate further investigations. At ﬁrst glance, it might seem that the text contains a jungle of complicated formulas. To some extent, this is typical for the subject. On the other hand, we believe that the ambitious reader ﬁnally will be delighted by the ways in which complex but beautiful formulas emerge from simple principles, albeit sometimes through non-trivial calculations. The disclosure of some of the hidden structures is one of the aims of this work. First and foremost, the book is a research monograph presenting a new theory. On the other hand, we have attempted of a self-contained presentation of the material so that it should be accessible for non-specialists. Although we strictly concentrate on the development of new ideas, we necessarily touch upon many of the recent developments in conformal diﬀerential geometry. Therefore, the text may also serve as an informal introduction to the subject. We hope that we have succeeded in ﬁnding some balance between the presentation of structural ideas and the discussion of full (calculational) details. In particular, we also included proofs of some results which might be considered as well-known by specialists. But since the various ﬁelds which are touched upon here do not have a common folklore, proofs are given if required for the sake of a coherent presentation. Also, due to varying conventions, it was sometimes easier to supply proofs than to refer to the

Preface

xiii

literature. The list of references is not representative for any of the numerous ﬁelds linked with the subject. The early phases of the work were ﬁnanced by a grant of the Swedish Research Council (VR) at Uppsala University. Since 2005 Sonderforschungsbereich 647 “Space-Time-Matter” at Humboldt-University, Berlin supported the research as part of a project initiated by H. Baum. Special thanks go to the participants of my courses and the seminars at Humboldt-University on the subject during the years 2005–2008. Over the years, I beneﬁted a lot from discussions with H. Baum, ˇ T. Branson, A. Cap, R. Gover, F. Leitner, T. Leistner, M. Olbrich, B. Ørsted, P. Somberg, V. Souˇcek, and from the stimulating annual conferences in Srni. In a series of lectures in Srni 2005, I had the privilege of presenting part of the results. In later stages of the project, discussions with R. Graham inﬂuenced the shape of the theory. Finally, I am grateful to the reviewers for valuable hints.

Berlin, autumn 2008

Chapter 1

Introduction 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. In this chapter we describe the main ideas, review the central results and discuss related perspectives. Full details are given in the later chapters. In the ﬁrst two sections, we start with a brief discussion of the origin of the new method in the spectral geometry of Riemannian manifolds of constant negative curvature (Kleinian manifolds). This is an extremely rich subject in its own which connects hyperbolic geometry and hyperbolic dynamics with conformal geometry and conformal dynamics. We use Selberg zeta functions of Kleinian manifolds to illustrate these relations. In particular, we relate zeros of Selberg zeta functions to automorphic distributions. Everything here is, basically, a consequence of the fact that the same Lie group appears as a group of isometries and as a conformal group. The geometric framework of the AdS/CFT-correspondence ([82]) is a Lorentzian analog of this situation. In that case, too, there is one Lie group which appears both as an isometry group of a space and as a conformal group of a suitable completion of it. This is the source of the links of the present theory to theoretical physics. Some observations on Selberg zeta functions and automorphic distributions (Section 1.2) naturally suggest the existence of one-parameter families of diﬀerential intertwining operators for principal series representations of the conformal group. These families are the ﬂat model cases of analogous families in conformal geometry which will dominate the investigation of Q-curvature. There are two main constructions of such families which will be called the residue families (Section 1.6) and the tractor families (Section 1.9). Both constructions generalize diﬀerent aspects of the ﬂat models, and coincide for conformally ﬂat metrics (Section 1.10). The main results of the book do not depend on Selberg zeta functions. In fact, their role can be ignored by those readers who are primarily interested in the

2

Chapter 1. Introduction

emerging perspective towards the structure of Q-curvature. On the other hand, the suggestive power of the zeta functions has not yet been exhausted, and similar arguments in other contexts are expected to have interesting consequences.

1.1 Hyperbolic geometry and conformal dynamics Let Bn+1 = x ∈ Rn+1 | |x| < 1 be the unit ball in Rn+1 with the Riemannian metric n+1 4 g= dx2 . 2 2 (1 − |x| ) i=1 i g has constant sectional curvature −1. (Bn+1 , g) is the ball model of hyperbolic geometry. The connected component Gn+1 = SO(1, n + 1)◦ of the identity in SO(1, n+1) acts isometrically on Bn+1 , and Bn+1 can be identiﬁed with the homogeneous space Gn+1 /K n+1 , where K n+1 = SO(n + 1) is the maximal compact subgroup which ﬁxes the origin. For background material on hyperbolic geometry see [206]. The non-compact Lie group Gn+1 acts transitively also on the geodesic boundary S n of Bn+1 , and S n can be identiﬁed with a homogeneous space G/P , where P is the isotropy group of a point. The action of Gn+1 on S n has an additional metric aspect: Gn+1 acts by conformal diﬀeomorphisms. Here a diﬀeomorphism κ of S n is called conformal with respect to the round metric gc , if κ∗ (gc )

and gc

are conformally equivalent, i.e., diﬀer by a nowhere-vanishing function κ∗ (gc ) = e2Φ(κ) gc ,

Φ(κ) ∈ C ∞ (S n ).

The action of Gn+1 on S n gives rise to the spherical principal series representations − nλ g∗ (vol(gc )) c ◦ g∗ , λ ∈ C πλ : g → vol(gc ) on C ∞ (S n ). Here g∗ denotes push-forward and vol(gc ) is the Riemannian volume form of gc . Alternatively, πλc can be identiﬁed with a representation which is induced by a character of the minimal parabolic subgroup P . For the role and the structure of principal series representations in harmonic analysis we refer to [156]. The fact that the same Lie group is an isometry group and a conformal group has numerous consequences. One of these is that the intertwining operator for spherical principal series πλc can be interpreted as a scattering operator of hyperbolic space. More precisely, for (λ) < − n2 the integral operator 1 ω(y) vol(gc ) Iλ : ω → 2λ+2n |x−y| n S

1.1. Hyperbolic geometry and conformal dynamics

3

is well deﬁned and satisﬁes the intertwining relation c Iλ ◦ πλc (g) = π−λ−n (g) ◦ Iλ

(1.1.1)

for all g ∈ Gn+1 . Iλ admits a meromorphic continuation to C with simple poles in − n2 + N0 . Its residue R2N = Resλ=− n2 +N (Iλ ), N ∈ N0 is a diﬀerential operator which satisﬁes c c n n R2N ◦ π− (g) = π− (g) ◦ R2N , g ∈ Gn+1 . 2 +N 2 −N

More explicitly, R2N is a constant multiple of the operator n 2 +N −1

(Δgc −j(n−1−j)) .

(1.1.2)

j= n 2

On the other hand, an eigenfunction u ∈ C ∞ (Bn+1 ) of the Laplace-Beltrami operator of the hyperbolic metric g, i.e., a solution of −ΔBn+1 u = λ(n−λ)u, (λ) =

n n , λ = , 2 2

has an asymptotics of the form u(r, b) ∼ (1−r2 )λ S(λ)(ω)(b) + (1−r2 )n−λ ω(b) + · · · , b ∈ S n for r → 1. u is determined by its boundary value ω. The quality of ω (smooth function, distribution, hyperfunction) reﬂects the growth of u for r → 1 ([140], [152], [233]). S(λ) is the scattering operator of the space Bn+1 . It is equal to the product (Γ(λ)/Γ(λ − n2 ))Iλ−n , up to a non-vanishing factor. Next, we discuss some aspects of spectral theory on Kleinian manifolds. This illustrates the rich relations between analysis on hyperbolic spaces and conformal invariants on spheres, and serves as a preparation of the considerations in Section 1.2. Let Γ ⊂ Gn+1 be a discrete subgroup without torsion, i.e., without nontrivial elements which are conjugate to elements of the maximal compact subgroup SO(n + 1). Γ acts properly discontinuously on Bn+1 , and the quotient X n+1 = Γ\Bn+1 is a smooth hyperbolic manifold (Kleinian manifold). For n = 2, Γ can be viewed as a discrete subgroup of P SL(2, C), i.e., Γ is a Kleinian group in the classical sense. The dynamics of discrete groups acting on the sphere by conformal transformations is very complex, in general. S n admits a Γ-invariant decomposition S n = Ω(Γ) ∪ Λ(Γ)

4

Chapter 1. Introduction

into the closed limit set Λ(Γ) and its complement Ω(Γ). Γ operates properly discontinuously on the proper set Ω(Γ). The limit set Λ(Γ) is a fractal-like set. Γ is called cocompact if X is compact. In that case, Ω(Γ) = ∅ and Λ(Γ) = S n . Γ is called convex-cocompact if it acts with a compact quotient on the convex hull of Λ(Γ). Then Γ\Ω(Γ) is compact, and the hyperbolic manifold X = Γ\Bn+1 can be compactiﬁed by adding Γ\Ω(Γ). For an account of the complexity and astonishing beauty of limit sets of Schottky groups acting on S 2 we refer to [184]. Associated to any convex-cocompact discrete group Γ there are two closely related objects: the Selberg zeta function ZΓ (λ) and the scattering operator SΓ (λ). The zeta function ZΓ is deﬁned by an Euler product over the prime closed oriented geodesics in X, i.e., by the prime periodic orbits of the geodesic ﬂow Φt of X on the unit tangent bundle SX. Let

(1.1.3) ZΓ (λ) = det id −S N (Pc− )e−λ|c| , (λ) > δ(Γ). p.p.o. c N ≥0

In (1.1.3), the ﬁrst product runs over the prime periodic orbits c of Φt . For any periodic orbit c, Pc− denotes the corresponding linear Poincar´e map on the stable tangent bundle T − (SX) ⊂ T (SX). S N (·) denotes the induced linear map on symmetric tensors. δ(Γ) is the critical exponent of Γ (topological entropy of Φt ). We refer to [153] for the notions of dynamics used here. In the special case of a cocompact Γ ⊂ P SL(2, R) (n = 2), the above deﬁnition reduces to (1 − e−(λ+N )|c| ), (λ) > 1 = δ(Γ). ZΓ (λ) = p.p.o. c N ≥0

Selberg ([213]) introduced and studied this function for coﬁnite Γ. For the details we refer to [139]. In (1.1.3), the zeta function is regarded as an invariant of the geodesic ﬂow on SX. Alternatively, ZΓ can be viewed as being associated to the action of Γ on S n . In fact, prime closed orbits of Φt correspond to attracting ﬁxed points, and the eigenvalues of Pc− and the quantity e−|c| can be read oﬀ from the linearization of the action of Γ at its ﬁxed points. ZΓ is a relative of Ruelle’s zeta function ([104], [209], [151]) ZΓR (λ) = (1 − e−λ|c| )−1 , (λ) > δ(Γ), p.p.o. c

which, in turn, should be viewed as an analog of Riemann’s zeta function ([230], [197]) (1 − p−λ )−1 , (λ) > 1. ζ(λ) = p

For a cocompact Γ, we have δ(Γ) = n, and ZΓ admits a meromorphic continuation to C. All poles are real (if there are any). ZΓ has inﬁnitely many zeros in

1.1. Hyperbolic geometry and conformal dynamics

5

the critical strip 0 < (λ) < n. All non-real zeros are on the line (λ) = n2 . These zeros have an interpretation in terms of hyperbolic geometry as well as in terms of conformal dynamics. In fact, λ0 ∈ R is a zero of ZΓ iﬀ λ0 (n−λ0 ) is an eigenvalue of the Laplacian −ΔX . Since X is compact, the spectrum of −ΔX consists only of eigenvalues of ﬁnite multiplicity. On the other hand, λ0 ∈ R is a zero of ZΓ iﬀ the space ω ∈ C −∞ (S n ) | πλc 0 −n (γ)ω = ω, γ ∈ Γ of Γ-automorphic distributions is non-trivial. Here πλc denotes the spherical principal series representation on distributions. These characterizations also concern multiplicities. The equivalence of both descriptions of non-real zeros follows from Helgason’s theory of Poisson transformations. The notion of automorphic distributions (for coﬁnite discrete groups) was coined in [210]. For beautiful applications to the functional equations of L-functions we refer to [182]. For a convex-cocompact Γ, the volume of X is inﬁnite (if Γ is not cocompact). −ΔX has a continuous spectrum [n2 /4, ∞) of inﬁnite multiplicity and a ﬁnite discrete spectrum in [0, n2 /4]. The scattering operator SΓ (λ) arises by averaging Iλ−n over Γ. It describes the asymptotics of generalized eigenfunctions (Eisenstein series). SΓ (λ) is a meromorphic family of pseudo-diﬀerential operators which acts on functional spaces on the compact manifold Γ\Ω(Γ). The poles of Iλ−n in n2 +N0 lead to poles of SΓ in n2 + N. ZΓ admits a meromorphic continuation to C. The zeros of ZΓ can be characterized in two diﬀerent manners. In order to simplify the discussion, we restrict here to non-real zeros. On the one hand, such zeros of ZΓ are related to non-real poles of the scattering operator SΓ . Such a characterization is natural if one regards ZΓ as an object in the spectral geometry of the Kleinian manifold Γ\Bn+1 . The following description, however, is more natural from the point of view of the dynamics of the action of Γ on the sphere S n : zeros of ZΓ correspond to non-trivial spaces c (γ)ω = ω, γ ∈ Γ (1.1.4) ω ∈ C −∞ (Λ(Γ)) | πλ−n of Γ-automorphic distributions on S n with support in the limit set Λ(Γ). In fact, the limit set Λ(Γ) is the closure of the set of all (attracting) ﬁxed points, and the local factors of the zeta function are deﬁned by quantities which describe the behaviour of the action at these ﬁxed points. The celebrated Patterson-Sullivan measure μP S is a canonical Γ-automorphic distribution with support in Λ(Γ). It satisﬁes the conformal invariance c (γ)μP S = μP S , γ ∈ Γ, πδ(Γ)−n

and is responsible for the simple zero of ZΓ at δ(Γ). For a convex-cocompact Kleinian group Γ, the set of all λ ∈ C for which the space (1.1.4) is non-trivial, is discrete, and deﬁnes a notion of a spectrum of such a Kleinian group Γ, viewed as an object of dynamics on the sphere.

6

Chapter 1. Introduction

For background information and details we refer to [50], [51], [26], [28], [148], [151], [163], [188], [196], [195], [201], [206], [224], [225], [226] and the references therein. The relation between divisors of zeta functions and automorphic distributions will be important in Section 1.2. Note that the attempts to extend the relations of hyperbolic geometry of Kleinian manifolds to conformally invariant structures on the sphere, to the setting of rational iteration on the Riemann sphere, are known under the name of Sullivan’s dictionary ([183], Chapter 5). In particular, the analogs of the Kleinian limit sets are the Julia sets of rational maps, Patterson-Sullivan measures (see [196]) are replaced by conformal measures (see [232] and the references therein), and hyperbolic laminations serve ([169]) as analogs of the hyperbolic 3-space. In the present book, the spectral geometry of Kleinian manifolds is used as a source of inspiration in another direction. We forget the dynamics but retain the geometry of Kleinian manifolds at inﬁnity, at least asymptotically. Convexcocompact Kleinian groups Γ give rise to Kleinian manifolds which can be seen as asymptotically hyperbolic Einstein manifolds with natural conformal compactiﬁcations (in the sense of [199]). The fundamental role of such spaces for conformal geometry was discovered in [99] (see also [96]). More recently, the constructions in [99] proved to be important also in connection with the so-called AdS/CFT-correspondence [171]. The original geometric setting of the AdS/CFT-correspondence was the conformal completion of the Lorentzian analog AdSn+1 of the real hyperbolic space Bn+1 . The universal covering of the anti de Sitter space AdSn+1 is a simply connected complete Lorentzian manifold of curvature −1 which has Einstein’s static universe Einn = (R × S n−1 , −dr2 + gS n−1 ) as a conformal boundary. O(2, n) operates by isometries on AdSn+1 and by conformal diﬀeomorphisms on the quotient S 1 × S n−1 with the metric −gS 1 + gS n−1 . Geometric questions concerning the dynamics of discrete groups Γ ⊂ O(2, n) and the conformal completions of other anti de Sitter space times were the subject of recent studies (see [102], [103]).

1.2 Automorphic distributions and intertwining families Let H + be the upper hemisphere of S n ⊂ Rn+1 (with its standard metric gc of curvature 1) with boundary S n−1 . In the present section, we use the relation between automorphic distributions and zeros of Selberg zeta functions to suggest the existence of certain polynomial families of diﬀerential intertwining operators C ∞ (S n ) → C ∞ (S n−1 ).

1.2. Automorphic distributions and intertwining families

7

Let Γ ⊂ Gn be a discrete torsion-free cocompact subgroup acting on the hyperbolic space Bn = Hn . The action of Gn on Hn extends to an action on the compactiﬁcation Hn = Hn ∪ S n−1 . Here S n−1 = ∂∞ (Hn ) is the geodesic boundary of Hn . Gn acts by conformal transformations on (S n−1 , gc ) (see the discussion in 1.1). By cocompactness of Γ, the limit set Λ(Γ) of its action on Hn coincides with S n−1 . Now consider S n−1 and Hn as the respective submanifolds of S n and Hn+1 deﬁned by the intersection with the plane xn+1 = 0. Then Gn is the subgroup of Gn+1 which preserves the subsphere S n−1 . If Γ is regarded as a subgroup of Gn+1 , it operates with an inﬁnite volume quotient space X n+1 on Hn+1 . Γ is then convex-cocompact. The limit set of its action on Hn+1 coincides with the limit set of its action on Hn . The sphere S n decomposes as S n = Λ(Γ) ∪ Ω(Γ) = S n−1 ∪ Ω+ (Γ) ∪ Ω− (Γ), where Γ operates properly discontinuously on the proper set Ω(Γ). The quotient manifold X n+1 = Γ\Hn+1 is a hyperbolic cylinder with cross section X n = Γ\Hn . We consider the geodesic ﬂows of X n and X n+1 on the corresponding sphere bundles. The geodesic ﬂow of a Riemannian manifold X is Hamiltonian on the cotangent bundle T ∗ X and hyperbolic (or Anosov) on the sphere bundle SX of X if the curvature is negative. All periodic orbits of the geodesic ﬂow of X n+1 are periodic orbits of the ﬂow of X n . In both cases, the periodic orbits give rise to a Selberg zeta function det(id −S N ((Pcm−1 )− )e−λ|c| ), (λ) > δ(Γ) Z m (λ) = p.p.o. c N ≥0

(m = n+1 and m = n). Although both zeta functions Z n+1 and Z n are deﬁned by the same set of prime periodic orbits, they do not coincide since the local factors of the Euler products contain the diﬀering contracting parts (Pcn−1 )− and (Pcn )− of the corresponding linear Poincar´e maps Pcn−1 and Pcn of periodic orbits on the sphere bundles. Here n − 1 and n indicate the dimensions of the corresponding stable tangent bundles. However, the identity m Z 1 −λ|c| (λ) = |c0 | , (λ) > δ(Γ) (1.2.1) m−1 − e Zm det(id −(P ) ) c p.o. c (here the sum runs over all periodic orbits, and c0 denotes the prime periodic orbit of c) and the relation (Pcn )− = (Pcn−1 )− ⊕ e−|c| id imply the factorization Z n+1 (λ) =

N ≥0

Z n (λ+N ).

(1.2.2)

8

Chapter 1. Introduction

Now Z n admits a meromorphic continuation to C. A streamlined proof of that fundamental fact can be built on a version of an Atiyah-Bott-Lefschetz ﬁxed point formula for the geodesic ﬂow of X n ([151]) by interpreting the right-hand side of (1.2.1) as an analog of the right-hand side of the Atiyah-Bott-Lefschetz formula ([13], [14]) for a diﬀeomorphism of a closed manifold. In that argument, Osborne’s fundamental character formula ([138]) plays a central role. Poles of Z m may appear only on the real line (at non-positive integers). The zeros of Z n oﬀ the real line are on the line (λ) = n−1 2 (Riemann hypothesis). They n 2 are of spectral nature: n−1 + is is a zero of Z iﬀ s2 + ( n−1 2 2 ) is an eigenvalue n+1 of the non-negative Laplacian −ΔX n . (1.2.2) implies that Z also admits a meromorphic continuation to C. That is actually only a very special case of the same result for a general convex-cocompact Γ ([195]). More precisely, any non-real n zero n−1 2 + is of Z generates a ladder n−1 + is − N0 2 of zeros for Z n+1 . These zeros can be interpreted as poles of the scattering operator (scattering resonances) of the inﬁnite-volume space X n+1 . From the dynamical point of view ([151]), it is more natural to regard the occurrence of a (non-real) zero λ of Z n+1 as a consequence of the existence of a non-trivial space c C −∞ (Λ(Γ))Γλ−n = ω ∈ C −∞ (S n ) | supp(ω) ⊂ Λ(Γ), πλ−n (γ)ω = ω, γ ∈ Γ of Γ-automorphic distributions on S n which are supported on the limit set of Γ (as opposed to a space of (generalized) eigenfunctions of the Laplacian on X n+1 ). An analogous complete characterization of the divisor of Z n+1 (i.e., including a description of the divisor on the real line) requires us to consider Γ-cohomology with values in spaces of distributions ([50], [151]). The above correspondence between non-real zeros of Z n+1 and automorphic distributions on the limit set Λ(Γ) = i(S n−1 ), of course, generalizes a correspondence between non-real zeros of Z n and Γ-automorphic distributions on S n−1 . Now the ladder structure of the set of non-real zeros of Z n+1 suggests that the Γ-automorphic distributions on Λ(Γ) = i(S n−1 ) arise as the result of the application of a series of universal, i.e., Γ-independent, Gn -equivariant families of local operators C −∞ (S n−1 )λ → C −∞ (Λ(Γ))λ−N −1 , N ≥ 0 to the Γ-automorphic distributions on S n−1 . Here the subscripts refer to the respective module structures deﬁned by the spherical principal series representations πλc on S n and S n−1 . Moreover, by taking adjoints, the existence of Gn -equivariant operator families c (λ) : C ∞ (S n )λ → C ∞ (S n−1 )λ−N DN which are compositions of diﬀerential operators on S n and S n+1 (and the restriction i∗ ), is suggested.

1.3. Asymptotically hyperbolic Einstein metrics. GJMS-operators

9

c The structure of the intertwining families DN (λ) will be clariﬁed in Section 1.4. It will be an important observation that these families can be described in terms of the intertwining operators P2N (gc ) (see (1.1.2)) on both spheres S n and S n−1 . In order to put this into the appropriate perspective, and as a preparation for the later sections, we continue in Section 1.3 with the introduction of the conformally covariant powers of the Laplacian (GJMS-operators) in full generality.

1.3 Asymptotically hyperbolic Einstein metrics. Conformally covariant powers of the Laplacian For a convex-cocompact Γ, the hyperbolic metric on X n+1 = Γ\Bn+1 has a secondorder pole at inﬁnity in the following sense. The complement of a compact convex core X0 of X is isometric to (0, ∞) × Σ (Σ ⊂ X a suitable hypersurface) with a metric of the form g = dt2 + e−2t h− + h0 + e2t h+ . Here h+ is a metric on Σ. It can be viewed as a metric on Γ\Ω(Γ). Substituting r = e−t , we ﬁnd that r2 g extends smoothly to r = 0, and pulls back to a conformally ﬂat metric on Γ\Ω(Γ). In other words, the metric g admits a conformal compactiﬁcation. The Feﬀerman-Graham construction of the Poincar´e-Einstein metric ([99], [96]) can be regarded as kind of an inverse. It associates to a given Riemannian manifold (Σ, h) a metric g = r−2 (dr2 + hr ) with a second-order pole on (0, ε) × Σ so that h arises by conformal compactiﬁcation of g: i∗ (r2 g) = h. Moreover, conformally equivalent metrics on Σ lead to diﬀeomorphic Poincar´e-Einstein (and ambient) metrics. Note that here we pass from the consideration of conformal diﬀeomorphisms (as in the previous sections) to the consideration of conformal classes [h] of metrics. The Poincar´e-Einstein metric and the equivalent Lorentzian Ricci ﬂat ambient metric are basic tools for systematic constructions of conformally invariant objects. One such construction is that of the conformally covariant powers of the Laplacian in [124]. For further results see [121]. In the case Σ = S n with the conformal class [hc ] of the round metric, the ambient metric is the usual Lorentzian metric −dx20 + dx21 + · · · + dx2n+1 on R1,n+1 which pulls back to hc on S n = x ∈ R1,n+1 | − x20 + · · · + x2n+1 = 0, x0 = 1 . The Lorentzian metric induces a metric on the hypersurface −x20 +x21 +· · ·+x2n+1 = −1, and the latter space is isometric to the ball model of hyperbolic geometry.

10

Chapter 1. Introduction

In the general case, the Lorentzian metric is replaced by the ambient metric, and the analog of the hyperbolic metric is the Poincar´e-Einstein metric gE = r−2 (dr2 +hr ) on the space (0, ε) × Σ. It is a solution of the Einstein equation Ric(g) + ng = 0 for small r. Here hr denotes a path of metrics on Σ so that h0 is the given metric h. gE is asymptotically hyperbolic in the sense that its sectional curvatures tend to −1 for r → 0. The basic facts on the spectrum of the Laplacian on general asymptotically hyperbolic manifolds were established in the pioneering works [179], [180] (see [181] for an overview of geometric scattering theory). For Poincar´e-Einstein metrics, geometric scattering theory yields a scattering operator S(h; λ) on C ∞ (Σ) which displays the transformation formula S(e2ϕ h; λ) = e−λϕ ◦ S(h; λ) ◦ e(n−λ)ϕ

(1.3.1)

under conformal changes h → e2ϕ h of the metric. S(h; λ) generalizes the scattering operator SΓ (λ) of convex-cocompact Kleinian groups (Section 1.1). Similarly as SΓ , for even n, the scattering operator S(h; λ) has simple poles in the set n2 , n2 +1, . . . , n (under some mild assumption), and its residues are constant multiples of conformally covariant powers P2N (h) of the Laplacian on Σ. (1.3.1) implies that n n (1.3.2) P2N (e2ϕ h) = e−( 2 +N )ϕ ◦ P2N (h) ◦ e( 2 −N )ϕ . (1.3.1) is a far-reaching generalization of the intertwining relation (1.1.1). In particular, the operator P2N (h) generalizes (1.1.2), and is a modiﬁcation of the power N ΔN g of the Laplacian by lower order terms, i.e., P2N (h) is of the form Δh + LOT . Here we use the following terminology. A natural diﬀerential operator on functions on a smooth manifold M is a rule g → D(M ; g) that associates to any Riemannian metric g on M a linear diﬀerential operator on C ∞ (M ) which is given locally by a universal polynomial formula in terms of the metric g, its inverse, the Levi-Civita connection and the curvature tensor using tensor products and contraction. A natural diﬀerential operator D(M ; g) on C ∞ (M ) is called conformally covariant of weight (a, b) if D(M ; e2ϕ g) = e−bϕ ◦ D(M ; g) ◦ eaϕ ,

(1.3.3)

for all ϕ ∈ C ∞ (M ) and certain conformal weights a, b. In that sense, the GJMSoperator P2N is conformally covariant of weight ( n2 − N, n2 + N ). The original construction ([124]) of the operators P2N rests on the FeﬀermanGraham ambient metric. In the following, these operators will be called GJMSoperators. The above relation to scattering theory was established in [128]. For more details we refer to Chapter 3. In a sense, the most important element in the sequence of GJMS-operators is the critical GJMS-operator Pn on M n . Its order coincides with the dimension of

1.4. Intertwining families

11

the underlying manifold, and its signiﬁcance comes from the fact that it appears in the transformation formula for the critical Q-curvature (Section 1.6). The ﬁrst two GJMS-operators P2 and P4 are well known. P2 is the conformal Laplacian or Yamabe operator Y = P2 = Δ −

n−2 τ 4(n−1)

(1.3.4)

and P4 is also known as the Paneitz operator P4 = Δ2 + δ ((n−2)Jg −4P) #d +

n−4 n 2 J −2|P|2 −ΔJ . 2 2

(1.3.5)

For the notation we refer to Section 2.5. For even n, and general metrics, the operators P2N exist for 2N ≤ n. However, the construction is obstructed at 2N = n. This reﬂects the fact that the very existence of conformally covariant operators of the form ΔN + LOT is obstructed at 2N = n. In fact, Graham ([118]) showed that there exists no natural conformally covariant modiﬁcation of Δ3g on manifolds of dimension 4. Later Gover and Hirachi ([112]) completed the picture by showing an analogous non-existence theorem for 2N > n. However, for speciﬁc conformal classes (as for locally conformally ﬂat metrics) GJMS-operators may exist for all orders ([96], [120]). On the other hand, there are no such obstructions for odd n.

1.4 Intertwining families Chapter 5 will be devoted to a detailed study of polynomial families of diﬀerential intertwining operators C ∞ (S n ) → C ∞ (S n−1 ). A profound understanding of their structure will be the basis of the later discussion of Q-curvature. The structure c (λ) : C ∞ (S n ) → C ∞ (S n−1 ) is inﬂuenced by an of the intertwining families DN underlying algebraic structure and by the curvature of the background metrics. In order to eliminate the inﬂuence of the curvature, we ﬁrst study the analogous intertwining families in the ﬂat case rn−1 → Rn . We shall describe such families in terms of homomorphisms of generalized Verma modules. This will be the key to the construction of curved analogs by applying a version of the curved translation principle. The latter method will yield the so-called tractor families (see Section 1.9). On the other hand, an interpretation of the families in the cases (Rn , Rn−1 ) and (S n , S n−1 ) in terms of the asymptotic analysis of eigenfunctions of Laplacians of associated conformally compact metrics will lead to the residue families (see Section 1.5). c (λ) in terms of The following result describes the intertwining families DN n n−1 GJMS-operators on S and S (for the round metric gc ). On the round sphere

12

Chapter 1. Introduction

(S n , gc ), the GJMS-operators P2N are given by the explicit formulas n 2 +N −1

P2N (gc ) =

(Δgc −j(n−1−j))

(1.4.1)

j= n 2

for all N ≥ 1 (if n is even) ([23], [31], [32], [110], [120]). In Section 1.1, we have seen that these operators are intertwining operators for spherical principal series representations of Gn+1 . We brieﬂy recall the relation between the conformal covariance and the intertwining property. For g ∈ Gn+1 , we have g∗ (gc ) = Φ2g gc for some non-vanishing Φg ∈ C ∞ (S n ). The conformal covariance of P2N states, in particular, that n

n

(Φg ) 2 +N ◦ P2N (g∗ (gc )) ◦ (Φg )− 2 +N = P2N (gc ), i.e.,

n

n

(Φg )− 2 −N ◦ P2N (gc ) ◦ (Φg ) 2 −N = g∗ ◦ P2N (gc ) ◦ g ∗ ,

i.e.,

n

n

P2N (gc ) ◦ ((Φg ) 2 −N g∗ ) = ((Φg ) 2 +N g∗ ) ◦ P2N (gc ).

The latter identity can be written in the form c c n n P2N (gc ) ◦ π− (g) = π− (g) ◦ P2N (gc ), 2 +N 2 −N

where

πλc (g)

=

g∗ (vol(gc ) vol(gc )

− nλ

(1.4.2)

◦ g∗ = Φ−λ g ◦ g∗ .

c (λ) is The main result on the intertwining families DN

Theorem 1.4.1. (i) For any N ∈ N0 , there exists a polynomial family c D2N (λ) = (−1)N λ2N i∗ + A2N −1 λ2N −1 + · · · + A1 λ + A0

(1.4.3)

of diﬀerential operators C ∞ (S n ) → C ∞ (S n−1 ) of order 2N which satisﬁes the intertwining relations c c c πλ−2N (g) ◦ D2N (λ) = D2N (λ) ◦ πλc (g), g ∈ Gn , λ ∈ C

for the spherical principal series on S n and S n−1 . The operator coeﬃcients in (1.4.3) are compositions of the GJMS-operators P2j (S n ) and P2j (S n−1 ) for j = 1, . . . , N , and the restriction i∗ . For n−1 n−1 λ ∈ − +N, . . . , − +2N −1 , 2 2

1.4. Intertwining families

13

c the normal order of D2N (λ) is 2N . If

λ=−

n−1 +N +k−1, k = 1, . . . , N, 2

c c then the normal order of D2N (λ) is 2k−2. In particular, D2N (− n−1 2 +N ) has n−1 normal order 0, i.e., is tangential to S . (ii) For any N ∈ N0 , there exists a polynomial family c N 2N +1 c D2N D1 + B2N λ2N + · · · + B1 λ + B0 +1 (λ) = (−1) λ

(1.4.4)

of diﬀerential operators C ∞ (S n ) → C ∞ (S n−1 ) of order 2N +1 which satisfy the intertwining relations c c c c n πλ−2N −1 (g) ◦ D2N +1 (λ) = D2N +1 (λ) ◦ πλ (g), g ∈ G .

Here D1c (λ) = D1 (S n , S n−1 ; gc ; λ) = i∗ ∇N (gc ) . The operator coeﬃcients in (1.4.4) are compositions of the GJMS-operators P2j (S n ), P2j (S n−1 ) for j = 1, . . . , N and D1c . For n−1 n−1 +N +1, . . . , − +2N , λ ∈ − 2 2 c the normal order of D2N +1 (λ) is 2N +1. If

λ=−

n−1 +N +k, k = 1, . . . , N, 2

n−3 c c then the normal order of D2N +1 (λ) is 2k−1. In particular, D2N +1 (− 2 +N ) has normal order 1.

Note that D0c (λ) = i∗ and D1c (λ) = D1 (S n , S n−1 ; gc ; λ) = i∗ ∇N (gc ) . From a more general point of view, this is a special case of (1.8.7) since S n−1 → S n is a totally geodesic embedding, and the second fundamental form of the submanifolds S n−1 vanishes. We emphasize again that in Theorem 1.4.1 equivariance is to be understood with respect to Gn and not with respect to Gn+1 . The proof of Theorem 1.4.1 uses, among other things, the following result. It provides a set of factorization identities which can be used to determine the families recursively. Analogous arguments will play a central role in the theory of residue families (see Section 1.7). c (λ) of even order satisfy the identities Theorem 1.4.2. The families D2N n

n

c c n − +j = D2N D2N −2j − −j P2j (S ) 2 2

and c D2N

n−1 n−1 c 2N −j − 2N −j − = P2j (S n−1 )D2N −2j 2 2

(1.4.5)

(1.4.6)

14

Chapter 1. Introduction

for j = 0, . . . , N . In particular, for j = N , we have the relations

n c D2N − +N = i∗ P2N (S n ), 2 n−1 c D2N − +N = P2N (S n−1 )i∗ . 2 c Similarly, the families D2N +1 (λ) of odd order satisfy the identities n

n

c c n D2N +1 − +j = D2N −2j+1 − −j P2j (S ) 2 2

and c D2N +1

n−1 n−1 n−1 c )D2N +1−2j 2N +1−j − 2N +1−j − = P2j (S 2 2

for j = 0, . . . , N . In particular, we have the relations

n c c n D2N +1 − +N = D1 P2N (S ), 2 n−3 c D2N +N = P2N (S n−1 )D1c . +1 − 2 Note that, for even order families, the special values for which the families factorize belong to the two non-intersecting intervals

n n n−1 n−1 − +N, − +2N . and − , − +N 2 2 2 2 Similarly, for odd order families, these special values are contained in the disjoint intervals

n n n−3 n−3 − , − +N and − +N, − +2N . 2 2 2 2 Theorem 1.4.2 can be applied as follows. In order to ﬁnd an explicit formula c (λ), we use the fact that it is given by a polynomial of degree 2N with leadfor D2N ing term (−1)N λ2N i∗ (Theorem 1.4.1/(i)). Therefore, it is enough to determine the remaining 2N operator coeﬃcients A0 , . . . , A2N −1 . Now the 2N non-trivial identic (λ) by using the known families ties (1.4.5), (1.4.6) can be used to determine D2N c c D2 (λ), . . . , D2N −2 (λ). A similar procedure applies to the odd order families. For low orders, it is easy to derive explicit formulas. Here we give such forc (λ) of order ≤ 3. The families of even order 0 and 2 are mulas for the families DN c ∗ D0 (λ) = i and D2c (λ) = −(2λ+n−3)i∗P2 (S n ) + (2λ+n−2)P2(S n−1 )i∗ n−2 n−3 ∗ − λ+ λ+ i . 2 2 This formula for D2c (λ) follows from the recursive relations.

(1.4.7)

1.4. Intertwining families

15

For odd orders 1 and 3, we have D1c (λ) = i∗ grad(H0 ) = i∗ ∂/∂n (this operator does not depend on λ since H = 0) and 1 1 D3c (λ) = − (2λ+n−5)D1c P2 (S n ) + (2λ+n−2)P2(S n−1 )D1c 3 3 n−5 n−2 − λ+ λ+ D1c . 2 2

(1.4.8)

Theorem 1.4.1 and Theorem 1.4.2 both follow from algebraic results using an induction construction. In fact, all operators are induced by homomorphism of generalized Verma modules. In order to formulate the result, we introduce a bit more notation. Let gm be the Lie algebra of Gm = SO(1, m)◦ . Let pm ⊂ gm be a parabolic subalgebra. We have the decompositions + gm = n − m ⊕ mm ⊕ a ⊕ nm

p m = mm ⊕ a ⊕

n+ m

(triangle decomposition), (Langlands decomposition).

Here a is abelian and 1-dimensional. The abelian subalgebras n± m have dimension m − 1. a operates by ad(X) = ±α(X) on n± . Each λ ∈ C deﬁnes a character ξλ of pm on C(λ) by ξλ (X) = λα(X), X ∈ a (ξλ operates by 0 on m ⊕ n+ ). Let U(gm ) be the universal enveloping algebra of gm . Now let Iλ (gm ) ⊂ U(gm ) ⊗ C(λ) be the left U(gm )-ideal which is generated by the elements X ⊗ 1 − 1 ⊗ ξλ (X)1 ∈ U(gm ) ⊗ C(λ), X ∈ pm . We consider the left U(gm )-module Mλ (gm ) = (U(gm ) ⊗ C(λ))/Iλ (gm ). Mλ (gm ) is a generalized Verma module for gm . It is generated by the highest weight 1 ⊗ 1. Theorem 1.4.3. For any N ∈ N0 , there exists a polynomial family of elements DN (λ) ∈ U(n− n+1 ) so that the map U(gn ) ⊗ C(λ−N ) T ⊗ 1 → i(T )DN (λ) ⊗ 1 ∈ U(gn+1 ) ⊗ C(λ) induces a family of homomorphisms DN (λ) : Mλ−N (gn ) → Mλ (gn+1 ) c (λ). Here i : U(gn ) → U(gn+1 ) is of U(gn )-modules. DN (λ) induces the family DN induced by the canonical inclusion i : gn → gn+1 .

The latter result (Theorem 5.1.5) is not only an existence result. Its proof yields explicit formulas for the families DN (λ). In terms of the basis {Yj− } of n− n+1 , we have the explicit formula − 2N −2 N + · · · + (Δ− D2N (λ) = a0 (λ)(Yn− )2N + a1 (λ)Δ− n−1 (Yn ) n−1 ) ,

16

Chapter 1. Introduction

where Δ− n−1 = aj (λ) =

n−1

− 2 j=1 (Yj )

and

N −1 N! (−2)N −j (2λ−4N +2k+n+1), j = 0, . . . , N −1. (1.4.9) j!(2N −2j)! k=j

The coeﬃcients of D2N (λ) can be recorded in an associated polynomial which turns out to be a classical orthogonal polynomial. In fact, the polynomial a0 (λ)x2N − a1 (λ)x2N −2 ± · · · + (−1)N x0 −λ− n−1

in x coincides with the Gegenbauer polynomial C2N 2 (x), up to a rational coeﬃcient in λ ([21], 3.15). Similarly, for odd N , we ﬁnd − 2N −1 N − D2N +1 (λ) = b0 (λ)(Yn− )2N +1 + b1 (λ)Δ− + · · · + (Δ− n−1 (Yn ) n−1 ) Yn ,

where bj (λ) =

N −1 N! (−2)N −j (2λ−4N+2k+n−1), j = 0, . . . , N−1. (1.4.10) j!(2N −2j +1)! k=j

In that case, the associated polynomial b0 (λ)x2N +1 − b1 (λ)x2N −1 ± · · · + (−1)N x −λ− n−1

is related to the Gegenbauer polynomial C2N +1 2 (x). The intertwining maps DN (λ) identify the U(gn )-module Mλ−N (gn ) with a submodule of Mλ (gn+1 ). This result suggests that we ask for a description of the decomposition of Mλ (gn+1 ) under the action of U(gn ). Here it seems natural to expect the multiplicity-free branching rule Mλ−N (gn ). (1.4.11) Mλ (gn+1 ) N ≥0

The homomorphisms in Theorem 1.4.3 induce families of diﬀerential operators C ∞ (Rn ) → C ∞ (Rn−1 ) which intertwine the non-compact models of the spherical principal series on Rn and Rn−1 . The even order families are given by the formula N 2N −2j nc D2N (λ) = aj (λ)ΔjRn−1 i∗ (∂/∂xn ) , (1.4.12) j=0

n−1 x → (x , 0) ∈ Rn and ΔRn−1 = j=1 ∂ 2 /∂x2i . In that case the where i : R background metric is ﬂat. The polynomial degree of these families coincides with the polynomial degree of the families DN (λ). It equals N for D2N (λ). This is in contrast to the case of the hemisphere, where the corresponding polynomial degree is 2N . We interpret the higher order of the polynomials in the latter case as an eﬀect of the non-trivial curvatures of the metrics of the two-sphere. n−1

1.5. The residue method for the hemisphere

17

1.5 The residue method for the hemisphere In Section 1.4, we have seen that certain families of homomorphisms of Verma modules are responsible for the existence of intertwining families of diﬀerential operators. In Section 1.9, this will be the starting point for an application of a version of Eastwood’s curved translation principle. It will enable us to construct curved analogs of the intertwining families. In the present section, we describe a second construction of the intertwining c (λ) : C ∞ (S n ) → C ∞ (S n−1 ). It rests on the asymptotic analysis of families DN eigenfunctions of Laplacians for the hyperbolic metric. That method will ﬁnd a far-reaching generalization in Section 1.6 in terms of the residue families. Although the present situation does not fall into the framework of residue families, the discussion here serves as an illustration of the idea of the construction of residue families. The ﬁrst step is to recognize that there is a canonical Gn -invariant metric gN on Ω(Γ) = H + ∪ H − so that (Ω(Γ), gN ) is isometric to two copies of hyperbolic n-space. The intrinsic deﬁnition of the Γ-invariant metric gN on Ω(Γ) for any geometrically ﬁnite Kleinian groups Γ is due to Nayatani ([185]). Its Gn -invariance and the identiﬁcation of Ω(Γ) with (two copies of) hyperbolic space is a consequence of the special situation here. An alternative (but not intrinsic) deﬁnition of gN is gN = |H0 |−2 gc , where H0 ∈ C ∞ (S n ) is the restriction of the height function xn+1 . H0 is a deﬁning function for S n−1 ⊂ S n . Then gN has constant negative curvature, and we have an isometry (Ω(Γ), gN ) S 0 × (Hn , gc ) {±} × (Hn , gc ). (Ω± (Γ), gN ) then are conformally compact manifolds, and gN is a Poincar´e metric with the round metric gc on the boundary S n−1 = ∂Ω± as conformal inﬁnity (see [128] and Chapter 3). Now we consider eigenfunctions for the Laplacian ΔN of gN . For simplicity we restrict attention to H + = Ω+ (Γ). Eigenfunctions for ΔN on H + correspond to hyperfunctions on S n−1 via Helgason’s theory of Poisson transformations ([140], [152]). In particular, any f ∈ C ∞ (S n−1 ) gives rise to an eigenfunction −ΔN u = μ(n−1−μ)u, u ∈ C ∞ (H + ) on H + . The boundary function f can be recovered from u via the leading coeﬃcients in its (radial) asymptotics near the boundary. More precisely, the asymptotics of u near the boundary has two leading terms, and all lower order terms are given by certain families of diﬀerential operators on S n−1 acting on these leading terms; the family parameter is the spectral parameter μ. These diﬀerential operators can be determined (at least in principle) by an iterative procedure. For certain values of μ, they reduce to multiples of GJMS-

18

Chapter 1. Introduction

c operators on S n−1 . In order to construct the families DN (λ), we shall combine these families with normal derivatives in a suitable way. Before we describe the method to ﬁnd these compositions, we take a closer n−1 . We introduce coordinates look at the resulting families for the subsphere S n−1 + 2 (ρ, x) → ( 1−ρ x, ρ) ∈ H ⊂ S n . ρ is the height function. In (0, 1) × S these coordinates, dρ2 1 2 + (1−ρ )gS n−1 , gN = 2 ρ 1−ρ2

and the Laplacian ΔN reads (1−ρ2 )ρ2

∂ ρ2 ∂2 ∂ − (n−2)ρ + − 2ρ3 ΔS n−1 . 2 ∂ρ ∂ρ ∂ρ 1−ρ2

We try to ﬁnd a formal eigenfunction u(ρ, x) ∼ ρμ+N aN (f )(x), x ∈ S n−1 . N ≥0

Note that the latter form of the asymptotics of the eigenfunction diﬀers from the usual radial asymptotics often used in harmonic analysis. The condition −ΔN u = μ(n−1−μ)u is equivalent to (μ+N )(μ+N −1)ρμ+N −2aN (f ) (1−ρ2 )ρ2 −2

N ≥0

N ≥0

⎛ +⎝

(μ+N )ρμ+N +2 aN (f ) − (n−2)

m≥1

⎞⎛ ρ2m ⎠ ⎝

(μ+N )ρμ+N aN (f )

N ≥0

⎞

ρμ+N ΔS n−1 (aN (f ))⎠ = −μ(n−1−μ)

N ≥0

ρμ+N aN (f ).

N ≥0

A comparison of the coeﬃcients of the powers of ρ shows that a0 (f ) is free, aodd(f ) = 0, and N (2μ+N −(n−1))aN (f ) = (μ+N −2)(μ+N −1)aN −2(f ) − ΔS n−1 (aN −2 (f ) + · · · + a0 (f )) for even N ≥ 2. Now assume that (μ) =

n−1 2 .

(1.5.1)

Since

2μ+N −(n−1) = 0 for N ≥ 2, the latter formula implies that all coeﬃcients aeven can be determined from a0 (f ) by applying a diﬀerential operator of order 2N . It is a polynomial in the Laplacian ΔS n−1 . Notice that the recursion formula (1.5.1) says that, in order to ﬁnd aN (f ), one needs to know all terms with smaller indices.

1.5. The residue method for the hemisphere

19

Now let u ∈ C ∞ (H + ) be a genuine eigenfunction of ΔN with eigenvalue μ(n−1−μ), (μ) = n−1 2 . It has an asymptotics of the form ρμ+N aN (f ) + ρn−1−μ+N bN (f ). u∼ N ≥0

N ≥0

The eigenfunction u gives rise to a holomorphic family of measures on the halfplane (λ) > −(n−1)/2 by H0λ uϕdb, ϕ ∈ C ∞ (S n ). Mu (λ), ϕ = H+

The support of these measures is contained in H + . Moreover, the restriction of Mu (λ) to the open H + can be identiﬁed with the holomorphic family of functions λ → H0λ u. But holomorphy of Mu (λ), ϕ on C is lost for test functions ϕ ∈ C ∞ (S n ) the support of which intersects with S n−1 . However, thanks to the controlled boundary behaviour of u in terms of its asymptotics, the family Mu (λ) admits a meromorphic continuation to C as a family of distributions. In order to study the continuation, we write 1 n−2 λ+μ+N ρ aN (f )(x)ϕ(ρ, x)dx (1−ρ2) 2 dρ Mu (λ), ϕ = N ≥0

+

N ≥0

S n−1

0

1

ρλ+n−1−μ+N S n−1

0

n−2 bN (f )(x)ϕ(ρ, x)dx (1−ρ2 ) 2 dρ.

n−2

Using a Taylor series for (1 − ρ2 ) 2 , and repeated partial integration in ρ, we construct a meromorphic continuation of Mu (λ) with simple poles in the disjoint ladders λ ∈ −μ−1−N0, λ ∈ −(n−1−μ)−1−N0. The respective residues are linear combinations of terms of the form j j ∂ ∂ i∗ ak (f ), i∗ bk (f ) ∈ C −∞ (Λ(Γ)). ∂ρ ∂ρ n

But the property H0 ∈ C ∞ (S n )G implies that the maps 1 C ∞ (S n−1 ) f → Res−μ−1−N (Mu (λ)) ∈ C −∞ (Λ(Γ)) are Gn -equivariant. Up to a renormalization, these families (in the spectral parameter μ) are the adjoint families of the Gn -equivariant operator families to be constructed. The above arguments show how to compose tangential families and normal derivatives to equivariant families. The real value of the method, however, is its capacity to generate the residue families in a much more general setting (Section 1.6).

20

Chapter 1. Introduction

1.6 Q-curvature, holography and residue families A construction of central signiﬁcance in spectral geometry is the zeta-regularized determinant or functional determinant of geometric diﬀerential operators such as the Laplace-Beltrami operator on functions and the Hodge-Laplacians on diﬀerential forms. Among other things, it gave rise to the fundamental notion of analytic torsion ([207]), and was shown to have remarkable extremal properties in dimension n = 2 ([191]) as well as in higher dimensions ([31], [48], [37], [72], [189]). The conformal covariance of an operator has essential inﬂuence on the behaviour of its functional determinant, and the work of Branson ([31], [32]) revealed the fundamental role of Q-curvature in this context. Q-curvature Qn of a Riemannian manifold (M, g) of even dimension n is a scalar nth -order Riemannian curvature invariant which satisﬁes the remarkable transformation property n

enϕ Qn (e2ϕ g) = Qn (g) + (−1) 2 Pn (g)(ϕ)

(1.6.1)

under conformal changes of the metric. This identity will be called the fundamental identity. Here Pn is the critical GJMS-operators of M (Section 1.3). On a manifold of dimension n, Qn appears together with its lower order relatives Q2,n , . . . , Qn−2,n . However, in many respects, the quantity Qn,n = Qn is the most important one. It will be called the critical Q-curvature. The critical Q-curvature generalizes Gauß curvature K of a surface and the Paneitz curvature quantity Q4 = 2(J2 − |P|2 ) − ΔJ =

1 −3| Ric |2 + τ 2 − Δτ 6

(1.6.2)

of a four-manifold. For closed M , the fundamental identity (1.6.1) implies that the total Qcurvature Qn vol (1.6.3) Mn

is a global conformal invariant. For n = 2, this is obvious since, by the GaußBonnet formula, the integral is proportional to the Euler characteristic. In dimension 4, the conformal invariance of the integral also follows from the GaußBonnet-Chern formula using the fact that the Pfaﬃan is a constant multiple of J2 − |P|2 + 18 |C|2 . The conformal invariance of the total Q-curvature has strong implications on the structure of Q-curvature itself. The question to describe all scalar local curvature invariants with a conformally invariant total integral is of signiﬁcance in quantum ﬁeld theory (anomalies). Deser and Schwimmer proposed a geometric classiﬁcation of these quantities in [81]. After preparations in [5] and [6], Alexakis accomplished a detailed proof of that classiﬁcation in the monumental work [3].

1.6. Q-curvature, holography and residue families

21

These studies are closely related to conformal anomalies of functional determinants of conformally covariant diﬀerential operators. In a series of works (starting with [45]), Branson and Ørsted explored the relation of the conformal anomaly of the functional determinant to the constant term in the heat kernel asymptotics. In such a context, Branson and Ørsted called the constant term an the conformal index density. The latter name is suggested by the role of analogous quantities in the heat equation approach to index theory ([107]). For closed M , the total integral of the conformal anomaly is conformally invariant. The search for universal structural results for conformal anomalies of determinants led Branson to the idea that the anomalies are deeply linked with Qn ([31]). For more details we refer to the discussion at the end of the present section. Now how is Qn deﬁned? In dimensions n = 2 and n = 4, it is given by the explicit formulas Q2 = K and (1.6.2). But in higher even dimension n, the Q-curvature quantities Q2N,n (2N ≤ n) are deﬁned only implicitly through the GJMS-operators P2N : for 2N < n, the constant term P2N (1) of P2N on an nmanifold has the form n

− N Q2N,n . (−1)N 2 The subcritical Q-curvature Q2N,n is a scalar Riemannian curvature invariant of order 2N . In order to simplify notation, we often write Q2N if the dimension of the underlying space is evident. In the critical case 2N = n, we set Qn = Qn,n . A comment on the nature of these deﬁnitions is in order. These deﬁnitions are extrinsic in the following sense. Q-curvature is derived from GJMS-operators. The GJMS-operators P2N (g) are induced by the powers of the Laplacian of the Feﬀerman-Graham ambient metric associated to g. This is an extrinsic deﬁnition since the interesting object is generated by a construction on a certain ambient space of two dimensions higher. Now (1.3.4) and (1.3.5) show that Q2,n = J and Q4,n =

n 2 J −2|P|2 −ΔJ. 2

In the respective critical cases n = 2 and n = 4, we recover Q2 = τ /2 = K and (1.6.2). Explicit formulas for Q6 and Q8 in terms of Riemannian invariants were derived in [116]. The complexity of such formulas increases exponentially with the order. It is one purpose of the present book to reveal the structure of these quantities. In the following, we apply a version of the residue method of Section 1.5 to deﬁne the so-called residue families. These families will be used to study the structure of Q-curvatures. Note that the residue families, however, do not cover the construction in Section 1.5. The relation of residue families to GJMS-operators and Q-curvature is as follows. For any metric h on M n and any integer N ∈ [1, n2 ], there is a residue

22

Chapter 1. Introduction

family of order 2N which, for a certain value of the family parameter, specializes to the GJMS-operator P2N (h). The conformal covariance of the latter operator is embedded in the conformal covariance of the corresponding residue family. The most interesting one is the critical residue family of order n. It is related to the critical Q-curvature. The critical residue family specializes to the critical GJMS-operators Pn at λ = 0, and the linear term of its Taylor series in λ “sees” Qn . The fundamental identity of Qn is a direct consequence of the conformal covariance of the family. Now let (M, h) be a compact Riemannian manifold of even dimension n. We view (M, h) as the conformal inﬁnity of an associated Poincar´e-Einstein metric g on (0, ε) × M in the normal form g = r−2 (dr2 +hr ), h0 = h with

¯ (n) log r + · · · . hr = h0 + r2 h(2) + · · · + rn h(n) + h

(1.6.4)

The metric g satisﬁes the Einstein condition Ric(g) + ng = 0

(1.6.5)

to high order near the boundary (r = 0). The condition (1.6.5) has the conse¯ (n) and the trace of h(n) are locally quence that the coeﬃcients h(2) , . . . , h(n−2) , h determined by h (see [122] and [96] for the details). This generalizes the case of the hyperbolic space Hn+1 of sectional curvature −1 with (a multiple of) the round metric on the sphere at inﬁnity. For odd n, the analogous series of hr does not contain log r-terms, and all coeﬃcients are determined by h = h0 . For Einstein g, the series (1.6.4) will be referred to as the Feﬀerman-Graham expansion. The following constructions will depend only on those terms in the Taylor series (1.6.4) which are determined by the leading term h0 . Moreover, for even n, ¯ (n) of log r will not play a role. Therefore, we apply the the trace-free coeﬃcient h convention that hr denotes the ﬁnite sum h0 + r2 h(2) + · · · + rn−2 h(n−2) + rn h(n) for some h(n) (with determined trace). This convention will be used throughout without further notice. Now let u ∈ ker(Δg + μ(n−μ)) ((μ) = n2 and μ = n2 ) be an eigenfunction of the Laplacian Δg with the formal asymptotics u∼

j≥0

rμ+2j a2j (h; μ) +

rn−μ+2j b2j (h; μ), r → 0

(1.6.6)

j≥0

with leading term a0 = f ∈ C ∞ (M ). f is called the boundary value of u. We extend the family of functions rλ u, λ ∈ C on (0, ε) × M through the boundary

1.6. Q-curvature, holography and residue families

23

r = 0 to a family of distributions on [0, ε) × M . More precisely, let Mu (λ) be deﬁned by rλ uϕ vol(dr2 +hr ), (λ) 0, ϕ ∈ C0∞ ([0, ε) × M ). Mu (λ), ϕ = [0,ε)×M

Then Mu (λ) admits a meromorphic continuation with simple poles in the ladder −μ−1−N0 (we ignore the analogous simple poles in the ladder −(n−μ)−1−N0 ). The residues are of the form f δN (h; μ)(ϕ) vol(h) Res−μ−1−N Mu (λ), ϕ = M

with certain families of diﬀerential operators δN (h; λ) : C ∞ ([0, ε) × M ) → C ∞ (M ). The coeﬃcients a2j (h; μ), j ≤ n2 are completely determined by h0 , h(2) , . . . , h(n−2) and tr(h(n) ), i.e., by h. Hence the families δN (h; λ), 2N ≤ n are natural in h. The res (h; λ) are certain renormalizations of δN (h; λ). residue families DN The evaluation of the above construction motivates the following deﬁnition (see Deﬁnition 6.6.2 for the general case). Deﬁnition 1.6.1 (Residue families). For even n and 2N ≤ n, let the family of diﬀerential operators res (h; λ) : C ∞ ([0, ε) × M n ) → C ∞ (M n ) D2N

be deﬁned by res (h; λ) = 22N N ! D2N

n

n − −λ+2N −1 · · · − −λ+N δ2N (h; λ+n−2N ) 2 2

with δ2N (h; λ) =

N j=0

∗ 1 2N −2j T (h; λ) ◦ v0 + · · · + T0∗ (h; λ) ◦ v2j ◦ i∗ ◦ (∂/∂r) , (2N −2j)! 2j

where i∗ restricts functions to r = 0. The rational families T2j (h; λ) are determined by the relation T2j (h; λ)f = a2j (h; λ), i.e., the operator T2j (h; λ) maps the leading coeﬃcient in (1.6.6) to the coeﬃcient of rλ+2j . The renormalized families

n

n −λ−1 · · · −λ−j T2j (h; λ) P2j (h; λ) = 22j j! 2 2

24

Chapter 1. Introduction

are holomorphic and satisfy P2j (·; λ) = Δj + LOT . Formal adjoints are deﬁned with respect to the metric h on M . The holographic coeﬃcients v2j ∈ C ∞ (M ) are the Taylor coeﬃcients of the volume function vr (r, ·) =

vol(hr ) = v0 + r2 v2 + · · · + rn vn + · · · , v0 = 1; vol(h)

(1.6.7)

here vol(hr ) denotes the Riemannian volume form of hr . The coeﬃcients v2j (j = 0, . . . , n2 ) in (1.6.7) are given by local formulas in terms of the curvature of the metric h (and its covariant derivatives). The coeﬃcient vn has found much attention in recent years. It is called the holographic anomaly of the asymptotic volume of the Poincar´e-Einstein metric g (see [119] and the discussion below). The relation n

P2N h; −N = P2N 2 and the self-adjointness of P2N imply that

n res (1.6.8) D2N h; − +N = P2N (h)i∗ , 2 i.e., the residue families specialize to GJMS-operators for appropriate values of the family parameters. This should be compared with the formula Res n2 +N (S) = −cN P2N (h) (Theorem 3.2.1). Dnres (h; λ) is called the critical residue family. It depends on the coeﬃcients h0 = h, h(2) , . . . , h(n−2) and the h-trace of h(n) . Since all these terms are determined by h, the family is completely determined by h. For odd n, residue families are deﬁned analogously and (1.6.8) continues to hold true. By parity reasons, there is no critical residue family in that case. It should be emphasized that the residue families are not families in the sense of Section 1.8 since they are deﬁned only for one speciﬁc background metric dr2+hr , which extends h into a neighbourhood. Instead, the critical residue family satisﬁes the transformation formula ∗ λ κ (r) −(λ−n)ϕ res ˆ res e ◦ Dn (h; λ) = Dn (h; λ) ◦ κ∗ ◦ , (1.6.9) r where ˆ h = e2ϕ h, and κ is the diﬀeomorphism which relates the corresponding Poincar´e-Einstein metrics κ∗ r−2 (dr2 +hr ) = r−2 (dr2 + ˆhr ) (1.6.10) ([99], Theorem 2.3, [126], Section 5 and [96], Theorem 4.4). Here we apply the above convention and (1.6.10) is to be understood as an identity up to terms in

1.6. Q-curvature, holography and residue families

25

O(rn−2 ) and vanishing trace tr(r−n+2 ·). κ is induced by the gradient ﬂow of the function ρ with the properties |dρ|2ρ2 g = 1, g = r−2 (dr2 +hr ) and

ˆ i∗ (ρ2 g) = h.

Note that the Taylor-coeﬃcients of κ∗ ◦ (κ∗ (r)/r)λ up to rn are completely determined by h. res (h; λ) under Similar transformation formulas describe the behaviour of D2N conformal changes h → e2ϕ h. The diﬀeomorphisms κ appear in the physics literature also under the name of PBH transformations ([147], [218]). Now (1.6.9) would be a consequence of an identity Dnres (h; λ) = Dn (dr2 +hr ; λ) for some conformally covariant family Dn (g; λ) as in Section 1.8. In fact, by the naturality of Dn (g; λ), Dn (κ∗ (dr2 +hr ); λ) = κ∗ ◦ Dn (dr2 +hr ; λ) ◦ κ∗ . Hence ˆ λ) = Dn (dr2 + ˆ Dnres (h; hr ; λ) 2 r ∗ 2 κ (dr +hr ); λ = Dn κ∗ (r) ∗ λ κ (r) = e(λ−n)ϕ ◦ Dn (dr2 +hr ; λ) ◦ κ∗ ◦ r ∗ λ κ (r) = e(λ−n)ϕ ◦ Dnres (h; λ) ◦ κ∗ ◦ r (see Remark 6.6.3). Now since Dnres (h; λ) satisﬁes the identity Dnres (h; 0) = Pn∗ (h; 0)i∗ = Pn∗ (h)i∗ = Pn (h)i∗ ,

(1.6.11)

it is reasonable to ask whether also n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h).

(1.6.12)

In the following, we analyze the meaning of the relation (1.6.12) and conﬁrm its validity (see Theorem 6.6.1). n In view of P˙n (h; 0)(1) = (−1) 2 Qn (h), (1.6.12) is equivalent to the formula n

2 −1 n n

2(−1) Qn (h) = δn (h)(1) + 2 T2j∗ (h; 0)(vn−2j ), ! −1 ! 2 2 j=0 n 2

n

(1.6.13)

26

Chapter 1. Introduction

where

def δn (h) = P˙n (h; 0)− P˙ n∗ (h; 0)

(Corollary 6.6.1). In terms of the operators P2j (h; 0), that identity reads n

2 −1 n ( n −j −1)! ∗ ! P2j (h; 0)(vn−2j ). (1.6.14) 2n−2j 2 2(−1) Qn (h) = δn (h)(1) + 2 j=0 j! n 2

Note that in (1.6.13) and (1.6.14), the nature of the contribution, which is deﬁned by δn (h) = P˙n (h; 0) − P˙n∗ (h; 0) (acting on 1), diﬀers from the remaining terms. But calculations up to n = 6 show that this contribution can be written also as a linear combination of the other terms. Indeed, this is a general fact (see Theorem 6.6.4). Theorem 1.6.1. For even n, n

2 −1

n n n −1 ! 2jT2j∗ (h; 0)(vn−2j ). ! nδn (h)(1) = −2 2 2 j=0

(1.6.13) and Theorem 1.6.1 imply that (1.6.12) is equivalent to n

n 2

(−1) nQn (h) = 2

n−1

2 −1

n n −1 ! (n−2j)T2j∗ (h; 0)(vn−2j ). ! 2 2 j=0

The latter formula, however, admits an independent proof (see [125] and Theorem 6.6.6). This proves (1.6.12). Theorem 1.6.2 (The holographic formula). The critical Q-curvature Qn (h) of (M n , h) is given by the formula n

n 2

(−1) nQn (h) = 2

n−1

2 −1 n n

! −1 ! (n−2j)T2j∗ (h; 0)(vn−2j ). 2 2 j=0

(1.6.15)

In terms of the operators P2j (0), Theorem 1.6.2 reads n 2

(−1) Qn =

n

2 −1

( n −j)! ∗ −1 ! P2j (0)(vn−2j ). 2n−1−2j 2 2 j! j=0

n

(1.6.16)

Theorem 1.6.2 relates Qn to the coeﬃcients v2j and the asymptotics of harmonic functions. In particular, it emphasizes the signiﬁcance of all holographic coeﬃcients v2j . We continue with the description of some direct consequences of Theorem 1.6.2. Further consequences will be discussed in Section 1.7.

1.6. Q-curvature, holography and residue families

27

Corollary 1.6.1. For closed M n and u ∈ ker(Pn ), the sum n 2 −1

In (u) =

j=0

(n−2j) Mn

vn−2j T2j (0)(u) vol

is conformally invariant. Note that ker(Pn ) is a conformally invariant space. Note also that the statement of Corollary 1.6.1 does not refer to Q-curvature. For compact M , integration of (1.6.15) yields the formula

n n n −1 ! ! 2 Qn vol = (−1) 2 2n vn vol . (1.6.17) 2 2 M M This relation was proved in [128] (see also [97]). It is a relation between two global conformal invariants. The more precise relation (1.6.14), of course, makes sense without the compactness assumption on M . n (1.6.16) shows that (−1) 2 Qn contains the contribution n

∗ (0)(J) = −Δ 2 −1 (J) + LOT. −Pn−2 n

It is well known that Δ 2 −1 (J) contributes in this way to Qn (see [31], Corollary 1.4 or [32], Corollary 1.5). For a conformally ﬂat metric h, the holographic coeﬃcients in (1.6.16) are given by v2j = (−2)−j tr ∧j (P) , where the Schouten tensor P is identiﬁed with the endomorphism P : T M → T M . Hence (Theorem 6.14.1) Theorem 1.6.3. Let (M n , h) be conformally ﬂat. Then Qn =

n

2 −1

( n −j)! ∗ n n −1 ! P2j (0) tr ∧ 2 −j (P) . (−1)j 2 2 −1−j 2 2 j! j=0

n

In particular, n

n

n −1 ! ! tr ∧ 2 (P) + divergence terms 2 2 n = (−1) 2 [(n−2)(n−4) · · · 2] Pf n +divergence terms, n

Qn = 2 2 −1

where Pf denotes the Pfaﬃan form. On the round sphere the divergence terms vanish. The fact that for a conformally ﬂat metric on a closed M n the integral of Qn is proportional to the Euler characteristic of M n goes back to [39]. Theorem 1.6.3 is a local version of that result.

28

Chapter 1. Introduction

Theorem 1.6.2 (or formula (1.6.16)) can be made more explicit for n ≤ 6. In particular, we ﬁnd the formula Q6 = −8 · 48v6 − 32P2∗ (0)(v4 ) − 2P4∗ (0)(v2 )

(1.6.18)

(see (6.10.4)) for the critical Q-curvature in dimension 6. Here 1 v2 = − tr(P), 2

v4 =

1 tr(∧2 P) 4

1 1 and v6 = − tr(∧3 P) − (P, B), 8 48

where B is a version of the Bach tensor. An alternative proof of (1.6.18) rests on the direct evaluation of (1.6.13) for n = 6. Theorem 1.6.2 deals only with the critical Q-curvature. In dimension n ≥ 6, analogous holographic formulas for the subcritical Q-curvatures Q2 , Q4 and Q6 can be summarized in the form ⎛ ⎞ ⎛ ⎞⎛ ⎞ −8 · 48 −32P2∗ ( n2 −3) −2P4∗ ( n2 −3) Q6 v6 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ∗ n 0 16 2P Q ( −2) v = (1.6.19) ⎝ 4⎠ ⎝ ⎠ ⎝ 4⎠ . 2 2 Q2 v2 0 0 −2 For the general case we refer to Conjecture 6.9.1. The relation between subcritical res Q2N and subcritical D2N (− n2 +N )(1) is more complicated than in the critical case. It will be discussed in Section 1.7. We refer to the identities (1.6.14), (1.6.15), (1.6.16) and (1.6.19) as holographic formulas for Q-curvature. These formulas describe Q-curvature in terms of the asymptotic geometry of an associated metric in one higher dimension. A similar terminology is used in physics in connection with the AdS/CFT-correspondence. It proposes relations between quantum ﬁeld theories and gravitational theories in one more dimension (see [171], [241], the reviews [172], [82], [29], [227], [146] and [11] for mathematical aspects in the case of pure gravity). Establishing a dictionary is a central part of the eﬀorts. Such relations between theoretical concepts in diﬀerent dimensions are reminiscent of the technical method of holography ([223]). We describe how the holographic formula (1.6.16) for Q-curvature is linked with ideas around the AdS/CFT-correspondence through the functional determinants of conformally covariant diﬀerential operators. The relations rest on the connection of both quantities Qn and vn to the conformal index densities an of such operators. From the physical perspective, an and vn are interpreted as conformal anomalies in quantum ﬁeld theories and gravitation, respectively. In order to describe the relation between an and vn in Figure 1.1, we recall that Witten ([241]) proposed to formulate the AdS/CFT-correspondence as identities of partition functions. In the case of pure gravity, a prototype of such a relation identiﬁes the partition function ZCF T [h] of an appropriate conformal ﬁeld theory on M (with the background metric h) with the quantity exp(−S[g]) which is given by the value of the Einstein-Hilbert action S[·] on a solution g of the Einstein equation (on a manifold X with boundary M ) with [h] as conformal inﬁnity. More

1.6. Q-curvature, holography and residue families

29

holographic formula / vn Q n bF o y< FF y y FF y y functional determinant FFF yy AdS/CFT-duality " |yy an Figure 1.1: Q-curvature and the conformal anomalies an and vn precisely, let

(τg − 2Λ) vol(g),

S[g] = − X

and let g satisfy the Einstein equation Ric(g) + ng = 0. Then τg = −n(n+1), and g is a solution of the equation Ric(g) −

τ g + Λg = 0 2

for Λ = − n(n−1) , i.e., g is critical for the action S with the cosmological constant 2 Λ. The value of S for such a metric is given by the divergent integral 2n vol(g). X

In order to renormalize the volume, holographic renormalization ([219], [80] and [11]) proceeds as follows. We assume that g is a global conformally compact Einstein metric on X \ ∂X with conformal inﬁnity [h]. A choice of a representing metric h on M gives rise to a deﬁning function r such that |dr|2r2 g = 1 near ∂X, and g can be written in the normal form r−2 (dr2 + hr ). For even n, we ﬁnd an asymptotics vol(g) = c0 ε−n + c2 ε−(n−2) + · · · + cn−2 ε−2 − L log ε + V + o(1) (1.6.20) r≥ε

for ε → 0. The renormalized (or asymptotic) volume of g with respect to h is deﬁned as the constant term V = V (g; h) in this asymptotics, and we set S[g; h] = 2nV (g; h). Although V (g; ·) is not conformally invariant, it satisﬁes ϕvn (h) vol(h) (d/dt)|t=0 V (e2tϕ h) =

(1.6.21)

(1.6.22)

M

for all ϕ ∈ C ∞ (M ), i.e., the holographic anomaly vn describes the failure of V being conformally invariant. Moreover, the functional L = M vn vol is conformally invariant. For the proofs we refer to [119].

30

Chapter 1. Introduction

Now the functional h → exp(−S[g; h]) is to be compared with the partition function of an appropriate conformal ﬁeld theory on M . On the level of conformal anomalies this leads to a comparison of the holographic anomaly vn with the conformal anomalies (1.6.23) (d/dt)|t=0 log det D(e2tϕ h) of the determinants of certain conformally covariant diﬀerential operators D which act on the various types of ﬁelds. Here the determinant is the zeta-regularized determinant, and the basic fact is that (1.6.23) is given by a local formula. These formulas generalize a formula of Polyakov ([203]) for the determinant of the Laplacian in dimension n = 2. We brieﬂy recall the structure of such formulas. Under appropriate technical conditions, for a (power of a) conformally covariant self-adjoint diﬀerential operator D of order m with positive deﬁnite principal symbol, the generalized Polyakov formula states that ϕan (D(h)) vol(h) (1.6.24) −(d/dt)|t=0 log det D(e2tϕ h) = m M ∞

for all ϕ ∈ C (M ). The coeﬃcient an (D) ∈ C ∞ (M ) is taken from the heat asymptotics n−j −tD )∼ t− m ϕaj (D) vol, t → 0. (1.6.25) tr(ϕe j≥0

M

Moreover, the conformal index theorem of Branson and Ørsted states that the integral M an (D) vol is conformally invariant. We call an (D) the conformal index density (of D). (1.6.24) says that the function an (D) describes the conformal anomaly of the determinant of D. For details we refer to [45], [46], [47] and also [31]. Note also that the coeﬃcient an−m (D) ∈ C ∞ (M ) of t−1 in (1.6.25) is conformally invariant ([194], [107], [198], [204], [36]). In dimension n = 4, Henningson and Skenderis ([141]) veriﬁed that the holographic anomaly v4 actually coincides with a speciﬁc linear combination of the conformal anomalies of determinants of diﬀerential operators on the various types of ﬁelds of supersymmetric Yang-Mills theory (for the details see Section 6.15). In order to study extremal properties of determinants, it is important to pass from the inﬁnitesimal Polyakov formulas (1.6.24) to formulas for the quotients ˆ det(D(h))/ det(D(h)).

(1.6.26)

Branson discovered the role of Q-curvature in that process. In fact, one expects ˆ = vol(h) and ker D is trivial) the quotients (1.6.26) can be written that (if vol(h) always in the universal form − log

ˆ det(D) = c(D) det(D)

! + ˆ n vol ϕ Qn vol +Q M

M

! , F vol −Fˆ vol

(1.6.27)

1.6. Q-curvature, holography and residue families

31

where F is a local scalar invariant, and Qn possibly has to be modiﬁed by a local conformal invariant (if h is not conformally ﬂat). A non-trivial kernel of D creates an additional term. Likewise, some more terms appear if the volume is not ﬁxed. For more details we refer to [41] and its references, [43], [31], [35] and [63]. The universal Q-curvature term is generated by the following arguments. We observe that 1 ˆ − log det D(h) = (d/dt) log det D(e2tϕ h) dt log det D(h) 0

1

= −m

ϕan (D(e2tϕ h)) vol(e2tϕ h)dt 0

M

using (1.6.24). Now we split oﬀ a multiple of Qn from an (D) and ﬁnd 1 1 n ϕQn (e2tϕ h)entϕ vol(h)dt = ϕ Qn (h) + (−1) 2 Pn (h)(tϕ) vol(h)dt 0 M 0 M 1 n 2 ϕ Qn (h) + (−1) Pn (h)(ϕ) vol(h) = 2 M

1 ˆ vol(h) ˆ . = ϕ Qn (h) vol(h) + Qn (h) 2 M Thus, in order to organize the terms as in (1.6.27), one needs to know that, up to a local conformal invariant, an (D) diﬀers from a multiple of Qn by a quantity G so that M ϕG(h) vol(h) is the conformal variation of M F (h) vol(h) for some local Riemannian invariant F . Conjecture 1 in [41] states that this is always possible. Any progress here would involve a deeper understanding of the structure of the divergence terms in an (D) and Qn . In that connection it also seems natural to deal with a notion of Q-curvatures which covers more than just Qn in dimension n (see [35]). All Q-curvature quantities in the extended sense admit a conformal transformation law which is governed by a linear diﬀerential operator with leading term a power of the Laplacian. For Qn , the holographic formula provides a description of the divergence contributions in terms of lower order holographic data: v2j for j < n2 . The structure of the divergence contributions to conformal index densities an (D), however, is much less clear. In particular, it is not known whether the lower order heat coeﬃcients play some role here. We close the present section with some comments on the case n = 2. Both v2 and a2 (−Δ) are constant multiples of the Gauß-curvature K. Hence 2(V (e2ϕ h) − V (h)) 1 ϕv2 (e2tϕ h)e2tϕ vol(h)dt = − =2

0 1

=− 0

1

ϕK(e2tϕ h)e2tϕ vol(h)dt 1 ϕ [K(h) − tΔh (ϕ)] vol(h)dt = − ϕ K(h) − Δh (ϕ) vol(h) 2 M M M

0

M

32

Chapter 1. Introduction

ˆ = K(h) − Δh (ϕ), i.e., using e2ϕ K(h) 2(V (e2ϕ h) − V (h))

1 1 ˆ ˆ =− 2K(h)ϕ + |dϕ|2h vol(h). ϕ K(h) vol(h) + K(h) vol(h) = − 2 M 2 M On the other hand, we have the Polyakov formula det(−Δe2ϕ h ) 1 log =− 2K(h)ϕ + |dϕ|2h vol(h) det(−Δh ) 12π M

(1.6.28)

ˆ = vol(h)). For S 2 with the round metric hc , an inequality of Onofri (if vol(h) implies that det(−Δe2ϕ hc ) ≤ det(−Δhc ) ˆ c ) = vol(hc )). Equality holds true exactly for those metrics which are (if vol(h conformally diﬀeomorphic to hc . The renormalized volume is maximal for these metrics. For detailed proofs we refer to [63]. In [229], the authors studied the asymptotic volume of Kleinian manifolds X 3 = Γ\H3 with compact boundary M 2 = Γ\Ω(Γ), and related it to the Liouville action functional (reﬁning [160]). Here the asymptotics of the volume of X 3 is deﬁned by a family of equidistant (Epstein) hypersurfaces which exhaust the ends of X 3 . Epstein used equidistant hypersurfaces some years before, in order to prove that for convex-cocompact Kleinian manifolds X = Γ\Hn+1 of odd dimension the coeﬃcient of log ε in the asymptotics of the volume is proportional to the Eulercharacteristic χ(Γ\Ω(Γ)), Γ\Ω(Γ) = ∂X (the proof is given in an appendix of [195]). That result played a key role in the identiﬁcation of the multiplicities of the topological singularities of Selberg zeta functions for such Kleinian groups (see [195] and [151] for a detailed discussion). Epstein did not analyze the renormalized volume, however. For further results on the asymptotic volume of hyperbolic 3manifolds we refer to [237], [161].

1.7 Factorization of residue families. Recursive relations In Section 1.6, we have seen how the critical residue families Dnres (h; λ) are related to the fundamental identity and the holographic formula for Qn . However, it is less clear whether the fundamental identity can be derived more directly from the holographic formula. In fact, it seems that these two results actually should be regarded as two diﬀerent facets of the more fundamental critical residue family. In the present section, we discuss one more of these facets: recursive formulas. The situation is summarized in Figure 1.2. c (λ) : C ∞ (S n ) → C ∞ (S n−1 ) According to Theorem 1.4.2, any family D2N satisﬁes a system of 2N factorization identities. These identities are consequences

1.7. Factorization of residue families. Recursive relations

33

holographic formula for Qn O evaluation ? ? Green’s formula Dnres (λ) RRR k k RRRfactorization covariancekkkkk RRR k k RRR k k k k RRR ukk ( fundamental recursive identity formulas Figure 1.2: 3 facets of the critical residue family of corresponding factorization identities for the families D2N (λ) of homomorphisms of Verma modules. The factorization identities continue to hold true for the residue families res (h; λ) of Section 1.6, at least for conformally ﬂat metrics. These identities D2N give rise to recursive relations for Q-curvatures,

Q-polynomials

and GJMS-operators.

Even for conformally ﬂat metrics, the complexity of explicit formulas for the critical Q-curvature Qn , in terms of Riemannian invariants, increases exponentially with the dimension n. This makes recursive relations interesting. We start with the formulation of the factorization identities of residue families for conformally ﬂat metrics (Theorem 6.11.1). res (h; λ) facTheorem 1.7.1. Let h be conformally ﬂat. Then the residue family D2N torizes for " # n+1 n n n+1 ,...,− +N ∪ − +N, . . . , − +2N λ∈ − 2 2 2 2

into products of (lower order ) residue families and GJMS-operators. More precisely, for j = 1, . . . , N , the identities

n n res res h; − +2N −j = P2j (h) ◦ D2N h; − +2N −j D2N −2j 2 2 and res D2N

n+1 n+1 res +j = D2N −2j h; − −j ◦ P2j (dr2 +hr ) h; − 2 2

hold true. Here P2j are GJMS-operators with respect to the metrics h and dr2 +hr . res In particular, we have the relation D2N h; − n2 +N = P2N (h)i∗ (see (1.6.8)). Theorem 1.7.1 oﬀers a recursive method to determine GJMS-operators. On res (h; λ) satisﬁes 2N factorization identities. On the other hand, the one hand, D2N

34

Chapter 1. Introduction

it is a polynomial of degree N , i.e., is determined by N + 1 coeﬃcients. Hence the right-hand sides of the factorization identities satisfy N − 1 linear relations. In particular, for n ≥ 6, Pn (h)i∗ can be written as a linear combination of the compositions n

res h; −1 Pn−2 (h) ◦ D2res (h; 1), . . . , P2 (h) ◦ Dn−2 2 and

n+3 res Dn−2 h; − ◦ P2 (dr2 +hr ), 2

n+5 res Dn−4 h; − ◦ P4 (dr2 +hr ). 2

res But the residue families D2N (h; λ) can be written recursively as linear combinations of compositions of GJMS-operators P2 (h), . . . , P2N (h) and the Yamabe operator P2 (dr2 +hr ). This follows from the system

n n res res h; − +2N −j = P2j (h) ◦ D2N h; − +2N −j , j = 1, . . . , N, D2N −2j 2 2 n−1 n+3 res res D2N h; − = D2N ◦ P2 (dr2 +hr ) −2 h; − 2 2

of N +1 identities. Thus Pn (h)i∗ can be written as a linear combination of compositions of the lower order GJMS-operators Pn−2 (h), . . . , P2 (h) and the Yamabe operator P2 (dr2 +hr ) and the Paneitz operator P4 (dr2 +hr ). In particular, for n = 6 and N = 3 we ﬁnd P6 (h)i∗ = αP4 (h)D2res (h; 1) + βP2 (h)D4res (h; 2) + γD4res (h; −9/2)P2 (dr2 +hr ) + δD2res (h; −11/2)P4 (dr2 +hr ) 1 for (α, β, γ, δ) = 21 (18, −5, −4, 12). An evaluation of the latter formula yields the following result (Corollary 6.11.6).

Theorem 1.7.2. For M 6 with a conformally ﬂat metric h, the critical GJMSoperator P6 can be written in the form P6 = 2(P4 P2 + P2 P4 ) − 3P23 + 2(L + L∗ ),

(1.7.1)

where the operator L is deﬁned by i∗ P2 (dr2 +hr ), P4 (dr2 +hr ) = L(h)i∗ . (1.7.1) actually holds true in all dimensions n ≥ 6. For the round sphere S n we ﬁnd L = 3P2 . For general metrics, L + L∗ is of second order. In fact, a formula for the Q-polynomial Qres 6 (λ) (see Theorem 1.7.7) leads to the alternative recursive formula (1.7.13) for P6 as the sum of the leading part 2(P4 P2 + P2 P4 ) − 3P23 and an explicit second-order operator.

1.7. Factorization of residue families. Recursive relations

35

Now we combine (1.6.12) with Theorem 1.7.1. We describe how this leads to the recursive formula Qn =

n

aI P2I (Qn−2|I| ) + (−1) 2 −1 (n)

I

(n−2)(n−4) · · · 2 ¯ n2 −1 ¯ P (Q2 ) (n−3)(n−5) · · · 1 2

(1.7.2)

for the critical Q-curvature of a conformally ﬂat metric (Conjecture 6.11.1). In (1.7.2), the sum runs over all ordered partitions I of integers in the interval [1, n2 −1] as sums of natural numbers. For I = (I1 , . . . , Im ), the operator P2I is deﬁned as the composition P2I1 ◦ · · · ◦ P2Im of GJMS-operators, and we set |I| = i Ii . Finally, we use the notation P¯2 (h) = P2 (dr2 +hr ),

¯ 2 (h) = Q2 (dr2 +hr ). Q

(1.7.2) expresses Qn in terms of lower order Q-curvatures and lower order GJMSn operators. The right-hand side of (1.7.2) contains 2 2 −1 terms. In general, the (n) coeﬃcients aI are not known explicitly, but admit algorithmic descriptions ([95]). The holographic formula (1.6.15) and the recursive relation (1.7.2) for Qn (h) use the corresponding Poincar´e-Einstein metric g = r−2 (dr2+hr ) in fundamentally diﬀerent ways: (1.6.15) involves holographic coeﬃcients and the asymptotics of harmonic functions of the Laplacian of g, (1.7.2) contains high order powers of the Yamabe operator of dr2 +hr . In order to describe the mechanism which generates (1.7.2), we restrict to the special cases n = 4 and n = 6. In these cases, the rational coeﬃcients aI are explicit and we actually prove (1.7.2) (even for general metrics). The following result is valid for all metrics (Theorem 6.11.2). Theorem 1.7.3. For (M 4 , h), the critical residue family D4res (h; λ) : C ∞ ([0, ε) × M ) → C ∞ (M ) satisﬁes the identities D4res (h; 0) = P4 (h)i∗ , D4res (h; 1) = P2 (h) ◦ D2res (h; 1), 3 7 res res h; − h; − D4 = D2 ◦ P2 (dr2 +hr ), 2 2 1 res D4 h; − = i∗ P4 (dr2 +hr ). 2 The ﬁrst three identities in Theorem 1.7.3 suﬃce to determine the family D4res (h; λ) in terms of their right-hand sides. But D2res (h; λ) can be written as a linear combination of P2 (h) and P¯2 (h). A combination of the resulting formula with the relation Q4 = −D˙ 4res (0)(1)

36

Chapter 1. Introduction

yields the recursive formula ¯ 2 (h). Q4 (h) = P2 (h)Q2 (h) − 2i∗ P¯2 (h)Q

(1.7.3)

(1.7.3) is a special case of (1.7.2). The recursive formula (1.7.3) has an interesting and important feature: although the arguments which yield (1.7.3) apply only in dimension n = 4, the formula holds true in all dimensions n ≥ 4 (Lemma 6.11.5). In other words, this type of formula is distinguished among other formulas for Q4 , and we refer to this property as universality. In that sense, (1.7.1) is universal. Next, we derive a universal formula for Q6 . The following result is valid for all metrics (Theorem 6.11.4). Theorem 1.7.4. For (M 6 , h), the critical residue family D6res (h; λ) satisﬁes the identities D6res (h; 0) = i∗ P6 (h), D6res (h; 1) = P4 (h) ◦ D2res (h; 1), D6res (h; 2) = P2 (h) ◦ D4res (h; 2), D6res (h; −5/2) = D4res (h; −9/2) ◦ P2 (dr2 +hr ). Now we proceed similarly as for Q4 . The four identities in Theorem 1.7.4 suﬃce to determine the family D6res (h; λ). Combining the resulting formula with the relation Q6 (h) = D˙ 6res (h; 0)(1), we ﬁnd that Q6 (h) coincides with a linear combination of D4res (h; −9/2) P2 (dr2 +hr ),

P2 (h)D4res (h; 2) and P4 (h)D2res (h; 1),

acting on the function u = 1. Since the families D4res (h; λ) and D2res (h; λ), in turn, can be written as linear combinations of compositions of P4 (h), P2 (h) and P2 (dr2 +hr ), it follows that Q6 (h) ﬁnally can be written as a linear combination of compositions of P4 (h), P2 (h) and the Yamabe operator P2 (dr2 +hr ) (acting on u = 1). This method yields the following special case of (1.7.2) (Theorem 6.11.5). Theorem 1.7.5. The critical Q-curvature of (M 6 , h) satisﬁes Q6 =

2 5 8 ¯ 2, (P2 Q4 + P4 Q2 ) − P22 Q2 + i∗ P¯22 Q 3 3 3

(1.7.4)

where all quantities are understood with respect to h. The proof of Theorem 1.7.5 uses the observation that (1.7.3) holds true also in dimension n = 6 (universality). This is the way how Q4 enters the formula for Q6 . Among other formulas for Q6 , (1.7.4) again is distinguished by the fact that, like (1.7.3), the same formula (with the same numerical coeﬃcients) holds true

1.7. Factorization of residue families. Recursive relations

37

for the subcritical Q-curvatures Q6,n in all (even) dimensions n > 6 (Theorem 6.11.6). In other words, (1.7.4) is universal. For general n, a proof of the recursive formula (1.7.2) along these lines would rest on the formula n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h), the ﬁrst half of the system of factorization identities in Theorem 1.7.1, and the identity n−1 n+3 res res Dn h; − (1.7.5) = Dn−2 h; − ◦ P¯2 (h). 2 2 In addition, such a proof uses the analogous factorization identities for lower order families, and identities which relate the subcritical Q-curvatures Q2N , 2N < n to the quantities P¯2N (1). For n ≤ 6, such formulas are provided by the universality of the respective recursive formulas. For general n, this property remains to be established. Now we turn to the discussion of the Q-polynomials def

N res Qres 2N (h; λ) = −(−1) D2N (h; λ)(1).

We start with the formulation of some results in low order cases. The proof of Theorem 1.7.5 rests (among other things) on the identity Q6 = D˙ 6res (0)(1). For the subcritical Q-curvatures Q4,n and Q6,n , there are analogous, but more complicated relations to the values n n

D˙ 4res − +2 (1) and D˙ 6res − +3 (1), 2 2 respectively. The following formulas hold true for all metrics and n ≥ 4 and n ≥ 6, respectively (Theorem 6.11.7). Theorem 1.7.6. n

n−4 (Q4,n + P2 Q2,n ), D˙ 4res − +2 (1) = −Q4,n − 2 2 n

n−6 3 3 2 res ˙ D6 − +3 (1) = Q6,n + Q6,n + P2 Q4,n − 2P4 Q2,n + P2 Q2,n . 2 2 2 2 Note that these relations are a bit surprising. In fact, we recall that in the critical case the relation between D˙ nres (h; 0)(1) and Qn (h) is suggested by the formula Dnres (h; 0) = Pn (h)i∗ and the transformation formula for the quantity D˙ nres (h; 0)(1) which follows from the conformal covariance of the family. Although n

res − +N = P2N i∗ D2N 2 res (see (1.6.8)), this argument does not yield information on the quantity D˙ 2N (− n2 + N )(1). On the other hand, the subcritical Q-curvatures do not obey transformation laws, under conformal changes h → e2ϕ h, which are linear in ϕ.

38

Chapter 1. Introduction

Theorem 1.7.6 and Theorem 1.7.4 imply the following result for the Qpolynomial Qres 6 (h; λ) (for all metrics h) (Theorem 6.11.8). Theorem 1.7.7. For even n ≥ 6, Qres 6 (λ) =

n n 1 λ λ+ −4 λ+ −5 Q6 2 2 2

n n 3 + λ λ+ −3 λ+ −4 P2 Q4 + P2 Q2 2 2 2

n n − λ λ+ −3 λ+ −5 P4 Q2 . 2 2

The critical case n = 6 of the latter result is a special case of the following recursive formula for the critical Q-polynomial Qres n (λ) (Theorem 6.11.10). Theorem 1.7.8. Let (M, h) be a Riemannian manifold of even dimension n with a conformally ﬂat metric. Assume that Qres 2j (0) = 0

for

j = 1, . . . ,

n , 2

(1.7.6)

and deﬁne the polynomials Qres 2j (λ) by res Qres 2j (λ) = λQ2j (λ).

Then Qres n (λ)

= (−1)

n 2 −1

λ −

n 2 −1

λ

n 2

k

k=1 n 2 −1

+λ

j=1

+k

(−1)

Qn

2 λ− n

j

k=1 k=j

n 2

+k k−j

P2j Qres n−2j

n 2

−j .

(1.7.7)

Here all quantities are understood with respect to h. Theorem 1.7.8 is a consequence of n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h)

and the ﬁrst half of the system of factorization identities in Theorem 1.7.1. The condition (1.7.6) is conjectured to be always satisﬁed (Remark 6.11.5). (1.7.7) relates the critical Q-polynomial Qres n (λ) to the critical Q-curvature Qn , lower order GJMS-operators and values of lower order Q-polynomials. It motivates us to ﬁnd analogs of (1.7.7) for subcritical residue families. The conjectural identities (6.11.38) provide such analogs. For their proofs it would be enough to establish appropriate generalizations of Theorem 1.7.6, i.e., relations

1.7. Factorization of residue families. Recursive relations

39

between the subcritical Q-curvatures Q2N,n (2N < n) and the respective values res (− n2 + N )(1) (Theorem 6.11.11). These formulas are subcritical analogs of the D2N n identity D˙ nres (0)(1) = −(−1) 2 Qn . A combination of (1.7.7) with its subcritical analogs implies a formula of the form cI (λ)P2I1 . . . P2Ir Qn−2|I| (1.7.8) Qres n (λ) = I

for the critical Q-polynomial in terms of Q-curvatures Qn , . . . , Q2 and GJMSoperators Pn−2 , . . . , P2 . Here the sum runs over all ordered partitions I of integers in [0, n2 − 1] as sums of non-negative integers, and the coeﬃcients cI (λ) are polynomials of degree n2 . The formula 2 Qres 4 (λ) = −λ(λ−1)Q4 − λ P2 Q2

and Theorem 1.7.7 (for n = 6) are the ﬁrst special cases. The identity (1.7.8) relates the holographic data of the bulk space Poincar´eEinstein metric, which deﬁne the polynomial Qres n (λ), to Q-curvatures and GJMSoperators of its conformal inﬁnity. In other words, it can be regarded as a generalization of the holographic formula for Q-curvature. More precisely, by comparing the coeﬃcients of powers of λ, the identity (1.7.7) is equivalent to a set of n2 identities which relate n n

n −1 !vn and Qn . ! (−1) 2 2n−1 2 2 The linear coeﬃcient is responsible for the identity Q˙ res n (0) = Qn , and the highest power of λ yields (Corollary 6.11.5) Corollary 1.7.1. Under the assumption (1.7.6), n 2 −1 n n

n res top −1 2 −1 P Qres −1 !(Qn ) = (−1) 2 Qn − −j . 2j n−2j j −1 2 2

n

(1.7.9)

j=1

Here the superscript “top” indicates the coeﬃcient of the highest power of λ. The main diﬀerence between (1.6.15) and (1.7.9) concerns the way in which both formulas explain the divergence terms. Whereas in (1.6.15) these contributions are formulated in terms of the operators T2j∗ (0) (which describe the asymptotics of harmonic functions of the Poincar´e-Einstein metric) acting on holographic coeﬃcients, in (1.7.9) these are covered by lower order GJMS-operators acting on lower order Q-curvatures. ∗ Among the terms which deﬁne the Q-polynomial Qres n (λ), the term Pn (λ)(1) plays a special role. In fact, Theorem 1.6.1 and Theorem 1.6.2 (together with the n identity P˙n (0)(1) = (−1) 2 Qn ) imply that P˙ n∗ (0)(1) is a linear combination of the other items. The following conjecture states how this eﬀect extends to the full polynomial Pn∗ (λ)(1) (see Conjecture 6.11.2 and (6.11.55)).

40

Chapter 1. Introduction

Conjecture 1.7.1. n

2

(n+2j)T2j∗ (λ)(vn−2j ) ≡ 0.

(1.7.10)

j=0

(1.7.10) can be veriﬁed for n ≤ 6 by direct calculation using explicit formulas for the terms involved. It can be applied to evaluate the left-hand side of (1.7.9). For n = 6, this leads to the recursive formula res 6 Q6 = [P2 (Qres 4 (2)) + 2P4 (Q2 (1))] − 6 [Q4 + P2 (Q2 )] Q2 − 2 3!v6 .

(1.7.11)

In Section 6.12, we will use this formula to derive the following recursive formula for the critical GJMS-operator P6 . Theorem 1.7.9. The critical GJMS-operator P6 is given by the recursive formula P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 48δ(P2 #du) − 8δ(B#du).

(1.7.12)

Here [·]0 denotes the non-constant part of the respective operator. Among other possible formulas for the critical P6 , (1.7.12) is distinguished by the property that all subcritical cubes P6 admit the analogous recursive presentations P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 48δ(P2 #du) −

n

16 δ(B#du) − −3 Q6 u, (1.7.13) n−4 2

where Q6 is given by (1.7.11) (Corollary 6.12.2). Formula (1.7.13) shows the role of the Bach tensor B, or rather of the conformally covariant second-order operator −δ(B#d) + (B, P) (in dimension n = 4), as an obstruction to the existence of P6 in dimension n = 4 ([118]). On the round spheres S n , the GJMS-operators are commuting intertwining operators, and (1.7.13) reduces to the non-linear relation P6 = 4P2 P4 − 3P23 + 12P2 .

(1.7.14)

Similarly, for n = 8 (assuming (1.7.10)) we ﬁnd the recursive formula res Q8 = − P2 (Qres 6 (3)) + 3P4 (Q4 (2)) + 3P6 (Q2 ) − 12 Q6 − P2 (Qres 4 (1)) − 2P4 (Q2 ) Q2 2 − 18 Q4 + P2 (Q2 ) + 4!3!27 v8

(1.7.15)

(Theorem 6.13.1). This formula emphasizes the signiﬁcance of the Q-polynomials.

1.7. Factorization of residue families. Recursive relations

41

Finally, Conjecture 1.7.1 implies the alternative formula n

nQres n (λ)

2 n

n ! λ(λ−1) . . . λ− +1 =2 2jT2j∗ (λ)(vn−2j ) 2 2 j=0

n

(Theorem 6.11.13) for the Q-polynomial Qres n (λ). When combined with a formula from [70], it yields a link between the quadratic term in the total critical Q-polynomial Qres λ → n (λ) vol M

of a closed manifold (M, h) and the renormalized volume of a global conformally compact Einstein metric with h being a representative of its conformal inﬁnity (Theorem 6.11.15). Theorem 1.7.10. Under Conjecture 1.7.1, the sum λ(λ−1) . . . (λ− n2 +1) n Qres (λ) vol + S(n−λ)(1) vol + λ2 V (−1) 2 c n2 n ( n2 −1)! M M vanishes up to order 2 at λ = 0. Theorem 1.7.10 should be compared with the result that, for odd dimension n, ˙ the renormalized volume V is given by the integral − M S(n)(1) vol ([97]). By the res recursive structure of Qn (λ), the additional contribution ¨ res (0) vol Q n M

can be expressed in terms of GJMS-operators acting on Q-curvatures. The latter result allows us to determine the inﬁnitesimal conformal variation of the quadratic coeﬃcient of the total Q-polynomial; we recall that the linear coeﬃcient is the conformal invariant M Qn vol. We prove (see Theorem 6.11.16 and the discussion following it) Theorem 1.7.11. For even n and closed M , •

n 1 ¨ res (0) vol [ϕ] = Q ϕ 2Qn − (−1) 2 c−1 n vn vol . n 2 2 M M Note that, by Theorem 1.6.2, the conformal anomaly on the right-hand side is the divergence of a natural one-form. Theorem 1.7.11 involves the conformal variation only of local quantities. It will be derived from the transformation law of Dnres (λ) without appealing to Conjecture 1.7.1. We ﬁnish with some comments concerning the extension of the above results beyond the conformally ﬂat case. The main technique in the proofs are the factorization identities. For general dimension n, and for general order, the full system

42

Chapter 1. Introduction

of factorization identities is proved for conformally ﬂat metrics (Theorem 1.7.1). The identities in the ﬁrst half of the system actually hold true for general metrics. It follows that Theorem 1.7.8 and Corollary 1.7.1 extend to general metrics: the n proofs rest only on these factorization identities and D˙ nres (0)(1) = −(−1) 2 Qn . For the proof of the recursive formula (1.7.2) it suﬃces to establish only one additional factorization identity: (1.7.5). In the cases n = 4 and n = 6, it is covered by Theorem 1.7.3 and Theorem 1.7.4. For more details we refer to the comments at the end of Section 6.11. The observation that residue families satisfy factorization identities, which are literally the same as those for families of homomorphisms of Verma modules, being responsible for the ﬂat case, suggests that the former are induced in some way by the latter. The holographic duality (Section 1.10) will give that idea a precise form.

1.8 Families of conformally covariant diﬀerential operators In Section 1.6, we associated to any Riemannian manifold (Σ, h) a canonical sequence of one-parameter families of linear diﬀerential operators res (h; λ) : C ∞ ([0, ε) × Σ) → C ∞ (Σ), 0 ≤ N ≤ n = dim Σ DN

which satisfy appropriate analogs of the transformation law (1.6.9) under conformal changes h → e2ϕ h of the metric, and which specialize to GJMS-operators on Σ for appropriate values of λ. We also pointed out that the transformation laws of res (·; λ) actually suggest that we ask for more general constructions the families DN which contain the residue families as a special case. More precisely, we are led to consider the following setting. Let M be a manifold and Σ an oriented codimension one submanifold. Let i : Σ → M be the embedding. We consider one-parameter families of natural linear diﬀerential operators DN (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ), λ ∈ C of order N ≥ 0 which are conformally covariant in the sense that ∗

e−(λ−N )i

(ϕ)

◦ DN (M, Σ; e2ϕ g; λ) ◦ eλϕ = DN (M, Σ; g; λ)

(1.8.1)

for all ϕ ∈ C ∞ (M ) and all λ ∈ C. Here a series of comments is in order. For any ﬁxed family parameter the operators DN (M, Σ; g; λ) are supposed to be compositions of diﬀerential operators on M and Σ with the restriction map i∗ . In that sense, we talk about diﬀerential operators which map functions on M to functions on a submanifold Σ. The order of a diﬀerential operator in a family may depend on the family parameter. The

1.8. Families of conformally covariant diﬀerential operators

43

order of the operators for a generic parameter is called the order of the family. In (1.8.1) it is, obviously, enough to assume that the functions ϕ are deﬁned in an open neighborhood of Σ. We emphasize that in (1.8.1) the order of the family is coupled in a speciﬁc way to the conformal weights: the order N coincides with the diﬀerence of the conformal weights λ and λ−N . In view of the hoped-for applications to Qcurvature, we restrict to the given version of conformal covariance. Other versions might be of interest in diﬀerent contexts. Finally, we do not specify the quality of the families as functions of the family parameter. There are two reasons for that omission: we will not discuss any aspect of a possible general theory of such families, and in the speciﬁc constructions of families, which are analyzed here, that quality varies from polynomial to rational. The problem of constructing and analyzing conformally covariant families DN (M, Σ; g; λ) should be viewed as a problem in conformal submanifold theory. In particular, the relation between conformally covariant families and Q-curvature motivates us to regard Q-curvature as a subject in conformal submanifold theory. We note that, to a large extent, conformal submanifold theory is concerned with invariants with respect to conformal diﬀeomorphisms of a ﬁxed background metric. A typical subject is the study of properties of immersions of surfaces in S n (n ≥ 3) which are invariant under the group of conformal diﬀeomorphisms of the round sphere (S n , gc ). Sometimes that area is called M¨obius geometry ([142]). One speciﬁc problem, which triggered its own large body of literature, was Willmore’s conjecture concerning embeddings i : T 2 → Rn of tori for which the functional W = T 2 |H|2 vol (H denotes the mean curvature vector) is minimal. Since the functional W is M¨ obius-invariant, this is a problem in conformal submanifold theory. For a surface i : Σ2 → M 3 in a general background metric g, the appropriate generalization of W is the conformally invariant functional Σ |L0 |2 vol (L0 is the trace-free part of the second fundamental form). Here conformal invariance refers to arbitrary conformal changes of the metric. It seems that the intrinsic aspect of conformal submanifold theory is still much less developed than the intrinsic theory on a given manifold. Notable exceptions, however, are Fialkow’s classical work [100] and the more recent work [52] which approaches classical problems of the subject using modern technology of conformal diﬀerential geometry. Now we describe a simple pattern by which critical conformally covariant families give rise to fundamental pairs. A conformally covariant family will be called critical if its order coincides with the dimension of the target space Σ. A fundamental pair consists of a curvature quantity and a linear diﬀerential operator so that under conformal changes of the metric these satisfy an analog of the fundamental identity for Q-curvature. For even n = dim Σ and a critical family Dn (g; λ), assume that the operator Dn (g; 0) lives on Σ, i.e., is of the form Dn (g; 0) = Pn (g)i∗

44

Chapter 1. Introduction

for some natural diﬀerential operator Pn on Σ. It is not required here that Pn (g) only depends on the induced metric i∗ (g) on Σ. If Pn (·) only depends on i∗ (g), it is a conformally covariant diﬀerential operator on Σ, i.e., enϕ Pn (e2ϕ g) = Pn (g) for all ϕ ∈ C ∞ (Σ). Next, we deﬁne the quantity n def Qn (g) = −(−1) 2 D˙ n (g; 0)(1) ∈ C ∞ (Σ).

(1.8.2)

It is a consequence of the conformal covariance of the family that n

enϕ Qn (e2ϕ g) = Qn (g) + (−1) 2 [Pn (g), ϕ](1)

(1.8.3)

for all ϕ ∈ C ∞ (M ). In fact, the operator D˙ n (·; 0) satisﬁes the identity enϕ D˙ n (e2ϕ g; 0) = D˙ n (g; 0) + [ϕ, Dn (g; 0)] , which for u = 1 implies (1.8.3). In order to simplify the formulas, we have suppressed here all pull-back operators i∗ . (1.8.3) is an analog of the fundamental identity n

n

enϕ Qn (e2ϕ g) = Qn (g) + (−1) 2 [Pn (g), ϕ](1) = Qn (g) + (−1) 2 Pn (g)(ϕ) for Qn (see (1.6.1)). Therefore, those critical families for which the fundamental pair (Pn , Qn ) actually reproduces the pair (Pn , Qn ) deserve particular interest. However, in general, there is no reason to expect that the quantity Qn only depends on the induced metric i∗ (g) on Σ even if Dn (g; 0) has this property. The above pattern is obviously suggested by the theory of the residue families. Therefore, of particular interest are those conformally covariant families which specialize to the families res (h; λ) for (M, Σ; g) = [0, ε) × Σ; dr2 +hr . DN c (λ) provide another test case for any construction of conforThe families DN mally covariant families. These are actually tractor families in the sense of Section 1.9 for (M, Σ) = (S n+1 , S n ). We recall that Dnc (0) = Pn (S n , gc )i∗ (Theorem 1.4.2), i.e., the operator Dnc (0) actually lives on the submanifold S n , and is given by the critical GJMS-operator for the round metric gc . Moreover, the quantity n −(−1) 2 D˙ nc (0)(1) equals Qn (S n , gc ) = (n−1)! (Lemma 6.1.2). The above derivation of (1.8.3) is an analog of a family proof of the fundamental identity for Qn via scattering theory ([128]). In fact, near λ = n the scattering operator S(h; λ) : C ∞ (∂M ) → C ∞ (∂M ) has the form

S(h; λ) = −

c(n)Pn (h) + S0 (h; λ) λ−n

1.8. Families of conformally covariant diﬀerential operators

45 n

with a holomorphic family S0 (h; λ) so that S0 (h; n)(1) = c(n)(−1) 2 Qn (h). c(n) is a constant which only depends on n. Hence the family Φ(h; λ) =

n−λ S(h; λ) c(n) n

˙ n)(1) = is holomorphic near n, and satisﬁes Φ(h; n) = Pn (h) and −(−1) 2 Φ(h; Qn (h). Moreover, under conformal changes of h the scattering operator satisﬁes S(e2ϕ h; λ) = e−λϕ ◦ S(h; λ) ◦ e(n−λ)ϕ , ϕ ∈ C ∞ (∂M )

(1.8.4)

(see (3.2.6)). The fundamental identity then follows by a similar argument as above (now applied to Φ(λ)). However, a signiﬁcant diﬀerence between both transformation rules (1.8.1) (for N = n) and (1.8.4) deserves to be mentioned: n is in (1.8.4) and in (1.8.1) the diﬀerence and the sum of the conformal weights, respectively. The above arguments also work for odd n, and yield a pair (Pn , Qn ) which satisﬁes the identity enϕ Qn (e2ϕ g) = Qn (g) + [Dn (g; 0), ϕ](1) for all ϕ ∈ C ∞ (M ). Here Qn ∈ C ∞ (Σ) is deﬁned as in (1.8.2) but without the n sign (−1) 2 . In that case, the independence of the fundamental pair (Pn , Qn ) of the metric near Σ is no issue. In fact, for (S n+1 , S n ) (with odd n) the operator Dnc (0) does not live on S n . Thus the fundamental pair (Pn , Qn ) has to be regarded as being associated to the embedding. Chang and Qing ([68]) discovered such a fundamental pair (P3 , T ) on a compact 4-manifold with boundary in their study of Polyakov-formulas for determinants of elliptic boundary value problems in dimension 4. We will not address a general theory of conformally covariant families. Instead, we restrict to the discussion of one construction of such families which is of special signiﬁcance: the tractor families. The tractor families form a sequence of conformally covariant families which are canonically associated to the given data (M, Σ; g) of a hypersurface Σ in M and a background metric. The conformal covariance of the tractor families is a direct consequence of its deﬁnition which rests on the conformally invariant tractor calculus. This explains its name. The tractor families are rational in λ, and the above construction of fundamental pairs faces the additional problem that the critical tractor families are not holomorphic at λ = 0. For more details we refer to Section 1.9 and Section 1.10. We close the present section with a brief discussion of a formula for a conformally covariant family of ﬁrst order. Assume that dim Σ ≥ 2. We introduce some more notation. We choose a unit normal vector ﬁeld N = N (g) on Σ. Let L(X, Y ) = −g(∇X (Y ), N ) = g(∇X (N ), Y ), X, Y ∈ X (Σ)

(1.8.5)

46

Chapter 1. Introduction

be the second fundamental form of Σ and H=

tr L ∈ C ∞ (Σ) n−1

(1.8.6)

be the corresponding mean curvature. In this convention, the mean curvature of the round sphere S n−1 in Rn for n ≥ 3 with respect to the exterior normal is 1. Now the family D1 (M, Σ; g; λ) = i∗ ∇N (g) − λH(g)i∗ : C ∞ (M ) → C ∞ (Σ)

(1.8.7)

is conformally covariant in the sense that ∗

D1 (e2ϕ g; λ) = e(λ−1)i

(ϕ)

◦ D1 (g; λ) ◦ e−λϕ

(1.8.8)

for all ϕ ∈ C ∞ (M ) and all λ ∈ C (Theorem 6.2.1). D1 (M, Σ; g; λ) is not critical since dim Σ ≥ 2. For a curve Σ in a surface M , the corresponding critical family is D1 (M, Σ; g; λ) = i∗ ∇N (g) − λκ(g)i∗ , where κ(g) = g(∇c˙ (c), ˙ N (g)) ∈ C ∞ (Σ) (c is a natural parametrization of Σ) is the corresponding geodesic curvature. The corresponding fundamental pair is (i∗ ∇N (g) , κ(g)). For a second-order example we refer to Theorem 1.9.1. This family interpolates the Yamabe operators on M and Σ. It is critical if dim Σ = 2.

1.9 Curved translation and tractor families In order to ﬁnd conformally covariant families in the sense of Section 1.8, direct calculations are eﬀective only for small orders. (1.8.7) gives a ﬁrst-order family. The following result yields a second-order critical family D2 (M, Σ; g; λ) for an oriented surface Σ in a manifold M of dimension 3 with an arbitrary background metric g. Theorem 1.9.1. The natural family D2 (M, Σ; g; λ) = −2λi∗ P2 (M, g) + (2λ+1)P2 (Σ, i∗ (g))i∗ + 2λ(2λ+1)H(g)i∗∇N (g) + λ(2λ+1) Q2 (Σ; i∗ (g))i∗ − i∗ Q2 (M ; g) − λH(g)2 i∗ is conformally covariant, i.e., it satisﬁes the relations ∗

e−(λ−2)i

(ϕ)

◦ D2 (M, Σ; e2ϕ g; λ) ◦ eλϕ = D2 (M, Σ; g; λ)

for all ϕ ∈ C ∞ (M ) and all λ ∈ C. Recall that Q2 = J.

1.9. Curved translation and tractor families

47

Theorem 1.9.1 extends to higher dimensions (Theorem 6.4.1). The resulting families D2 (M, Σ; g; λ) specialize, for instance, to D2c (λ) (see (1.4.7)), and generalize the residue family D2res (h; λ) since D2 (dr2 +hr ; λ) = D2res (h; λ), where Σ is considered as the hypersurface r−1 (0) in M = [0, ε)×Σ (Theorem 6.7.1). Although D2 (M, Σ; g; λ) is polynomial of degree 3 in λ, under certain curvature conditions its degree can be smaller. In fact, if H = 0, then the degree is ≤ 2, and it is 1 iﬀ H = 0 and Q2 (M ; g) restricts to Q2 (Σ; i∗ (g)). Similarly, D1 (M, Σ; g; λ) is polynomial of degree 1 for general metrics, but does not depend on λ iﬀ H = 0. In order to ﬁnd higher order families, more systematic constructions are required. Such a tool is Eastwood’s curved translation principle ([90], [20], [88]). We describe the idea of the method, and how it will be applied here. It is a curved version of Zuckerman translation of homomorphisms of Verma modules ([243]). Zuckerman translation constructs homomorphisms of Verma modules from given ones by tensoring with ﬁnite-dimensional representations. The construction translates the inﬁnitesimal character of the Verma modules involved (whence the name). We will not go here into the details of that powerful method in any greater generality (see [236]). For our purpose, it will be enough to deal with it in a very special case. Its curved version mimics the construction in the framework of conformally covariant diﬀerential operators. In fact, the idea is to rephrase the construction in terms of diﬀerential intertwining operators in the conformally ﬂat case of the sphere, and to ﬁnd a conformally covariant substitute of the resulting formula. A key point is that the splitting operators, which realize the embedding of irreducibles into tensor products and projections to subquotients of tensor products, are diﬀerential operators of small order. In order to apply curved translation, we prove that all families DN (λ) arise by iterated translation of two extremely simple families: D0 (λ) = i and D1 (λ) = Yn− i, where i : gn → gn+1 . D0 and D1 generate all even and odd order families, respectively, by iteration. The construction of even order families is as follows. We deﬁne families Θ2N (λ) : Mλ−2N (gn ) → Mλ (gn+1 ) by Θ2N (λ) = p(λ) ◦ [Θ2N −2 (λ−1) ⊗ In )] ◦ i(λ−(2N −1)),

Θ0 (λ) = i

using the embeddings i(λ) : U(gn ) ⊗ C(λ−1) → U(gn ) ⊗ C(λ) ⊗ Fn and projections p(λ) : U(gn+1 ) ⊗ C(λ−1) ⊗ Fn+1 → U(gn+1 ) ⊗ C(λ),

(1.9.1)

48

Chapter 1. Introduction

which induce families of homomorphisms Mλ−1 (gn ) → Mλ (gn ) ⊗ Fn ,

Mλ−1 (gn+1 ) ⊗ Fn+1 → Mλ (gn+1 ).

Here Fm denotes the ﬁnite-dimensional standard representation of Gm . In (1.9.1), In : Fn → Fn+1 denotes the Gn -equivariant embedding which corresponds to the embedding Gn → Gn+1 . The rational families i(λ) and p(λ) are given in Lemma 6.18.2. Then the translation Θ2 (λ) : Mλ−2 (gn ) → Mλ (gn+1 ) of Θ0 is induced by right multiplication with (n−3+λ)D2 (λ), up to a rational multiple which is caused by the chosen normalizations of i(λ) and p(λ). Here − D2 (λ) = −(n−3+2λ)Δ− n + (n−2+2λ)Δn−1 (Theorem 6.18.2). More generally, we identify all iterated translates of D0 in (Theorem 6.18.7) Theorem 1.9.2. The family Θ2N (λ) : Mλ−2N (gn ) → Mλ (gn+1 ) is induced by right multiplication with ⎤ ⎡ ⎣

N

(n−1−(j +N )+λ)⎦ D2N (λ),

(1.9.2)

j=1

up to a rational multiple which is caused by the chosen normalizations of i(λ) and p(λ). The proof of Theorem 1.9.2 rests on Theorem 1.4.3. The construction actually lifts to the semi-holonomic category of Verma modules (see Section 6.17 and [92]). This has the consequence that it can be combined with the mechanisms of inductions via Cartan connections ([19]). Explicit formulas in the semi-holonomic category, however, quickly become very complicated with increasing order. The semi-holonomic lifts of D2N (λ) are rational in λ with at most simple poles at the zeros of the product in (1.9.2). One can regard the existence of poles as an algebraic fact which points to the possible non-existence of certain curved analogs. Now we write D2N (λ) as an N th Zuckerman translate, and replace each factor in the resulting formula by a conformally covariant operator. Such substitutes are given in terms of tractor D-operators. The simplest of these operators is given by ⎞ ⎛ 1 n−2+2λ (Δg +λJ(g))u ⎠ ∈ Γ(T ∗ M ) DM (g; λ) : C ∞ (M ) u → ⎝ du 2λu (Theorem 6.20.1). It replaces the projection p(λ) : Mλ−1 ⊗ F → Mλ

1.9. Curved translation and tractor families

given by

49

⎧ 1 − ⎪ ⎨1 ⊗ 1 ⊗ v− → n−2+2λ (Δn ⊗ 1), p(λ) : 1 ⊗ 1 ⊗ vi → Yi− ⊗ 1, ⎪ ⎩ 1 ⊗ 1 ⊗ v+ → 2λ ⊗ 1,

(Lemma 6.18.2). For the sphere S n with the round metric gc , the conformal covariance of DM (g; λ) generalizes the equivariance of the intertwining map C ∞ (G, C(λ))P → (C ∞ (G, C(λ−1)) ⊗ F ∗ )P induced by p(λ). Analogous tractor D-operators are used on the hypersurface Σ. An additional ingredient is a substitute for the embedding In : Fn → Fn+1 . It is given by a conformally covariant projection operator ΠΣ : Γ(T ∗ M ) → Γ(T ∗ Σ) (Theorem 6.20.6). The formal deﬁnitions of the conformally covariant tractor families T (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ) DN

are given in Deﬁnition 6.21.1, Deﬁnition 6.21.2 and Deﬁnition 6.21.3. More precisely, the ﬁrst two deﬁnitions describe what is meant by iterated curved translation of the embedding i : Σ → M and of (versions of) the conformally covariant T (M, Σ; g; λ) are normal derivative D1 (M, Σ; g; λ). Then the tractor families DN deﬁned by division of the resulting compositions by certain polynomials in λ (Definition 6.21.3). This is the curved analog of the factorization in Theorem 1.9.2. In general, the tractor families have simple poles with non-vanishing residues. For a detailed discussion of the role of these poles we refer to the introduction of Chapter 6. For (M, Σ) = (S n+1 , S n ), the tractor families give rise to intertwining famT c (S n+1 , S n ; gc ; λ) = DN (λ). In particular, these ilies. More precisely, we have DN special cases can be used as test cases for general statements on tractor families. We recall that the value of the critical family Dnc (λ) at λ = 0 lives on S n , and is given by the critical GJMS-operator Pn (S n , gc ) on the sphere. Moreover, n D˙ nc (0)(1) = −(−1) 2 Qn (S n ; gc ). This generalizes (1.6.12) and ﬁts with the pattern formulated in Section 1.8. In contrast to the general case, the tractor families c (λ) do not have poles. DN c (λ) : C ∞ (S n+1 ) → C ∞ (S n ) can be constructed The equivariant families D2N in two ways: either by induction by homomorphisms of Verma modules or by the residue method of Section 1.5. On the other hand, they are not covered by the general theory of residue families outlined in Section 1.6. This is a consequence of the c res (λ) and D2N (S n ; hc ; λ) observation that the respective polynomial degrees of D2N c are 2N and N . The fact that D2N (λ) is a tractor family, which also admits a residue type construction, suggests to ask whether such constructions exist for other tracb (λ) : C ∞ (Rn+1 ) → C ∞ (S n ) tor families as well. One such case are the families DN (Section 5.4). These families intertwine (the non-compact model of) spherical principal series on Rn+1 with spherical principal series on S n for those conformal maps

50

Chapter 1. Introduction

c b of Rn+1 ∪{∞} which leave the sphere S n invariant. Like DN (λ) the families DN (λ) are induced by homomorphisms of Verma modules. We note that the polynomial c b (λ) and D2N (λ) are 2N and 3N , respectively. degrees of the induced families D2N These diﬀerent degrees reﬂect the properties of the second fundamental form L in both cases: (S n+1 , S n ) and (Rn+1 , S n ) both are totally umbilic, but the mean curvatures diﬀer. A consequence of the latter observation concerning degrees is that, in general, tractor families are not determined recursively by (analogs of) factorization identities. If Σ is the boundary of a compact manifold M , the tractor families, for speciﬁc choices of the parameter λ, were used before by Branson and Gover ([40]) for the construction of conformally covariant elliptic self-adjoint boundary value problems. Such boundary value problems give rise to conformally covariant pseudodiﬀerential operators on Σ (see also [111]). The tractor families are curved versions of those semi-holonomic lifts of DN (λ) which are constructed by iterated translation. In general, there exist semiholonomic lifts which diﬀer from this speciﬁc construction (Section 6.18). The tractor families specialize to residue families (at least) in the conformally ﬂat case. This and further aspects of the construction will be discussed in Section 1.10.

1.10 Holographic duality. Extrinsic Q-curvature. Odd order Q-curvature Associated to a Riemannian manifold (Σ, h) (dim Σ = n) we have the sequence res DN (h; λ) : C ∞ ([0, ε) × Σ) → C ∞ (Σ)

(N ≤ n for even n) of residue families and the sequence T DN (dr2 +hr ; λ) : C ∞ ([0, ε) × Σ) → C ∞ (Σ)

(N ≤ n for even n) of tractor families. Both sequences are completely determined by the metric h. Moreover, both sequences coincide in the ﬂat case (Σn , h) = (Rn , hc ). This follows from the interpretation of the residue families as homomorphisms of Verma modules which, in turn, can be decomposed as products of tractor operators for the Euclidean metrics on Rn+1 and Rn . The holographic duality concerns generalizations of that relation to the curved case (see Theorem 6.21.2). Theorem 1.10.1 (Holographic duality). For any conformally ﬂat Riemannian manifold (Σ, h), res T (h; λ) = DN (dr2 +hr ; λ), N ∈ N. DN

1.10. Holographic duality. Extrinsic Q-curvature. Odd order Q-curvature

51

Theorem 1.10.1 deals with residue families of any order. The odd order families are deﬁned similarly as the even order ones (Deﬁnition 1.6.1). We compare the res (λ) rests inputs of both sides of the duality. The deﬁnition of the residue family DN on the holographic coeﬃcients v2j and a sequence of diﬀerential operators which describe the asymptotics of an eigenfunction of the Laplacian for the Poincar´eT (dr2 +hr ; λ) is deEinstein metric of h. On the other hand, the tractor family DN ﬁned in terms of the tractor connections of the conformal compactiﬁcation dr2+hr of the Poincar´e-Einstein metric of h and of the metric h. The relation of both constructions gives tractor formulas for the eigenfunctions. Note that, in contrast to general metrics h, the families in Theorem 1.10.1 exist for all orders N , since, for conformally ﬂat h, the Taylor series of hr breaks oﬀ at r4 . The holographic duality can be regarded either as a formula for the complicated product on the right-hand side, or as the statement that the left-hand side is the specialization of a conformally covariant family to the metric dr2+hr . Theorem 1.10.1, in particular, implies that the rational tractor families are polynomial in λ. We expect that the duality extends to the pseudo-Riemannian case. The following consequences of Theorem 1.10.1 are of special importance. Corollary 1.10.1. For a conformally ﬂat metric h, Dnres (h; 0) = DnT (dr2 +hr ; 0), Dnres (h; λ)(1) = DnT (dr2 +hr ; λ)(1). Corollary 1.10.1 covers the critical cases of even and odd order. For even n, it implies the relation DnT (dr2 +hr ; 0) = Dnres (h; 0) = Pn (h)i∗

(1.10.1)

(by (1.6.11)) and the relation n

T 2 2 Qres n (h; λ) = −(−1) Qn (dr +hr ; λ)

(1.10.2)

of the Q-polynomials n

res 2 Qres n (h; λ) = −(−1) Dn (h; λ)(1)

def

and QTn (g; λ) = DnT (g; λ)(1).

˙ res Since Qres n (h; 0) = 0 and Qn (h; 0) = Qn (h) (see (1.6.12)), Theorem 1.10.1 says that n Q˙ Tn (dr2 +hr ; 0) = −(−1) 2 Qn (h). This is a tractor formula for the critical Q-curvature Qn (h). (1.10.1) is a tractor formula for the critical GJMS-operator Pn (h) (for a conformally ﬂat metric h). It is a challenge to extend the holographic duality to metrics which are not conformally ﬂat. We prove analogs of Corollary 1.10.1 for second-order families (in the non-critical case n ≥ 3), for order 3 families (in the case n ≥ 3) and

52

Chapter 1. Introduction

for fourth-order families (in the critical case n = 4) ((6.22.5), Theorem 6.25.4, Theorem 6.22.3 and Theorem 6.22.4). Theorem 1.10.1 implies that for conformally ﬂat metrics the tractor families T (dr2 +hr ; λ) satisfy systems of factorization identities: these are consequences DN of corresponding identities for the residue families (Theorem 1.7.1). It would be interesting to determine to which extent, and in which form, the full theory of factorization identities generalizes to tractor families. The conformally covariant critical tractor family DnT (M, Σ; g; λ) (dim Σ = n) is deﬁned for any hypersurface i : Σ → M . DnT (M, Σ; g; λ) is rational in λ with a possible simple pole at λ = 0. However, for even n, it is conjectured to be regular at λ = 0 (as in Theorem 1.10.1). Then the pair

DnT (M, Σ; g; 0), D˙ nT (M, Σ; g; 0)(1) is well deﬁned. In the situation of Corollary 1.10.1, this pair is given by n Pn (h)i∗ , −(−1) 2 Qn (h) . In particular, it only depends on the metric h = i∗ (g), g = dr2 + hr . In general, the situation is more complicated, and the behaviour of the critical tractor family near λ = 0 is described in the following conjecture. Conjecture 1.10.1 (Decomposition). The operator PnT (·, Σ; g) = DnT (·, Σ; g; 0) admits a decomposition PnT (·, Σ; g) = Pn (Σ, i∗ (g))i∗ + Pne (·, Σ; g) into conformally covariant diﬀerential operators Pn : C ∞ (Σ) → C ∞ (Σ),

Pne : C ∞ (M ) → C ∞ (Σ),

n

where Pn has leading term Δ 2 . The operator Pne depends on the metric in a small neighborhood of Σ, but lives on Σ. Similarly, QTn (·, Σ; g) = D˙ nT (·, Σ; g; 0)(1) admits a decomposition n

−(−1) 2 QTn (·, Σ; g) = Qn (Σ, i∗ (g)) + Qen (·, Σ; g). Moreover, the pairs (Pn , Qn ) and (Pne , Qen ) satisfy the respective fundamental identities enϕ Qn (ˆ g ) = Qn (g) − Pn (g)(ϕ) on Σ and g ) = Qen (g) − Pne (g)(ϕ). enϕ Qen (ˆ

1.10. Holographic duality. Extrinsic Q-curvature. Odd order Q-curvature

53

Qen will be called the extrinsic Q-curvature. For n = 4, we conﬁrm Conjecture 1.10.1 by proving the following result (see Section 6.22). Theorem 1.10.2. For (M 5 , Σ4 ; g), the operator P4T (g; 0) and the function QT4 (g) decompose as D4T (g; 0) = P4 (i∗ (g))i∗ + P4e (g)i∗ and

−QT4 (g) = Q4 (i∗ (g)) + Qe4 (g),

respectively. Here (P4 , Q4 ) is the Paneitz pair on Σ for the metric i∗ (g). The extrinsic pair (P4e , Qe4 ) is given by the formulas P4e = −4δ(J #d) and Qe4 = 4δ(δJ )−2Δ(tr J ), where J is the conformally invariant symmetric bilinear form 1 J = i∗ (PM ) − PΣ + HL − H 2 g 2 with trace tr(J ) =

(1.10.3)

1 |L0 |2 . 6

Here L0 is the trace-free part of the second fundamental form L. The pair (P4e , Qe4 ) satisﬁes the fundamental identity ˆ e = Qe + P e (ϕ). e4ϕ Q 4 4 4 Moreover, for the conformal compactiﬁcation g = dr2+hr of the Poincar´e-Einstein metric extending the metric h on Σ, the extrinsic pair (P4e , Qe4 ) vanishes since the invariant J vanishes. Theorem 1.10.2 implies that for any metric g, which satisﬁes the conformally invariant condition J (g) = 0, the tractor family D4T (·, Σ4 ; g; λ) yields the pair (P4 , Q4 ) of Σ with the induced metric. In a conformally ﬂat background g, the invariant J can be written in the form 1 1 |L20 |g . L20 − n−2 2(n − 1) (Lemma 6.23.3). Thus J relates to L20 in the same way as the Schouten tensor relates to the Ricci tensor.

54

Chapter 1. Introduction

An obvious problem is to extend that picture to general dimensions by determining a conformally invariant condition on g which forces the extrinsic pair (Pne , Qen ) to vanish. In the case (S 5 , S 4 ) with the round metric, we easily ﬁnd J = 0, i.e., the extrinsic pair vanishes. The tractor construction of P4 requires a choice of some embedding i : Σ → M with a vanishing J -tensor and a formal calculation. One should compare this with the construction of P4 using Poincar´e-Einstein metrics. It requires us to ﬁnd the Poincar´e-Einstein extension of the given metric h on Σ (i.e., to determine h(2) and tr h(4) in terms of h) followed by a calculation of the ﬁrst two terms in the asymptotics of eigenfunctions. For odd n, the critical tractor family DnT (M, Σ; g; λ) has a simple pole at λ = 0. Its residue is a conformally covariant diﬀerential operator. If it vanishes, then the fundamental pair

DnT (M, Σ; g; 0), D˙ nT (M, Σ; g; 0)(1) (1.10.4) is well deﬁned and satisﬁes a fundamental identity. The ﬁrst component is a conformally covariant odd order diﬀerential operator C ∞ (M ) → C ∞ (Σ). The second component QTn (M, Σ; g) = D˙ nT (M, Σ; g; 0)(1) (1.10.5) deﬁnes a notion of odd order Q-curvature. In contrast to even n, the pair (1.10.4) is not an invariant of the metric i∗ (g), but depends on the embedding. It is the analog of the extrinsic Q-curvature deﬁned above for even n. For odd n, there is no intrinsic Q-curvature. In the case n = 3, the critical tractor family D3T (M, Σ; g; λ) is regular at λ = 0 if L0 vanishes, and the resulting pair P3T (g) = D3T (g; 0) and QT3 (g) = D˙ 3T (0)(1) is related to a pair which was discovered by Chang and Qing ([68]) in connection with Polyakov-formulas for boundary value problems on 4-manifolds. In particular, the curvature quantity QT3 is the T -curvature of [68]. T -curvature generalizes geodesic curvature of the boundary of a surface. It gives rise to interesting geometric ﬂows on 4-manifolds with boundary ([186]) which supplement Q-curvature ﬂows ([49]). T (M, Σ; g; λ) give rise to The rational tractor families DN T QTN (M, Σ; g; λ) = DN (M, Σ; g; λ)(1) ∈ C ∞ (Σ). ∞ These are analogs of the Q-polynomials Qres N (Σ; h; λ) ∈ C (Σ). For a conformally ﬂat h, we have (Corollary 1.10.1)

DnT (dr2 +hr ; λ)(1) = Dnres (h; λ)(1),

1.11. Review of the contents

55

i.e., in this case the critical tractor Q-polynomial coincides with the critical Qpolynomial Qres n (h; λ). The main problems here are to determine to what extent res the recursive structure of the polynomials Qres 2N (h; λ) (and the families D2N (h; λ)) (discussed in Section 1.7) continues to exist for the tractor Q-polynomials (it is not yet known that these are polynomials), and which information on the embedding (Σ → M, g) is encoded in QTn (M, Σ; g; λ) and the lower order analogs. c T (λ) = DN (S n+1 , S n ; gc ; λ), the corresponding QFor the tractor families DN polynomials are determined in Lemma 5.2.11 and Lemma 5.2.14. In particular, for even n, n QTn (S n+1 , S n ; gc ; λ) = (−1) 2 λ(λ−1) . . . (λ−n+1). Theorem 1.10.2 emphasizes the submanifold perspective towards Q-curvature in an example. We close the present section with a description of the main features of that perspective. The critical Q-curvature is viewed as a quantity which naturally appears in (certain) critical families DnT (M, Σ; g; λ) of conformally covariant diﬀerential operators. These families live in a neighborhood of Σ → X. In that sense, Q-curvature emerges from an extrinsic point of view. The tractor families, in general, only lead to a Q-like curvature, and the actual relation to the Q-curvature remains to be determined. The approach shares the extrinsic point of view (in one form or another) with other approaches. We recall that the Feﬀerman-Graham ambient metric construction ([99]) is the basis of the construction of the GJMS-operators Pn ([124]). The ambient metric construction is a speciﬁc embedding of a given conformal manifold into an ambient space with a Ricci ﬂat (ambient) metric. The powers of the Laplacian of the ambient metric induce the GJMS-operators on the original manifold (Section 3.1). Here we adopt a diﬀerent point of view. Although we work with an ambient framework as well, we do not pose an apriori condition on the extension of the metric. Instead, we choose a speciﬁc conformally covariant construction for all metrics which pull back to a given one under the embedding. Separating contributions which depend on the extension from those which do not depend on it, ﬁnally leads to intrinsic results. For speciﬁc background metrics (as those, for example, which are related to the ambient metric or rather the associated Poincar´e-Einstein metric), the extrinsic contributions vanish (if Σ has even dimension).

1.11 Review of the contents In Chapter 2, we introduce basic notation and summarize background material. We describe Lie groups and their actions which are related to various models of hyperbolic spaces (Section 2.1). Of central signiﬁcance are suitable models for the spherical principal series representations. These are usually deﬁned as induced representations from parabolic subgroups. We relate these deﬁnitions to geometric

56

Chapter 1. Introduction

deﬁnitions which are adapted to their role in conformal geometry (Section 2.3). These issues are closely related to various versions of Helgason-Poisson transformations. A construction of Nayatani ([185]) provides us with an intrinsically deﬁned canonical invariant metric on the complement of limit sets of Kleinian groups (Section 2.4). In Section 2.5, we ﬁx our conventions in Riemannian geometry, and prove the basic facts concerning the conformal transformation properties of various curvature quantities. The construction of conformally covariant powers of Laplacian (GJMS-operators) is reviewed in Chapter 3. Here we describe results of Graham et al. ([124]), Branson ([32]) and Graham-Zworski ([128]). In Chapter 4, we discuss in detail the example of the fourth-order Paneitz operator and the related Q-curvature. In particular, we prove the conformal transformation law of Q4 . The main algebraic result of the book is Theorem 1.4.3. It yields the families DN (λ) ∈ U(n− n+1 ). Its proof splits into two parts according to the parity of the order N (Theorem 5.1.1 for even N and Theorem 5.1.3 for odd N ). We characterize these families in terms of Gegenbauer polynomials (Theorem 5.1.2, Theorem 5.1.4). The families DN (λ) induce families of equivariant diﬀerential operators c nc DN (λ) : C ∞ (S n ) → C ∞ (S n−1 ) and DN (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ). The induction and the mutual relations between these families are discussed in Section 5.2. nc (λ) directly follows from the corresponding formula The explicit formula for DN for DN (λ). We prove Theorem 1.4.1 by using the recursive method which rests on the system of factorization identities. The fact that the families DN (λ) can be read oﬀ from the asymptotics of eigenfunctions of the Laplacian for the hyperbolic metric on the upper half-space is proved in Theorem 5.2.5 and Theorem 5.2.6. In Section 5.1.3, we prove that Theorem 5.1.1 and Theorem 5.1.3 ﬁnd their natural interpretations as statements about homomorphism of Verma modules. c (λ) In Section 5.3, we derive explicit formulas for the intertwining families DN of order ≤ 4. In Section 5.4, we discuss a further series of examples. We use the relation between the residue method and geometric induction in order to ﬁnd b (λ) : C ∞ (Rn ) → C ∞ (S n−1 ) which are explicit formulas for low order families DN induced by DN (λ). We use the asymptotics of eigenfunctions of the Laplacian of the Poincar´e metric on the ball Bn in order to derive explicit formulas for the b families of order at most 3. In that case, the polynomial degree of D2N (λ) is 3N . The detailed presentation serves also as a preparation for the later discussion of curved analogs in Chapter 6. As an application, we prove that the operator D3b (0) : C ∞ (B4 ) → C ∞ (S 3 ) is proportional to the conformally covariant ChangQing operator P3 ([68]) specialized to the present conformally ﬂat situation. The general case is treated in Section 6.26. c (λ) in the theory of In Section 5.5, we describe the role of the families DN automorphic distributions and Selberg zeta functions.

1.11. Review of the contents

57

In the extensive Chapter 6, we leave the framework of group-induced families and conformally ﬂat situations. For a detailed description of the contents of Chapter 6 we refer to the introduction of that chapter. Here we only review the main points. In Section 6.2, we discuss a ﬁrst-order family. In Section 6.3, we prove Theorem 1.9.1. In Section 6.4, we generalize that result to codimension one oriented submanifolds Σ of manifolds M of dimension n ≥ 4 and arbitrary background metrics (Theorem 6.4.1). In Section 6.24, the same families will be constructed using tractor calculus. Section 6.5 and Section 6.6 are devoted to a discussion of the residue method in the curved case. Here we construct the residue families and discuss the holographic formulas (1.6.14), (1.6.15) and (1.6.16) for the critical Q-curvature. In Section 6.7, we prove that the second-order residue family, actually coincides with the family discussed in Section 6.4, specialized to the background metric given by the conformal compactiﬁcation of a Poincar´e-Einstein metric. In the same spirit, we identify in Section 6.8 the value at λ = 0 of the order 3 residue family with the corresponding Chang-Qing operator P3 (dr2 + hr ) (which will be discussed in detail in Section 6.26). Section 6.9 and Section 6.10 contain alternative direct proofs of the holographic formulas for n ≤ 6. These results provide additional cross-checks of the holographic formulas. For that purpose, we determine the ﬁrst three terms in the Feﬀerman-Graham expansion (Theorem 6.9.1). These are used to derive explicit formulas for the holographic coeﬃcients v2 , v4 and v6 (Theorem 6.9.2). Section 6.10 is devoted to a detailed discussion of Q-curvature Q6 in all dimensions n ≥ 6. In Section 6.11, we discuss the recursive structures outlined in Section 1.7. The central tools are the factorization identities for residue families (Theorem 1.7.1). In particular, we describe the algorithm which leads to the recursive formula (1.7.2) for the critical Q-curvature, prove Theorem 1.7.3 and Theorem 1.7.4, derive the explicit formula for Q6 directly from the factorizations, describe the recursive structure of the Q-polynomials, prove the alternative recursive formulas (1.7.11) for Q6 , and relate its quadratic coeﬃcient to the renormalized volume of Poincar´eEinstein metrics. In the following two sections, we discuss two applications of some results in Section 6.11. In Section 6.12, we derive the recursive formula (1.7.12) for P6 by inﬁnitesimal conformal variation of (1.7.11) and observe its universality. In Section 6.13, we derive the recursive formula (1.7.15) for Q8 . In Section 6.14, we determine the holographic coeﬃcients v2j for conformally ﬂat metrics. We describe the consequences of (1.6.16) in the conformally ﬂat case. In particular, we detect the Euler-form in the Q-curvature. We also show that for conformally ﬂat metrics the explicit formula for Q6 was derived in [12]. Finally, we emphasize similarities between holographic renormalization of the Einstein-Hilbert action and Weyl’s tube formula.

58

Chapter 1. Introduction

In Section 6.15, we describe the Henningson-Skenderis test of the AdS/CFT duality. It interprets the holographic conformal anomaly v4 as a linear combination of quantum conformal anomalies. Here the ﬁeld content of the SU (N ) super YangMills multiplet suggest the correct choice of operators. The calculations rest on explicit formulas for heat-equation coeﬃcients. In Section 6.16, we demonstrate that the holographic formula implies Gover’s formula for the Q-curvature of an Einstein metric. In Section 6.17 and Section 6.18, we recall basic ideas on geometric induction using Verma modules and Cartan connections. That approach cannot be used to construct the critical GJMS-operators. The obstruction comes from the non-existence of a certain lift of a homomorphism of Verma-modules to a homomorphism of semi-holonomic Verma modules. In Section 6.19, we prepare the translation from the Lie algebra framework to the curved framework. Some tools from conformal tractor calculus are developed in Section 6.20. The discussion in these sections is self-contained, and can be read as an introduction to tractor calculus. For more background material on tractor calculus see [17], [88], [54], [109] and the references in [41]. In Section 6.21, we formulate the principles around the construction of the T tractor families DN (·, Σ; g; λ). In particular, we introduce the notion of extrinsic Q-curvature of a codimension one embedding, formulate the holographic duality, and deﬁne an odd order version of Q-curvature. In Section 6.22, we analyze the constructions of Section 6.21 in special cases, and supply evidence for the conjectures. In particular, we determine the extrinsic pair (P4e , Qe4 ) (Theorem 1.10.2). In Section 6.24, we recover the order 2 family constructed in Section 6.4 by the general tractor construction. In Section 6.25 and Section 6.26, we prove that the order 3 tractor family contains the Chang-Qing pair (P3 , T ) (i.e., an order 3 operator together with a curvature invariant) if L0 = 0.

1.12 Some further perspectives The residue method of the present work applies Poincar´e-Einstein metrics in a neighborhood of the submanifold Σ → M . Another application of Poincar´eEinstein metrics is suggested by the following observations. The case (M, Σ) = (S n , S n−1 ) with the canonical metrics gc on spheres can be viewed as the boundary situation associated to an isometric embedding Hn → Hn+1 of hyperbolic spaces. Thus (S n−1 , [gc ]) is the conformal inﬁnity of the submanifold Hn of Hn+1 with conformal inﬁnity (S n , [gc ]). A natural problem is to construct the families c DN (S n , S n−1 ; λ) from that point of view using asymptotic analysis of eigenfunctions of Laplacians on hyperbolic spaces. The appropriate mathematical tool here,

1.12. Some further perspectives

59

of course, is the theory of Poisson transformations ([140]). These group-equivariant transforms can be used to detect the counterpart in the spectral theory on hyperbolic spaces of the family theory on the boundary. Here Gegenbauer polynomials play a central role, and this connection actually can be viewed as the reason for their occurrence in the boundary theory ([150]). The above situation is a typical special case of those considered in connection with the AdS/CFT correspondence for submanifold observables. Here the expectation value of a k-dimensional submanifold Σ of M (a k-brane) is calculated using a path integral with a brane wrapped on a k + 1-dimensional submanifold with boundary Σ of an Einstein space with M as boundary. Note that the renormalized volume of a surface observable leads to the Willmore functional ([127], [7]). That perspective suggests that we construct conformally covariant families by using submanifolds of Poincar´eEinstein spaces associated to the conformal inﬁnity (M, Σ; [g]). A closely related problem is to ﬁnd ambient metric constructions. It seems natural to extend the theory to submanifolds Σ of higher codimension. A typical case again is that of a standard embedding S m → S n of spheres. The theory of the Selberg zeta function for Kleinian groups with limit set S m again suggests the existence of Gm+1 -equivariant families of diﬀerential intertwining operators C ∞ (S n ) → C ∞ (S m ). An extension of the residue method would construct such families from an asymptotic analysis of generalized eigenfunctions of the Laplacian on the complement Ω of S m in S n . The ﬁrst obvious question is which metric is to be used on Ω. For the application to zeta functions and related automorphic distributions, that metric must be invariant under the conformal group of the complement. This group is isomorphic to SO(1, m + 1) × SO(n − m), and a canonical choice of such a metric in the conformal class of gc is the Nayatani metric ([162], [185]). It provides an isometry S n \ S m → Hm+1 × S n−m−1 with the direct product of an Einstein metric with negative scalar curvature and an Einstein metric with constant positive curvature. In the case m = n − 1 discussed in the present work, the second factor degenerates to S 0 = {±}, i.e., Ω is isometric to two copies of hyperbolic space. The generalized eigenfunctions of the Laplacian of that metric are determined by functions (or hyperfunctions) on the geodesic boundary S m of Hm+1 and eigenfunctions of the Laplacian on the second factor. Spherical harmonics then serve as an additional parameter (besides the spectral parameter) for the families C ∞ (S n ) → C ∞ (S m ). The product space H5 × S 5 (or rather AdS5 × S 5 ) is the framework of a basic special case of the AdS/CFT correspondence which proposes a duality between gravity on the ten-dimensional AdS5 × S 5 and conformal ﬁeld theory on the boundary S 4 (or rather the compactiﬁed Minkowski space M 4 ) of H5 (or rather AdS5 ). The metric on the product is the direct product of the canonical metrics (see [172] and [241]), i.e., it coincides with the Nayatani metric.

60

Chapter 1. Introduction

As an example of a conformally covariant family in a higher codimension situation, we note the following generalization of Theorem 1.9.1. For any isometric embedding Σm → M n (m < n) in the background metric g, the family D2 (M, Σ; g; λ) = −(m−2+2λ)i∗P2 (M, g) + (n−2+2λ)P2(Σ, g)i∗ + (n−2+2λ)(m−2+2λ)i∗∇H n−2 m−2 + 2 λ+ Q2 (Σ; g) − Q2 (M ; g) − λ|H|2 i∗ λ+ 2 2 is conformally covariant. Here H denotes the mean curvature vector. In the critical case m = 2, we ﬁnd D2 (M ; Σ; g; 0) = (n−2)P2 (Σ; g) and D˙ 2 (M, Σ; g; 0)(1) = (n−2)Q2(Σ; g). The main constructions (residue families and tractor families) extend to differential forms, and yield conformally covariant families of local operators on pforms. In particular, the residue families yield Q-curvature operators (p)

Qn−p (h) : ker(d | Ωp (M )) → Ωp (M ),

Q(0) n = Qn

on closed forms which satisfy fundamental identities, and are given by holographic formulas. The details and the relation to Q-curvature operators as constructed in [42] using ambient metric and tractor constructions will be discussed elsewhere. More generally, versions for other vector bundles are of interest, too. Such families should combine with Bernstein-Gelfand-Gelfand-sequences on M and Σ ([60]). Although the present work only deals with the Riemannian case, it is clear that most of the arguments will generalize to the broader setting of metrics with non-trivial signature. In particular, the asymptotic behaviour of eigenfunctions, the relation to Verma module theory ([16]) and the tractor constructions generalize appropriately. In fact, Q-curvature is deﬁned, and the holographic formula actually is valid in the general case ([125]). In another direction, the conformal setting can be enlarged to the setting of parabolic geometry ([54], [109], [222]). In particular, this would cover CRmanifolds ([98], [215], [214], [143], [144]). The holographic formulas for Q-curvature oﬀer natural ways to organize the Riemannian invariants which contribute. Explicit formulae are important for applications to problems in geometric analysis such as generalized Polyakov-formulas, Q-curvature prescription problems, Q-curvature ﬂows, geometric relevance of Qcurvature etc. ([63], [64], [66], [71], [74], [84], [49]). It would be interesting to see applications in such directions beyond 4 dimensions. Finally, we note that it is actually not too surprising that, starting from Selberg zeta functions, we came close to ideas in the AdS/CFT-correspondence and holography. In fact, the idea of studying the Selberg zeta function of a discrete subgroup Γ of SO(1, n)◦ in terms of the conformal action of Γ on the boundary of

1.12. Some further perspectives

61

hyperbolic space can be viewed as a version of a holography principle. For other discussions of relations between Kleinian groups and holography we refer to [154] and [160]. Such relations for Schottky groups in hyperbolic 3-space are naturally connected also with arithmetic geometry ([174]). Patterson’s conjecture concerning the divisor of Selberg zeta functions for Kleinian groups is another natural part of the picture ([50], [190], [151]). As described in Section 1.1, any Kleinian group Γ induces a hyperbolic quotient space X = Γ\Hn (Kleinian manifold), as well as a conformal dynamics on S n−1 and a continuous dynamics (geodesic ﬂow of X). The interplay between these three pictures has been the source of many exciting developments ([79], [151] and the references therein). In the arithmetical special case P SL(2, Z), it led to the work [61], [167].

Chapter 2

Spaces, Actions, Representations and Curvature 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 lightcone model, we derive formulas for its actions on S n−1 and Rn−1 ∪ {∞} using stereographic projection. These actions induce two models of the spherical principal series representations. We deﬁne these in a form which emphasizes the conformal nature of the geometric actions with respect to canonical metrics in each model. Moreover, we prove the equivariance of the corresponding Poisson transformations. These intertwine the principal series representations with eigenspace representations for the Laplacian on hyperbolic space. Finally, we derive the standard transformation laws for the Riemannian curvature, Ricci curvature and scalar curvature with respect to conformal changes of the metric.

2.1 Lie groups, Lie algebras, spaces and actions Let O(1, n) be the Lie group of all linear automorphisms of (R1,n , q), where q is the quadratic form q(x, y) = −x0 y0 + x1 y1 + · · · + xn yn . Let SO(1, n) = {T ∈ O(1, n) | det T = 1}, and let Gn = SO(1, n)◦ be the connected component of the identity in SO(1, n). Gn is a real simple Lie group of real rank one. Let gn be the Lie algebra of Gn . We write elements in Gn = SO(1, n)◦ as matrices of the block forms ⎛

⎞ 1×1 1×1 1 × n−1 ⎝ 1×1 1×1 1 × n−1 ⎠ , n−1 × 1 n−1 × 1 n−1 × n−1 K n = SO(n) embeds into Gn by T →

1 0

0 T

1×1 n×1

1×n . n×n

as a maximal compact subgroup.

64

Chapter 2. Spaces, Actions, Representations and Curvature

Let A be the group of the matrices ⎛ cosh t at = ⎝ sinh t 0

sinh t cosh t 0

Its Lie algebra a is generated by

⎞ 0 0⎠ , t ∈ R. 1 ⎛

0 1 def H0 = d/dt|t=0 (at ) = ⎝1 0 0 0

⎞ 0 0⎠ . 0

Let M n be the centralizer of a in K n . Then M n ⊂ K n is the group of all matrices of the form 1 0 , T ∈ SO(n−1). 0 T 0 0 The matrices , i, j = 1, . . . , n−1, i < j such that 0 Mij (Mij )rs = δir δjs − δis δjr form a basis of the Lie algebra mn of M n . The restricted root system Δ(g, a) consists of two elements ±α0 . We ﬁx a positive root α0 by the condition α0 (H0 ) = 1. Let a+ = {X ∈ a | α0 (X) > 0} be the positive Weyl chamber. The Weyl group W = W (g, a) consists of two elements. Let w be the non-trivial one. gn decomposes as − gn = n + n ⊕ mn ⊕ a ⊕ nn , + − where n± n = {Y ∈ gn | [X, Y ] = ±α0 (X)Y, X ∈ a}. The Lie algebras nn and nn consist of the respective matrices ⎛ ⎞ ⎛ ⎞ 0 0 Y+ 0 0 Y− ⎝ 0 0 Y +⎠ , ⎝ 0 0 −Y − ⎠ + t + t − t − t (Y ) −(Y ) 0 (Y ) (Y ) 0 ± where Y ± = (y1± , . . . , yn−1 ) ∈ Rn−1 . n± n are the Lie algebras of the abelian Lie + − groups Nn and Nn . We choose ⎞ ⎞ ⎛ ⎛ 0 0 ej 0 0 ej 0 ej ⎠ , Yj− = ⎝ 0 0 −ej ⎠ , j = 1, . . . , n−1 (2.1.1) Yj+ = ⎝ 0 t t ej −ej 0 etj etj 0 n−1 as standard bases of n± has the components (ej )k = δjk . In these n . Here ej ∈ R terms, the following commutator relations hold true: + − Yi , Yj = 2δij H0 + 2Mij , Mij , Yr± = δjr Yi± − δir Yj± , (2.1.2) H0 , Yj± = ±Yj± ,

[H0 , Mij ] = 0 .

2.1. Lie groups, Lie algebras, spaces and actions

65

The abelian groups (N n )± are realized as ⎧ ⎞⎫ ⎛ s 1 + |s|2 /2 ∓|s|2 /2 ⎪ ⎪ ⎪ ⎪ ⎨ ⎟⎬ ⎜ n ± ± 2 2 ⎟ , /2 1 − |s| /2 ±s ±|s| (N ) = ns = ⎜ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ st ∓st 1(n−1,n−1) n−1 where s = (s1 , . . . , sn−1 ) ∈ Rn−1 and |s| = j=1 s2j . Now let C(−1) = ξ = (ξ0 , ξ ) ∈ R1,n | |ξ |2 − ξ02 = −1, ξ0 > 0 . Gn = SO(1, n)◦ operates transitively on C(−1), and as homogeneous spaces C(−1) SO(1, n)◦ /SO(n), where SO(n) is the isotropy group of (1, 0). SO(1, n)◦ also operates transitively on the upper light cone C(0) = ξ = (ξ0 , ξ ) ∈ R1,n | |ξ |2 − ξ02 = 0, ξ0 > 0 . Since the action Gn × C(0) → C(0) commutes with the action ξ → λξ, λ > 0 on C(0), it induces an action on the sphere S n−1 (1, ξ ) ∈ R1,n | |ξ | = 1 ⊂ C(0) via the composition g(1, ξ ) g(1, ξ ) R ⊃S ξ → (1, ξ ) → g(1, ξ ) → 1, ∈ S n−1 . → g(1, ξ )0 g(1, ξ )0 (2.1.3) n−1 n n n As homogeneous spaces, S G /P , where P is the parabolic subgroup which ﬁxes the ray R+ (1, 1, 0, . . . , 0). The Gn -equivariant identiﬁcation S n−1 Gn /P n with P = M AN + shows that the geometric G-action on S n−1 can be identiﬁed with the G-action n

n−1

g : kM → κ(g −1 k)M on K n /M n = SO(n)/SO(n−1) which is induced by the Iwasawa decomposition G = KAN + . Here κ(g) exp H(g)n+ (g) is the Iwasawa decomposition of g. d c For g = (with arrows b, c of length n and a scalar d), we have bt A g(1, x) = (d + (c, x), b + xAt ), i.e., g induces the map x → on S n−1 ⊂ Rn .

xAt + b (c, x) + d

(2.1.4)

66

Chapter 2. Spaces, Actions, Representations and Curvature

n The restriction of the Lorentzian metric −dζ02 + i=1 dζi2 on R1,n to S n−1 coincides with the round metric gc induced by the embedding S n−1 → Rn . Moreover, (2.1.3) implies that the maps (2.1.4) are conformal with respect to the metric gc : 1 d c g∗ (gc ) = gc , g = t . (2.1.5) b A (d − (b, x))2 The action (2.1.4) of Gn on S n−1 extends the isometric action on the unit ball Bn in Rn with the metric n 4 dx2 . (1 − |x|2 )2 i=1 i

(2.1.6)

In order to see that, we use the fact that C(−1) (ξ0 , ξ ) →

ξ ∈ Bn 1 + ξ0

is an isometry ([206], Section 4.5). Its inverse is given by B x → n

These formulas show that

x →

d bt

1 + |x|2 2x , 1 − |x|2 1 − |x|2

∈ C(−1).

c operates on Bn by A

2xAt + b(1 + |x|2 ) . 1 − |x|2 + 2(c, x) + d(1 + |x|2 )

(2.1.7)

For x ∈ S n−1 = ∂Bn , the latter formula reproduces (2.1.4). Remark 2.1.1. The above conventions diﬀer from those in part of the conformal geometry literature ([17], [88], [109]). In fact, several authors deﬁne SO(1, n) by the quadratic form 2x0 xn + x21 + · · · + x2n−1 . In that situation it is convenient to write elements in the block form ⎛ ⎞ 1×1 1 × n−1 1×1 ⎝n−1 × 1 n−1 × n−1 n−1 × 1⎠ . 1×1 1 × n−1 1×1 Then M and A are the groups of matrices ⎛ ⎞ 1 0 0 ⎝0 T 0⎠ , T ∈ SO(n−1), 0 0 1

⎛

et ⎝0 0

⎞ 0 0 1 0 ⎠ , t ∈ R. 0 e−t

2.2. Stereographic projection

In terms of ⎛

1 0 H0 = ⎝ 0 0 0 0

⎞ 0 0 ⎠, −1

67

Yj+

⎛ 0 = ⎝0 0

ej 0 0

⎞ 0 −etj ⎠ , 0

⎛

Yj−

0 = ⎝etj 0

0 0 −ej

⎞ 0 0⎠ 0

the commutator relations of that model are + − Yi , Yj = δij H0 + Mij , Mij , Yr± = δjr Yi± − δir Yj± , H0 , Yj± = ±Yj± , [H0 , Mij ] = 0. The choice of the ﬂat model will later be reﬂected, for instance, in the explicit form of the conformally invariant tractor calculus (Chapter 6).

2.2 Stereographic projection Let i denote the mappings Rn−1 x → (x , 0) ∈ Rn and Rn x → (x, 0) ∈ Rn+1 . The second map induces a map i : S n−1 → S n of the standard spheres. Then S n−1 ⊂ S n is the intersection of S n with the hyperplane xn+1 = 0, i.e., H0 (x) = xn+1 is a deﬁning function. H0 will be called the height. Let κ : Rn−1 → S n−1 \ {N }, κ : Rn → S n \ {N } be the stereographic projections with respect to the pole N = (1, 0, . . . , 0) ∈ S n−1 ⊂ S n . Then 2 2x |x | − 1 , ∈ S n−1 κ = κN : x = (x1 , . . . , xn−1 ) → |x |2 + 1 1 + |x |2

and κ = κN : x = (x1 , . . . , xn ) →

|x|2 − 1 2x , |x|2 + 1 1 + |x|2

∈ Sn.

The diagram i

S n−1 −−−−→ / ⏐ ⏐κ i

Sn / ⏐κ ⏐

Rn−1 −−−−→ Rn commutes. We have κN (∞) = N and κN (0) = S = −N . The inverse of κN is given by 1 (x2 , . . . , xn+1 ). κ−1 N (x1 , . . . , xn+1 ) = 1 − x1

68

Chapter 2. Spaces, Actions, Representations and Curvature

We use κ to transport the round metric gc on S n to a metric on Rn . The identity κ∗

∂ ∂xi

2(1 + |x|2 )δij−1 − 4xi xj−1 ∂ ∂ 4xi + , i = 1, . . . , n 2 2 (1 + |x| ) ∂x1 j=2 (1 + |x|2 )2 ∂xj n+1

=

gives κ∗ (gc )

∂ ∂ , ∂xi ∂xj

=

4δij , i, j = 1, . . . , n, (1 + |x|2 )2

(2.2.1)

i.e., κ is an isometry if we use on Rn the spherical metric gs =

n 4 dx2 . (1 + |x|2 )2 i=1 i

(2.2.2)

The pull-back κ∗ (H0 ) of the height function H0 ∈ C ∞ (S n ) will be denoted also by H0 . Then 2xn H0 (x) = ∈ C ∞ (Rn ) 1 + |x|2 and dH0 (x) = −

n−1 j=1

4xj xn 2(1+|x|2 −2x2n ) dx + dxn , j (1+|x|2 )2 (1+|x|2 )2

and the gradient of H0 with respect to the metric gs is given by gradgs (H0 ) = −

n−1

∂ 1 ∂ + (1+|x|2 −2x2n ) . ∂xj 2 ∂xn

(2.2.3)

1 ∂ 11 (1+|x|2 ) . 2 ∂xn xn =0

(2.2.4)

xj xn

j=1

In particular, i∗ gradgs (H0 ) = It follows that κ

∗

(H0−2 gc )

=

1 + |x|2 2xn

2

n n 4 2 −2 dx = x dx2i , i n (1 + |x|2 )2 i=1 i=1

i.e., κ realizes an isometry of the upper hemisphere H + with the metric H0−2 gc and the hyperbolic upper half-space Hn = {(x , xn ) ∈ Rn | xn > 0} with the metric n −2 2 xn i=1 dxi . We use κ to transport the action of SO(1, n + 1) on S n to an action on n R ∪ {∞}. For g ∈ SO(n + 1) ⊂ SO(1, n+1) of the form ⎛ ⎞ 1 0 0 g = ⎝0 d c ⎠ , 0 bt T

2.2. Stereographic projection

we ﬁnd x →

69

2xT t + b(|x|2 − 1) . 1 + |x|2 − 2(c, x) + d(1 − |x|2 )

(2.2.5)

In particular, if b = c = 0, d = 1 and T ∈ SO(n), the action is given by x → xT t , i.e., T operates by standard rotation. Next, a calculation shows that κ−1 (at κ(x)) = et x,

κ−1 (n+ s κ(x)) = x + s.

(2.2.6)

Similar formulas hold true for κ . The action of SO(1, n) 0 ⊂ SO(1, n+1) 0 1 ± S n−1 on S n obviously preserves the hemispheres H ± and the hypersphere = ∂H . SO(1, n) 0 ⊂ Gn+1 Hence under the action on Rn ∪ {∞} the subgroup Gn = 0 1 preserves the half-spaces Rn± and their Rn−1 ∪{∞}. Moreover, the action boundary n −2 2 + preserves the hyperbolic metric xn i=1 dxi on Rn . In fact, for

⎛ 1 g = ⎝0 0

0 T 0

⎞ 0 0⎠ ∈ SO(1, n+1), T ∈ SO(n) 1

this is clear since g preserves H0 ∈ C ∞ (S n ) and the round metric gc on H + . On the other hand, shows that n+ (s) ∈ (N n )+ and at ∈ A preserve n (2.2.6) −2 2 the metric xn i=1 dxi . This proves the claim using the Iwasawa decomposition Gn = K n A(N n )+ . (2.2.6) also implies that stereographic projection can be interpreted naturally in terms of the Bruhat decomposition. We recall that GnS = (P n )− = M n A(N n )−

and GnN = (P n )+ = M n A(N n )+ .

We consider the stereographic projections κS : Rn−1 → S n−1 \ {S} and κN : Rn−1 → S n−1 \ {N } with respect to the poles S = (−1, 0, . . . , 0) = −N and N . Then κS (x ) =

1 − |x |2 2x , 2 1 + |x | 1 + |x |2

∈ S n−1 \ {S}.

Note that κS (x ) = −κN (−x ). Now G = N − P + ∪ wP + = N + P − ∪ wP − ,

70

Chapter 2. Spaces, Actions, Representations and Curvature

where w ∈ SO(n) is a rotation so that N + = wN − w−1 and we have w(S) = N . These decompositions of G are the Bruhat decompositions. It follows that the sphere S n−1 decomposes as S n−1 = G/GN = (N n )− (N ) ∪ wN = (N n )− (N ) ∪ S

(2.2.7)

S n−1 = G/GS = (N n )+ (S) ∪ wS = (N n )+ (S) ∪ N .

(2.2.8)

and + Lemma 2.2.1. n− s (N ) = κS (s) and ns (S) = κN (s).

Proof. (2.2.6) implies (κN )−1 (n+ s κN (0)) = s.

Now κN (0) = S implies the second assertion. The ﬁrst follows from this by conjugation with w. Finally, the stereographic projection κS : Rn−1 ∪ {∞} → S n−1 \ S,

S = (−1, 0, . . . , 0) 1 0 is related to the restriction of the action of σ0 = ∈ Gn+1 , where 0 T0 ⎛

0 0 ⎜−1 0 ⎜ T0 = ⎜ ⎜ 0 −1 ⎝ 0 0

... ... ... ... ...

⎞ 1 0⎟ ⎟ 0⎟ ⎟ ∈ SO(n+1). ⎠ −1 0 0 0 0

(2.2.9)

In fact, (2.2.5) yields σ0 : Rn x = (x1 , . . . , xn ) 1−|x|2 −2x1 −2xn−1 → , , . . . , ∈ Rn . 1+|x|2 −2xn 1+|x|2 −2xn 1+|x|2 −2xn Rn+

The map x → σ0 (τ (x)), where τ (x , xn ) = (x , −xn ) realizes an isometry → Bn for the respective hyperbolic metrics gc =

n 1 2 dx , x2n i=1 i

gc =

n 4 dx2 . (1−|x|2 )2 i=1 i

In fact, |σ0 (x , xn )|2 < 1 iﬀ 0 < −xn (|x |2 +(xn −1)2 ). Hence (σ0 )∗ (vol(Rn− , gc )) = vol(Bn , gc ),

2.3. Poisson transformations and spherical principal series

71

i.e., we obtain the identity (σ0 )∗ (xn ) =

1−|x|2 2

(σ0 )∗ (dx1 . . . dxn ) dx1 . . . dxn

1/n (2.2.10)

which will be useful in Section 5.4. For xn = 0, it restricts to 2 |x | − 1 2x n−1 R , x → − = −κN (x ) = κS (−x ) ∈ S n−1 . |x |2 + 1 |x |2 + 1

2.3 Poisson transformations and spherical principal series It will be convenient to ﬁx the following convention concerning the notation of standard metrics on Euclidean spaces Rm , on the spheres S m and on the models Hm , Bm of hyperbolic space. In all cases, the respective ﬂat, curvature 1 and curvature −1 metric will be denoted by gc (c for canonical). The meaning of gc will be always clear from the context. Thus 2 m • gc = m i=1 dxi for the Euclidean space R , m ∗ 2 m−1 • gc = i → Rm , i=1 dxi for the sphere i : S m 2 m • gc = x−2 m i=1 dxi for the upper half-space H , m • gc = 4(1−|x|2 )−2 i=1 dx2i for the unit ball Bm . First, we consider the ball-model Bm with boundary S m−1 . Gm operates by isometries on Bm and on S m−1 by conformal diﬀeomorphisms of gc . Theorem 2.3.1. The Poisson kernel def

P (x, b) =

1 − |x|2 , x ∈ Bm , b ∈ S m−1 |x − b|2

(2.3.1)

of the ball Bm satisﬁes the identities P (g · O, b) =

g∗ (vol(gc )) vol(gc )

1 m−1

= eg·O,b

(2.3.2)

for g ∈ Gm . Here vol(gc ) = vol(S m−1 , gc ) is the volume form of gc on S m−1 and x, b for x ∈ Bm , b ∈ S m−1 is the (signed ) distance of the unique horosphere ξb (x) through x with normal b from the center O of the ball. For a proof see [140]. Note that the ﬁrst equality in (2.3.2) follows from the version g∗ (gc ) = P (g · O, ·)2 gc , g ∈ Gm

(2.3.3)

72

Chapter 2. Spaces, Actions, Representations and Curvature

of the conformality of the action with respect to gc . In fact, using (2.1.7) we ﬁnd d c g · O = b/(1 + d) for g = , and a calculation (using −d2 + |b|2 = −1) bt A shows that 1 P (g · O, y) = , y ∈ S m−1 . (d − (b, y)) Thus (2.3.3) coincides with (2.1.5). For λ ∈ C, we deﬁne a Gm -module structure on C ∞ (S m−1 ) by def

πλc (g)u =

g∗ (vol(gc )) vol(gc )

λ − m−1

g∗ (u).

(2.3.4)

The representation πλc is called the compact model of the spherical principal series representation of Gm . The space C ∞ (S m−1 ) equipped with the Gm -action πλc will be denoted by C ∞ (S m−1 )λ . By the second identity in (2.3.2), the deﬁnition is equivalent to (2.3.5) πλc (g)u(b) = e−λg·O,b u(g −1 (b)). The latter deﬁnition, in turn, can be rephrased in group-theoretical terms. We identify S m−1 SO(m)/SO(m−1) = K m /M m Gm /P m , where P m is the parabolic subgroup which ﬁxes (1, 0, . . . , 0) ∈ S m−1 . It has the Langlands decomposition P m = M m A(N m )+ . Then (2.3.5) can be identiﬁed with the representation πλc (g)u(k) = eλH(g

−1

k)

u(κ(g −1 k))

(2.3.6)

on C ∞ (K)M , where g = κ(g) exp H(g)n+ (g) ∈ KAN + is the Iwasawa decomposition. Here we identify H(g) ∈ a with the corresponding multiple of H0 . The deﬁnitions (2.3.4), (2.3.5) and (2.3.6) emphasize diﬀerent aspects of the representation πλc . The version (2.3.4) emphasizes the conformal action of Gm on (S m−1 , gc ). (2.3.5) uses the relation to the hyperbolic space and (2.3.6) emphasizes that it can be regarded as an induced representation. The proof of the following result demonstrates the convenience of deﬁnition (2.3.4). Corollary 2.3.1. For the Poisson transformation def ∞ m−1 ) ω(b) → Pλ (ω)(x) = P (x, b)λ ω(b) vol(gc ) C (S S m−1

we have c (g) = g∗ ◦ Pλ . Pλ ◦ πλ−(m−1)

(2.3.7)

2.3. Poisson transformations and spherical principal series

73

Proof. Let vol = vol(gc ). (2.3.2) implies λ (g g) (vol) m−1 ∗ Pλ (ω)(g g · O) = ω vol vol S m−1 λ ∗ λ g (vol) m−1 g (vol) − m−1 ∗ g ∗ (ω vol) = vol vol S m−1 λ ∗ λ g (vol) m−1 g (vol) 1− m−1 ∗ g ∗ (ω) vol = vol vol S m−1 using g ∗ Hence

(g g) (vol)

g∗ (vol) g ∗ (vol) −1 g∗ (vol) ∗ = . = ∗ vol g (vol) vol vol c Pλ (ω)(g g · O) = Pλ (πλ−(m−1) (g −1 )ω)(g · O).

The proof is complete.

Now we consider the case of a totally umbilic hypersphere Σ in S m . Such spheres play a central role in classical conformal submanifold theory, where one associates to any hypersurface of the sphere the mean curvature sphere congruence or central sphere congruence consisting of those totally umbilic subspheres which are tangential of ﬁrst order and have the same mean curvatures as the hypersurface at the points of contact ([142]). It is a classical result that all totally umbilic hypersurfaces are of the form σ(S m−1 ) for some σ ∈ Gm+1 . Σ inherits a Riemannian metric gc from the sphere and we deﬁne λ − m−1 g∗ (vol(Σ, gc )) g∗ (2.3.8) πλ (Σ, gc )(g) = vol(Σ, gc ) for all g ∈ Gm+1 which preserve Σ (vol(Σ, gc ) ∈ Ωm−1 (Σ) denotes the Riemannian volume form). The spheres σ(S m−1 ) is totally umbilic since S m−1 → S m is and σ is a conformal diﬀeomorphism which preserves the trace-free part of the second fundamental form L. For Σ = σ(S m−1 ), the representations πλ (Σ, gc ) and πλ (S m−1 , gc ) are related by the obvious identity

σ∗ (vol(S m−1 , gc )) vol(Σ, gc )

λ − m−1

◦ σ∗ ◦ πλ (Σ, gc )(g) λ − m−1 σ∗ (vol(S m−1 , gc )) m−1 −1 , gc )(σgσ ) ◦ ◦ σ∗ . = πλ (S vol(Σ, gc )

In addition to the compact model, we also need the non-compact model of the principal series. It is induced by the action of Gm+1 on N − Rm which is induced by the Bruhat decomposition. We begin with the discussion of the following analog of Theorem 2.3.1 for the upper half-space.

74

Chapter 2. Spaces, Actions, Representations and Curvature

Theorem 2.3.2. The Poisson kernel def

P (x, ζ) =

xm , |(x , xm ) − (ζ, 0)|2

m−1 x = (x , xm ) ∈ Rm +, ζ ∈ R

of the upper half-space satisﬁes the identity dζ 1

m−1 g∗ (1+|ζ|2 )m−1 , g ∈ Gm , ζ ∈ Rm−1 , P (g · O, ζ) = dζ i.e., P (g · O, ζ)(1 + |ζ| ) = 2

g∗

dζ (1+|ζ|2 )m−1 dζ (1+|ζ|2 )m−1

1

m−1

(2.3.9)

(2.3.10)

= eg·O,ζ ,

(2.3.11)

where O = (0, 0, . . . , 1) and dζ is the (translation invariant ) Riemannian volume form for the Euclidean metric on Rm−1 . Proof. By the Iwasawa decomposition G = KAN + , it suﬃces to prove (2.3.10) for g in the subgroups K SO(m), A R and N + Rm−1 . We use the fact that at operates on Rm by x → et x (see (2.2.6)). Now (at )∗ i.e.,

⎛ ⎝

(at )∗

dζ (1 + |ζ|2 )m−1

dζ (1+|ζ|2 )m−1

dζ

=

1

⎞ m−1

⎠

=

e−(m−1)t dζ , (1 + e−2t |ζ|2 )m−1

e−t et = 2t . −2t 2 1 + e |ζ| e + |ζ|2

On the other hand, we have P (at · O, ζ) =

|(0, et )

et et = 2 . 2 − (ζ, 0)| |ζ| + e2t

Next, we use the fact that N + operates by translations on Rm (see (2.2.6)), i.e., (x , xm ) → (x + y , xm ) and ζ → ζ + y . The latter operation pushes forward the volume form

dζ (1+|ζ|2 )m−1

dζ . (1 + |ζ − y |2 )m−1 On the other hand, for the Poisson kernel we get P ((y , 1), ζ) =

1 . |(y , 1) − (ζ, 0)|2

into

2.3. Poisson transformations and spherical principal series

Finally, for g ∈ SO(m) the assertion is equivalent to dζ dζ = g , ∗ (1 + |ζ|2 )m−1 (1 + |ζ|2 )m−1

75

(2.3.12)

i.e., the SO(m)-invariance of the volume form dζ . (1 + |ζ|2 )m−1

But this is a consequence of (2.2.2). Note that (2.3.9) implies lim P (x, ζ)(1 + |ζ|2 ) = xm ,

ζ→∞

which is consistent with (2.3.11) in view of xm = x, ∞. Since Gm+1 operates on (S m , gc ) by conformal diﬀeomorphisms, the action (2.2.5) consists of conformal diﬀeomorphisms of the metric gs = κ∗ (gc ). Hence it acts by conformal diﬀeomorphisms of the Euclidean metric on Rm . The following deﬁnition emphasizes the conformality of this action. Let def πλnc (g) =

g∗ (vol(gc )) vol(gc )

λ − m

g∗ ,

(2.3.13)

where vol(gc ) denotes the Riemannian volume of the Euclidean metric gc on Rm . (2.3.13) is well deﬁned on the representation space 1 x def ∞ m Vλ (Rm ) = u ∈ C ∞ (Rm ) 1 |x|2λ u (R ) . (2.3.14) ∈ C |x|2

Note that the function x → |x|2λ u

x |x|2

is C ∞ only for x = 0. The condition in (2.3.14) requires that it extends through 0 as a C ∞ function. An equivalent condition is ⎞ ⎛ −1 0 0 ⎠ ∈ SO(m+1). πλnc (w)u ∈ C ∞ (Rm ) for w = ⎝ 0 −1 0 1(m−1,m−1) Note that w(N ) = S. In order to prove the equivalence, we observe that by (2.2.5) w(x) =

(−x1 , x2 , . . . , xn ) . |x|2

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Chapter 2. Spaces, Actions, Representations and Curvature

Hence, by the SO(m+1)-invariance of gs , 2 w∗ (gs ) w∗ (1 + |x|2 ) 1 w∗ (gc ) = = . 2 gc gs 1 + |x| |x|4 Thus

w∗ (vol(gc )) vol(gc )

− nλ

u(w(x)) = |x|2λ u

(−x1 , x2 , . . . , xn ) |x|2

.

This implies the equivalence. Corollary 2.3.2. For the Poisson transformation def ω → Pλ (ω)(x) = P (x, ζ)λ ω(ζ)dζ

(2.3.15)

Rm−1

we have nc Pλ ◦ πλ−(m−1) (g) = g∗ ◦ Pλ , g ∈ Gm .

The Poisson transformation is well deﬁned on the space Vλ−(m−1) (Rm−1 ). The following proof again demonstrates the convenience of the deﬁnition (2.3.13). Proof. Similarly as in the proof of Corollary 2.3.1, we calculate Pλ (ω)(g g · O) = P (g g · O, ζ)λ ω(ζ)dζ m−1 R λ (g g) (vol(g )) m−1 ∗ s = Φ−λ m−1 ω vol(gc ) vol(gs ) Rm−1 λ ∗ λ g (vol(g )) m−1 g (vol(gs )) − m−1 ∗ s g ∗ (Φ−λ = m−1 ω vol(gc )) vol(gs ) vol(gs ) Rm−1 λ ∗ λ g (vol(g )) m−1 g (vol(gc )) 1− m−1 ∗ s = vol(gs ) vol(gc ) Rm−1

λ g ∗ (Φ m−1 ) ∗ × g ∗ (Φ−λ m−1 )g (ω) vol(gc ) Φm−1 λ g (vol(g )) m−1 ∗ s nc = πλ−(m−1) (g −1 )(ω)Φ−λ m−1 vol(gc ) vol(gs ) Rm−1 nc = P (g · O, ζ)λ πλ−(m−1) (g −1 )(ω) vol(gc ) Rm−1

using gs = 4Φ−2 n gc .

In the proof of Corollary 2.3.2, we have used that gs = 4Φ−2 n gc for Φn (x) = 1+|x|2 implies g∗ (Φ−n g∗ (vol(gs )) n vol(gc )) = . vol(gs ) Φ−n n vol(gc )

2.3. Poisson transformations and spherical principal series

Hence

g∗ (vol(gs ) vol(gs )

− nλ =

g∗ (Φn ) Φn

λ

77

g∗ (vol(gc ) vol(gc )

− λn .

Therefore, the obvious relation κ∗ ◦ πλ (Rn , gs )(g) = πλ (S n , gc )(g) ◦ κ∗

(2.3.16)

can be written in the form λ g∗ (Φn ) n ◦ πλ (R , gc )(g) = πλ (S n , gc )(g) ◦ κ∗ , κ∗ ◦ Φn i.e.,

n n −λ (κ∗ ◦ Φ−λ n ) ◦ πλ (R , gc )(g) = πλ (S , gc )(g) ◦ (κ∗ ◦ Φn ).

(2.3.17)

n

Similarly as for S , we can also consider principal series representations on (generalized) hyperspheres Σ in Rn . These can be written in the form Σ = σ(Rn−1 ) for some σ ∈ Gn+1 (acting on Rn ). Then we deﬁne πλ (Σ, gc )(g) =

g∗ (vol(Σ, gc )) vol(Σ, gc )

λ − n−1

◦ g∗ ,

(2.3.18)

for all g ∈ Gn+1 which preserve Σ. Here gc is the restriction of the Euclidean metric on Rn , and vol(Σ, gc ) ∈ Ωn−1 (Σ) denotes the associated Riemannian volume form. The group σ0 Gn σ0−1 operates on Σ0 = S n−1 = ∂Bn since Gn leaves the subspace Rn−1 = ker(xn ) ⊂ Rn invariant. As a special case of (2.3.18), we obtain the principal series representation πλ (Σ0 , gc )(g) =

g∗ (vol(Σ0 , gc )) vol(Σ0 , gc )

λ − n−1

g∗

for g ∈ σ0 Gn σ0−1 and the induced metric gc on Σ0 . It is related to πλ (Rn−1 , gc ) by the obvious relation

(σ0 )∗ (vol(Rn−1 , gc )) vol(Σ0 , gc )

λ − n−1

◦ (σ0 )∗ ◦ πλ (Rn−1 , gc )(g) λ − n−1 (σ0 )∗ (vol(Rn−1 , gc )) −1 ◦ (σ0 )∗ = πλ (Σ0 , gc )(σ0 gσ0 ) ◦ vol(Σ0 , gc )

for g ∈ Gn , i.e., (σ0 )∗

vol(Rn−1 , gc )) σ0∗ (vol(Σ0 , gc ))

λ − n−1

◦ πλ (Rn−1 , gc )(g) λ − n−1 vol(Rn−1 , gc )) −1 . (2.3.19) = πλ (Σ0 , gc )(σ0 gσ0 ) ◦ (σ0 )∗ ◦ σ0∗ (vol(Σ0 , gc ))

78

Chapter 2. Spaces, Actions, Representations and Curvature

We illustrate these matters in the case n = 2. We use complex notation. Then B2 = D = {z ∈ C | |z| < 1} is the unit disk with the hyperbolic metric 4 (dx2 + dy 2 ). (1−|z|2)2 ˆ by G3 = SO(1, 3) = SL(2, C) operates on C az + b a b . : z → c d cz + d ˆ = R ∪ {∞} and the halfThe subgroup G2 = SO(1, 2) = SL(2, R) preserves R planes H+ = {z ∈ C | y > 0}, H− = {z ∈ C | y < 0}. Now 1 σ0 = √ 2

1 −i ∈ SL(2, C), −i 1

i.e., z → (z − i)/(−iz + 1) maps ˆ → S 1 = ∂D H+ → D, R so that 0 → −i, ∞ → i and i → 0. It is an isometry if we use the metric y −2 (dx2 + dy 2 ) on H+ . The restriction σ0 : R → S 1 relates to the stereographic projection κS : R → S 1 by κS = iσ0 . In fact, σ0 (x) =

2x x−i 1−x2 = − i −ix+1 1+x2 1+x2

and κS (x) =

1−x2 2x +i . 2 1+x 1+x2

The group SU (1, 1) = σ0 SL(2, R)σ0−1 ⊂ SL(2, C) preserves S 1 and D. Now the non-compact models of the principal series of the groups SL(2, C) and SL(2, R) are deﬁned by πλnc (g)

= πλ (C, gc )(g) =

and

¯ g∗ (dζ ∧ dζ) ¯ dζ ∧ dζ

πλnc (g) = πλ (R, gc )(g) =

g∗ (dx) dx

− λ2

g∗ , g ∈ SL(2, C)

−λ

Lemma 2.3.1.

g∗ , g ∈ SL(2, R).

πλnc (g)u(ζ) = |− cζ +a|2λ u

dζ −b −cζ +a

2.3. Poisson transformations and spherical principal series

for g =

a c

b d

∈ SL(2, C), ζ ∈ C. Similarly, πλnc (g)u(x) = |− cx+a|2λ u

for g =

a c

b d

79

dx−b −cx+a

∈ SL(2, R), x ∈ R.

ˆ Proof. By the holomorphy of the action of g on C, 1 −1 12 1 ∂g (ζ) 1 ¯ ¯ 1 dζ ∧ dζ. g∗ (dζ ∧ dζ) = 11 ∂ζ 1 But using

we get

1 −1 12 1 ∂g (ζ) 1 1 1 1 1 ∂ζ 1 = |−cζ +a|4 ,

¯ − 2 g∗ (dζ ∧ dζ) = |−cζ +a|2λ , dζ ∧ dζ¯ λ

and the proof is complete for SL(2, C). We omit the analogous argument for SL(2, R). The action of SU (1, 1) on Σ0 = S 1 = ∂D gives rise to the representation πλ (Σ0 , gc )(g) = (g∗ (dθ)/dθ)

−λ

g∗ .

Lemma 2.3.2. ¯ +α|2λ u πλ (Σ0 , gc )(g)u(ζ) = |− βζ

for g =

αζ −β ¯ +α −βζ

α β and ζ ∈ S 1 . β¯ α ¯

Proof. Let ζ = eiθ . In order to calculate g∗ (dθ)/dθ, we note that dζ/ζ = idθ and hence g∗ (dθ) g∗ (dζ) ζ = . dθ dζ g∗ (ζ) But since g∗ (dζ) = d(g −1 ζ) = d(

αζ ¯ −β 1 ¯ + α ) = (−βζ ¯ + α)2 dζ −βζ

80

Chapter 2. Spaces, Actions, Representations and Curvature

we get g∗ (dθ) dθ =

1 ¯ (−βζ +α)2

ζ αζ−β ¯ ¯ −βζ+α

=

ζ 1 1 = = ¯ ¯ ¯ ¯ (−ζ +α)(¯ αζ −β) (−ζ +α)(¯ α −β ζ) |− βζ +α|2

and the proof is complete. The relation between πλ (Σ0 , gc ) and πλ (R, gc ) is given by (σ0 )∗ Φ−λ ◦ πλ (R, gc )(g) = πλ (Σ0 , gc )(σ0 gσ0−1 ) ◦ (σ0 )∗ Φ−λ 1 1 for g ∈ SL(2, R) (see (2.3.19)) since Φ1 (x) = (1+x2 )/2 = vol(R1 , gc )/σ0∗ (vol(σ0 , gc ))

(see (2.2.1)). That identity can be proved directly. In fact, it is equivalent to πλ (R, gc )(g) ◦ Φλ1 σ0∗ = Φλ1 σ0∗ ◦ πλ (Σ0 , gc )(σ0 gσ0−1 ). (2.3.20) We calculate the left-hand side. We have

2λ πλnc (g)Φλ1 σ0∗ u(x) = πλnc (g) 2−λ |−ix+1| σ0∗ (u) (x) 12λ 1 1 1 dx−b )+111 g∗ σ0∗ (u)(x) = 2−λ |−cx+a|2λ 11−i( −cx+a = 2−λ |(a+ib)−x(c+id)|

2λ

for g =

a c

g∗ σ0∗ (u)(x)

b ∈ SL(2, R). On the other hand, using d σ0 gσ0−1 =

1 a+d+i(b−c) b+c+i(a−d) α β = β¯ α ¯ 2 b+c−i(a−d) a+d−i(b−c)

we obtain 1 1 ¯ −1 (x)+α12λ (gσ −1 )∗ u(x) 1−βκ 0 12λ 1 1 1 x−i = 2−λ |−ix+1|2λ 11−β¯ +α11 (gσ0−1 )∗ u(x) −ix+1 1 1 −λ 1 ¯ ¯ 12λ g∗ σ ∗ u(x). =2 (α + iβ) − ix(α − iβ)

Φλ1 σ0∗ πλ (Σ0 , gc )(σ0 gσ0−1 ) = 2−λ |−ix+1|

2λ

0

Now the explicit formulas for α and β imply α + iβ¯ = a + ib and α − iβ¯ = c + id. The proof of (2.3.20) is complete.

2.4. The Nayatani metric

81

2.4 The Nayatani metric We brieﬂy describe the construction of a canonical metric on the complement of a round sphere S m ⊂ S n viewed as the limit set of a Kleinian group Γ ⊂ Gn+1 = SO(n + 1, 1)◦ . We ﬁrst recall that there is a canonical Γ-invariant measure μP S which is supported on the limit set Λ(Γ) = S m ⊂ S n (Patterson-Sullivan measure) ([188], [178]). We consider μP S as a distribution in C −∞ (S n ) with support in Λ(Γ), i.e., μP S ∈ C −∞ (Λ(Γ)), and describe its invariance in terms of the (distributional) spherical principal series representation πλc of Gn+1 . In these terms, c (γ)μP S = μP S , γ ∈ Γ. πm−n

(2.4.1)

Since the limit set is a smooth submanifold, μP S can be written explicitly as 2 μP S (ϕ) = i∗ (ϕ) vol(S m , gc ) vol(S m , gc ), ϕ ∈ C ∞ (S n ), Sm

Sm

where volumes are Riemannian volume form with respect to the standard metric gc . μP S is actually not only Γ-invariant (as for more general Γ) but Gm+1 -invariant, where Gm+1 ⊂ Gn+1 is the subgroup which leaves the subsphere invariant. We prove this fact in order to illustrate the formalism used in the present work. By deﬁnition, 4 3 4 3 c c (g −1 )ϕ , g ∈ Gm+1 πm−n (g)μP S , ϕ = μP S , π−m and c π−m (g −1 )ϕ =

on S m . Thus 4 3 c πm−n (g)μP S , ϕ =

g ∗ (voln ) voln

Sm

m n

g ∗ (ϕ) =

g ∗ (volm ) ∗ g (ϕ) volm

2 g ∗ (i∗ (ϕ))g ∗ (volm )

Sm

volm = μP S , ϕ .

Nayatani [185] observed that the measure μP S induces a canonical Γ-invariant metric on the proper set Ω(Γ) as follows. Here the induced metric is actually Gm+1 invariant. In order to construct the metric, we use the intertwining operator ϕ(y) def Iλ : C ∞ (S n )λ → C ∞ (S n )−λ−n , Iλ (ϕ)(x) = voln (y) (2.4.2) |x − y|2λ+2n n S for the spherical principal series. It satisﬁes the intertwining relation c Iλ ◦ πλc (g) = π−λ−n (g) ◦ Iλ

for g ∈ Gn+1 which extends to distributions. Now deﬁne m+1

Φ = Im−n (μP S ) ∈ C −∞ (S n )G −m . def

(2.4.3)

82

Chapter 2. Spaces, Actions, Representations and Curvature

The restriction resΩ(Γ) (Φ) of Φ to Ω(Γ) = S n \S m is C ∞ since Iλ does not increase singular support. More explicitly, we have 6 5 1 Φ(x) = μP S (y), |x − y|2m for x ∈ Ω(Γ). The integral is well deﬁned since x is in the complement of the support of μP S . Moreover, resΩ(Γ) (Φ) > 0. Hence n

1

0 < Φ0 = (resΩ(Γ) (Φ)) m ∈ C ∞ (Ω(Γ))G −1 .

(2.4.4)

n

Now we deﬁne the G -invariant Nayatani metric on Ω(Γ) by gN = Φ20 gc .

(2.4.5)

c In the case of the equatorial subsphere S n−1 ⊂ S n , the π−1 (Gn )-invariant function Φ0 has also a simple extrinsic description. On the upper hemisphere H + ⊂ S n the function Φ−1 is a constant multiple of the restriction H0 of the height function 0 xn+1 to S n . This follows from the simple

Lemma 2.4.1.

5 μP S (y),

1 |x − y|2(n−1)

6

=c

1 |H0 (x)|

n−1

for x ∈ Ω(Γ). Thus the metric

1 gc H02

on H + is a constant multiple of the Nayatani metric. For m < n− 1, the corresponding Nayatani metric on S n \ S m plays a natural role in an analogous theory of equivariant families C ∞ (S n ) → C ∞ (S m ).

2.5 Riemannian curvature and conformal change In the present section we derive the basic identities which describe the behaviour of the Levi-Civita connection ∇, the Riemann curvature tensor R, the Ricci curvature tensor Ric and the scalar curvature τ with respect to conformal changes of the metric. In this connection, it is natural to introduce the rescaled scalar curvature J and the Schouten tensor P. In addition, we associate a symmetric bilinear form G to a hypersurface. We recall ﬁrst how the Riemann curvature tensor of (M n , g) transforms under ˆ denote the corresponding Levi-Civita connections. Then ([24], g → gˆ. Let ∇ and ∇ [85]) the Koszul formula 2g(∇X Y, Z) = X (g(Y, Z)) + Y (g(X, Z)) − Z (g(X, Y )) + g ([X, Y ], Z) + g ([Z, X], Y ) + g ([Z, Y ], X)

2.5. Riemannian curvature and conformal change

83

implies ˆ X Y = ∇X Y + X(ϕ)Y + Y (ϕ)X − g(X, Y ) grad ϕ ∇

(2.5.1)

for all vector ﬁelds X, Y ∈ X (M ). For the curvature R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,

R(X, Y, Z, W ) = g(R(X, Y )Z, W ),

it follows that ˆ R(X, Y )Z = R(X, Y )Z + g(∇X (grad ϕ), Z)Y − g(∇Y (grad ϕ), Z)X + ∇Y (grad ϕ)g(X, Z) − ∇X (grad ϕ)g(Y, Z) + Y (ϕ)Z(ϕ) − g(Y, Z)| grad ϕ|2 X − X(ϕ)Z(ϕ) − g(X, Z)| grad ϕ|2 Y + (X(ϕ)g(Y, Z) − Y (ϕ)g(X, Z)) grad ϕ, i.e., ˆ ˆ e−2ϕ R(X, Y, Z, W ) = e−2ϕ gˆ(R(X, Y )Z, W ) = R(X, Y, Z, W ) + Ξ(X, Z)g(Y, W ) − Ξ(Y, Z)g(X, W ) + Ξ(Y, W )g(X, Z) − Ξ(X, W )g(Y, Z) (2.5.2) for 1 def Ξ(X, Y ) = Ξ(g,ϕ) (X, Y ) = g(∇X (grad ϕ), Y ) − X(ϕ)Y (ϕ) + | grad ϕ|2 g(X, Y ). 2 Here Hess(ϕ)(X, Y ) = g(∇X (grad ϕ), Y ) = ∇X (dϕ), Y

(2.5.3)

is the covariant Hessian of ϕ. For the Ricci tensor Ric(X, Y ) = R(X, ei , ei , Y ) = g(R(X, ei )ei , Y ), i

i

we obtain 7 Ric(X, Y ) = Ric(X, Y ) + Ξ(X, ei )g(ei , Y ) + Ξ(ei , Y )g(X, ei ) − Ξ(ei , ei )g(X, Y ) − Ξ(X, Y )g(ei , ei ) i

i.e., 7 Ric(X, Y ) = Ric(X, Y ) − (n−2)Ξ(X, Y ) − g(X, Y )

Ξ(ei , ei )

i

= Ric(X, Y ) − (n−2)g(∇X grad ϕ, Y ) + g(X, Y )Δϕ − (n−2)| grad ϕ|2 g(X, Y ) + (n−2)X(ϕ)Y (ϕ)

(2.5.4)

84

Chapter 2. Spaces, Actions, Representations and Curvature

using the identity

g(∇ei (grad ϕ), ei ) = −Δϕ.

(2.5.5)

i

(2.5.4) yields for the scalar curvature τ = i Ric(ei , ei ) the transformation rule (2.5.6) τˆ = e−2ϕ τ + (2n−2)Δϕ − (n−1)(n−2)| grad ϕ|2 , i.e.,

ˆJ = e−2ϕ J + Δϕ − n−2 | grad ϕ|2 2

for J=

τ . 2(n−1)

(2.5.7)

(2.5.8)

These results imply the transformation formula 1 ˆ P(X, Y ) = P(X, Y ) − | grad ϕ|2 g(X, Y ) − g(∇X (grad ϕ), Y ) + X(ϕ)Y (ϕ) 2 = P(X, Y ) − Ξ(X, Y ) (2.5.9) for the Schouten tensor P=

1 (Ric −Jg) . n−2

(2.5.10)

(2.5.1) implies that the Hessian (2.5.3) satisﬁes the conformal transformation law 7 Hess(u)(X, Y ) = Hess(u)(X, Y ) − du, Xdϕ, Y − dϕ, Xdu, Y + (du, dϕ)g(X, Y ). (2.5.11) We write (2.5.2) also in the form ˆ =R+Ξg e−2ϕ R

(2.5.12)

in terms of the Kulkarni-Nomizu product (Ξ g)(X, Y, Z, W ) = Ξ(X, Z)g(Y, W ) − Ξ(Y, Z)g(X, W ) + Ξ(Y, W )g(X, Z) − Ξ(X, W )g(Y, Z) (see [24], Deﬁnition 1.110). Now for the Weyl tensor C being deﬁned by R = C−Pg we obtain

(2.5.13)

ˆ gˆ = R + Ξ g + (P − Ξ) g = R + P g = C. (2.5.14) ˆ = e−2ϕ R ˆ+P e−2ϕ C

2.5. Riemannian curvature and conformal change

85

In particular, we have ˆ 2 = |C|2 . e4ϕ |C|

(2.5.15)

A general reference for the basic formulas (2.5.12), (2.5.4) and (2.5.7) is [24], Theorem 1.159. Note that in [24] the curvature tensor has opposite sign, but Ricci curvature and scalar curvature coincide with ours, of course. The (somewhat) unusual sign in (2.5.13) is due to the conventions. A contraction of (2.5.13) in the second and third argument yields Ric on both sides since C is trace-free. Finally, we ﬁx some notational conventions. For a symmetric bilinear form B on a vector space V with a scalar product g, we deﬁne its g-trace as follows. B gives rise to a linear map B : V → V by g(B (X), Y ) = B(X, Y ) for X, Y ∈ V . In particular, g = id. We set trg (B) = tr(B ).

(2.5.16)

Then for an arbitrary basis e1 , . . . , en of V , trg (B) = g rs Brs , where Brs = B(er , es ) = g(B (er ), es ) and grs = g(er , es ). Here we use the Einstein sum convention, and raise and lower indices as usual using g. In particular, for a g-orthonormal basis, g(B (ei ), ei ) = B(ei , ei ). trg (B) = i

Note that

i

tr((B )2 ) = tr((g −1 B)2 ).

(2.5.17)

In fact, we calculate tr((B )2 ) = g rs g((B )2 (er ), es ) = g rs Brt g(B (et ), es ) = g rs Brt Bts = (g rs Brq )(g qt Bts ) = tr((g −1 B)2 ). In order to simplify formulas, we shall often write tr(B) instead of trg (B), if the metric is clear from the context. l . Then The components of the curvature tensor R will be denoted by Rijk r Rijkl = grl Rijk , r

Ricij =

r,s

Rirsj g rs =

Rrijs g rs =

r,s

and τ = trg (Ric) = tr(Ric ) =

r Rrij

r

i,j

g ij Ricij .

86

Chapter 2. Spaces, Actions, Representations and Curvature

For a hypersurface Σ ⊂ M with the unit normal vector ﬁeld N , we shall write ∇N , or just N , for the corresponding normal derivative. We deﬁne the symmetric bilinear form G(X, Y ) = R(X, N, N, Y ) = g(R(X, N )N, Y ) (2.5.18) on T (Σ) and let T = Ric −G. Then tr(G) = Ric(N, N ) and

tr(T ) = τM − 2 Ric(N, N ).

We shall also use the notation F = Ric(N, N ) = RicN .

(2.5.19)

Chapter 3

Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory 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.

3.1 GJMS-operators and Q-curvature In the present section, we review the construction of conformally covariant powers of the Laplacian which are known as GJMS-operators. These generalize the conformal Laplacian or Yamabe operator P2 = Y : g → Δg −

n−2 τg 4(n−1)

on a manifold of dimension n. P2 is of conformal weight ( n2 − 1, n2 + 1), i.e., it satisﬁes the transformation formula n

n

P2 (e2ϕ g) = e−( 2 +1)ϕ ◦ P2 (g) ◦ e( 2 −1)ϕ for all ϕ ∈ C ∞ (M ). Here we use the convention that −Δg is non-negative. A theorem of Graham, Jenne, Mason and Sparling ([124]) yields analogous conformally covariant modiﬁcations of powers of the Laplacian by lower order terms on any manifold M . More precisely, Theorem 3.1.1 ([124]). Let n ≥ 3 and let 8 1 ≤ N ≤ n2 if n is even, N ≥1 if n is odd.

88

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

Then on any manifold M of dimension n there exists a natural conformally covariant diﬀerential operator P2N of the form ΔN + LOT and of conformal weight ( n2 − N, n2 + N ). In other words, P2N (g) diﬀers from the power ΔN g of the Laplacian by lower order terms and satisﬁes the transformation formula n n P2N (e2ϕ g) = e−( 2 +N )ϕ ◦ P2N (g) ◦ e( 2 −N )ϕ (3.1.1) for all ϕ ∈ C ∞ (M ). On Rn with the Euclidean metric gc the operator P2N (gc ) coincides with ΔN Rn . Note that there is no uniqueness assertion in Theorem 3.1.1. The GJMSoperator P2N in Theorem 3.1.1 provides one speciﬁc construction of an operator with the listed properties. On the other hand, extending a result of Graham ([118]), Gover and Hirachi ([112]) proved that the statement is sharp in the sense that for general metrics the condition on the order (for even dimension n) cannot be relaxed. Theorem 3.1.1 holds true for pseudo-Riemannian manifolds. In the following, we write P2N,n if there is reason to emphasize both the dimension of the underlying manifold and the order of the GJMS-operator. For even dimension n, the operator Pn = Pn,n will be called the critical GJMS-operator. This terminology makes it natural to refer to the operators P2N,n , 2N < n (for even n) as to the subcritical GJMS-operators. We brieﬂy describe the main ideas of the proof ([124]) of Theorem 3.1.1. It rests on the Feﬀerman-Graham ambient metric introduced in [99]. For a Riemannian manifold (M, g), we form the space GM = {(x, tgx ) | t > 0} ⊂ S 2 (T ∗ M ). GM is a ray bundle on M . Its sections are the metrics which are conformally equivalent to g, i.e., GM is an invariant of the conformal class c of g. GM admits a canonical R+ -action δs : (x, gx ) → (x, s2 gx ) which enables us to deﬁne spaces of homogeneous functions EM (λ) = u ∈ C ∞ (GM ) | δs∗ (u) = sλ u, s > 0 for any λ ∈ C. Elements of EM (λ) can be identiﬁed with sections of the associated line bundle on M which is deﬁned by the R+ - character sλ . A choice of a metric in the conformal class c induces trivializations of these line bundles and isomorphisms EM (λ) C ∞ (M ). The ambient manifold G˜M is GM × (−1, 1). Let ι : GM → G˜M , ι : p → (p, 0) be the canonical embedding. The action δs extends naturally to G˜M . The metric g induces a tautological symmetric bilinear form g on the ray bundle GM which satisﬁes δs∗ (g) = s2 g.

3.1. GJMS-operators and Q-curvature

89

The Feﬀerman-Graham ambient metric g˜ is a (formal pseudo-Riemannian) metric on G˜M along the hypersurface GM (i.e., it exists in a suﬃciently small neighbourhood) so that (i) ι∗ (˜ g ) = g, i.e., it extends g, (ii) δs∗ (˜ g ) = s2 g˜, s > 0 and (iii) Ric(˜ g) = 0 along GM . Theorem 2.1 in [99] states that if n = dim M is odd then, up to R+ equivariant diﬀeomorphisms which ﬁx GM , there is a unique formal power series solution of (i), (ii), (iii). For even n the condition (iii), in general, cannot be satisﬁed. However, up to R+ -equivariant diﬀeomorphisms which ﬁx GM , and up to terms which vanish of order n2 , there is a unique formal power series solution g˜ so that Ric(˜ g ) vanishes of order n2 − 2 along GM and the tangential component of Ric(˜ g) vanishes of order n2 − 1 along GM . For full details see [96]. ˜. Now for N ≥ 1 let ΔN g ˜ be a power of the Laplacian of an ambient metric g It induces operators ˜ ˜ ΔN g ˜ : EM (λ) → EM (λ−2N ) on homogeneous functions on G˜M , and for a speciﬁc value of the weight λ it descends to a diﬀerential operator on homogeneous functions on GM . More precisely, if N is subject to the conditions in Theorem 3.1.1, then it induces a diﬀerential operator

n n EM − +N → EM − −N 2 2 by the composition u → ι∗ ΔN u), g ˜ (˜ where u ˜ denotes an arbitrary homogeneous extension of u. Here the crucial observation is that for λ = − n2 + N , the composition does not depend on the choice of the extension of u. By the uniqueness of g˜, up to diﬀeomorphisms which ﬁx the boundary, the resulting operator is an invariant of the conformal class c of g. It corresponds to the conformally covariant diﬀerential operator P2N (g) on C ∞ (M ). The condition N ≤ n2 (for even n) arises from the obstructions to the existence of g˜ satisfying the condition (iii) to high order. By the convention −Δg ≥ 0, the operator P2N (g) diﬀers from the operator P2N (g) in [124] by the sign (−1)N . The operators P2N (g) are formally self-adjoint ([128]). Branson ([32]) proved that the operator P2N (g) is of the form 0 + (−1)N P2N = P2N

n 2

−N Q2N ,

(3.1.2)

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

90

0 where Q2N,n = Q2N is a local scalar invariant and P2N annihilates constants. Due N to the sign (−1) in (3.1.2), this deﬁnition of Q2N (g) coincides with that in [32]. (3.1.2) implies that

n −N (−1)N Q2N (g). P2N (g)(1) = (3.1.3) 2

(3.1.1) applied to u = 1 gives n

n

e( 2 +N )ϕ P2N (e2ϕ g)(1) = P2N (g)(e( 2 −N )ϕ ).

(3.1.4)

0 (g)(1) = 0 yield But now (3.1.2) and P2N

n 2

n −N e( 2 +N )ϕ Q2N (e2ϕ g) n

0 (g)(e( 2 −N )ϕ −1) + = (−1)N P2N

We divide by

n 2 −N

and set N =

n 2

n 2

n −N e( 2 −N )ϕ Q2N (g).

(3.1.5)

(for even n). This yields

n

n

enϕ Qn (e2ϕ g) = (−1) 2 Pn0 (g)(ϕ) + Qn (g) = (−1) 2 Pn (g)(ϕ) + Qn (g) since Pn (for even n) has no constant term. The identity n

enϕ Qn (e2ϕ g) = (−1) 2 Pn (g)(ϕ) + Qn (g)

(3.1.6)

will be called the fundamental identity. The latter formal argument is due to Branson and Ørsted ([48], [31], [32]), and known as analytic continuation in dimension. Deﬁnition 3.1.1 (Q-curvatures). Let n ≥ 2 and N be as in Theorem 3.1.1. Then for a Riemannian manifold (M, g) of dimension n, the quantities Q2N (g) = Q2N,n (g) will be called the Q-curvatures of (M, g). For manifolds of even dimension n, we distinguish between the critical Q-curvature Qn (g) = Qn,n (g) and the subcritical Q-curvatures Q2N (g) = Q2N,n (g), 2N < n. For 2N < n, (3.1.5) gives n 2

−N Q2N (e2tϕ g) n

n

0 (g)(e( 2 −N )tϕ −1) + = e−( 2 +N )tϕ (−1)N P2N

n 2

−N e−2N tϕ Q2N (g).

Diﬀerentiating at t = 0 yields n n

n 0 −N Q•2N (g)[ϕ] = (−1)N −N P2N −N 2N ϕQ2N (g), (g)(ϕ) − 2 2 2 i.e.,

0 Q•2N (g)[ϕ] = (−1)N P2N (g)(ϕ) − 2N ϕQ2N (g).

(3.1.7)

3.2. Scattering theory

Here

91

Q•2N (g)[ϕ] = (d/dt)|t=0 (Q2N (e2tϕ g))

denotes the inﬁnitesimal conformal variation of Q2N . (3.1.6) yields the analogous result for 2N = n. But whereas the inﬁnitesimal conformal transformation law for the critical and the subcritical Q-curvatures are analogous, the subcritical Q-curvatures do not satisfy a conformal transformation law which is governed by a linear diﬀerential operator. The relation vol(e2ϕ g) = enϕ vol(g) and (3.1.6) imply that for a closed manifold M n the total Q-curvature Qn (g) vol(g) M

is conformally invariant. For the proof we use that Pn (g) is self-adjoint and annihilates constants. Partial integration yields Qn (e2ϕ g) vol(e2ϕ g) = Qn (e2ϕ g)enϕ vol(g) n n M M n = (−1) 2 Pn (g)(ϕ) vol(g) + Qn (g) vol(g). Mn

Mn

But the ﬁrst integral vanishes. Thus 2ϕ 2ϕ Qn (e g) vol(e g) = Mn

Mn

Qn (g) vol(g).

(3.1.8)

The fundamental identity (3.1.6) generalizes the Gauß curvature prescription equation e2ϕ K(e2ϕ g) = −Δg (ϕ) + K(g) on a surface M 2 . For N = 1 and n ≥ 3, (3.1.4) states n n−2 n−2 2ϕ (n +1)ϕ τ (e g)e 2 τ (g) (e( 2 −1)ϕ ), − = Δg − 4(n−1) 4(n−1) i.e., (−Δg +

n+2 n−2 n−2 τ (g))(u) = τ (e2ϕ g)u n−2 4(n−1) 4(n−1)

(3.1.9)

n

for u = e( 2 −1)ϕ . The latter identity is known as the Yamabe equation.

3.2 Scattering theory A major development ([128]) was the identiﬁcation of the GJMS-operators P2N (g) and the Q-curvature Qn (h) on a closed manifold M n in the scattering operator S(h; λ) : C ∞ (M ) → C ∞ (M )

92

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

of the Laplacian for an associated Poincar´e-Einstein metric via the formulas Res n2 +N (S(h; λ)) = −cN P2N (h) and n

S(h; n)(1) = (−1) 2 c n2 Qn (h) (Theorem 3.2.1 and Theorem 3.2.2). The ﬁrst formula generalizes the relation between the operators P2N on the round sphere (S n , gc ) and the residues of the standard Knapp-Stein intertwining operators for the spherical principal series viewed as the scattering operator of the free hyperbolic space with boundary S n at inﬁnity. The scattering operator is a rather complicated object even in simple cases and Qn is only a very small piece of information coded in that family of pseudodiﬀerential operators. A more direct formula for Qn (h) is n

(−1) 2 P˙ n (h; 0)(1), where Pn (λ) is a certain family of diﬀerential operators which arises by formally solving the eigenequation −Δg u = λ(n − λ)u (see (3.2.20)). That result is also due to [128]. Eliminating all superﬂuous constructions Feﬀerman and Graham ([97]) later found a simpler characterization of Qn in term of the Poincar´e metric. That result will play an important role in Section 6.6. Let M n be the boundary of a compact manifold X n+1 (if M does not bound any X one can take X = [0, ε] × M with ∂X = M M ). We consider metrics of the form (3.2.1) g = ρ−2 g, where ρ ∈ C ∞ (X) is a deﬁning function of ∂X in X, and g is a smooth metric on X. g is called a conformally compact metric on X and g a conformal compactiﬁcation. If we assume, in addition, that |dρ|g = 1 on ∂X, then g is asymptotically hyperbolic, i.e., the sectional curvatures of g approach −1 at ∂X. A conformally compact g has many conformal compactiﬁcations and induces a conformal class i∗ (ρ2 g) , i : ∂X → X. This class is called the conformal inﬁnity of g. Conversely, for a given conformal class [h] of metrics on ∂X, one can associate a distinguished class of conformally compact metrics on X with [h] as conformal inﬁnity. These are known as Poincar´e metrics since they generalize the correspondence between the round metric on the sphere S n and the constant curvature Poincar´e metric on the unit ball with S n as its sphere at inﬁnity. Poincar´e metrics correspond to ambient metrics ([99], [96]). Let gAH be an asymptotically hyperbolic metric on X with conformal inﬁnity [h]. We choose a representing metric h on ∂X. Then there exists a unique ρ ∈ C ∞ (X) near ∂X so that |dρ|ρ2 gAH = 1 near ∂X

3.2. Scattering theory

93

and i∗ (ρ2 gAH ) coincides with h on T (∂X). The gradient ﬂow of ρ for the conformal compactiﬁcation, ρ2 gAH , deﬁnes a diﬀeomorphism α : [0, ε) × ∂X → X onto a collar neighbourhood of ∂X. Then α∗ (ρ) = r and α∗ (gAH ) =

1 (dr2 +hr ), h0 = h r2

with a curve hr of metrics on ∂X. For more details we refer to [119]. Now we pose additional conditions on the metric. We require that the metric on (0, ε) × ∂X is Einstein, Ric(g) + ng = 0 (3.2.2) (n = dim ∂X), i.e., τ (g) = −n(n + 1), i.e., the scalar curvature of g coincides with that of the real hyperbolic space of dimension n. The condition (3.2.2) corresponds to the Ricci-ﬂatness of the related ambient metric ([99], [96]). Evaluating (3.2.2) for a formal power series in r leads to the following consequences. If n is odd, then there is a formal Taylor series hr ∼ h(0) + r2 h(2) + · · · + rn−1 h(n−1) + rn h(n) + · · · ,

(3.2.3)

where all powers up to n − 1 are even. Here the coeﬃcients h(2) , . . . , h(n−1) are locally determined by the leading coeﬃcient (conformal inﬁnity) h(0) = h. Moreover, h(n) has vanishing h-trace. hr is formally unique if assumed to be even in r (in that case h(n) = 0). For even n, the Taylor-type series of hr is of the form ¯ (n) + · · · , hr ∼ h(0) + r2 h(2) + · · · + rn−2 h(n−2) + rn h(n) + log rh (3.2.4) ¯ (n) where all powers up to n are even. Here the coeﬃcients h(2) , . . . , h(n−2) , h and the h-trace of h(n) are locally determined by the conformal inﬁnity h(0) = h. ¯ (n) has vanishing h-trace. The logarithmic terms obstruct the existence Moreover, h of a formal smooth solution. Note also that the condition Ric(g) + ng = O(ρn−2 ) uniquely determines the coeﬃcients h(2) , . . . , h(n−2) , and the additional condition tr(Ric(g) + ng) = O(ρn+2 ) can be satisﬁed and uniquely determines the h-trace of h(n) . See [99], [119], [128], [122]. Now a (true) metric on X is called a Poincar´e-Einstein metric associated to [h] if 8 if n is odd, O(ρ∞ ) Ric(g) + ng = n−2 ) if n is even, O(ρ

94

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

and induces the given class [h]. Moreover, it is required to be even in ρ if n is odd, and satisﬁes the vanishing trace condition tr(Ric(g) + ng) = O(ρn+2 ) if n is even (see [128] and the references therein). Geometric scattering theory ([181]) on X with an asymptotically hyperbolic metric g deﬁnes a scattering operator S(h; λ) : C ∞ (∂X) → C ∞ (∂X). For λ so that (λ) =

n 2,

the scattering operator is determined by the property S(h; λ)f = g,

where near the boundary F, G ∈ C ∞ (X)

u = ρn−λ F + ρλ G,

(3.2.5)

describes the boundary asymptotic behaviour of an eigenfunction −Δg u = λ(n − λ)u of the Laplacian of g, and f = F |∂X , g = G|∂X . Varying h in its conformal class, results in a conjugate scattering operator. More precisely, S(e2ϕ h; λ) = e−λϕ ◦ S(h; λ) ◦ e(n−λ)ϕ .

(3.2.6)

Theorem 3.2.1 ([128]). Let g be a Poincar´e metric associated to h. Then S(h; λ) has a meromorphic continuation to C. Let λ = n2 +N , where 8 1, 2, . . . , n2 if n is even, N∈ N if n is odd, and assume that λ(n − λ) is not an L2 -eigenvalue of −Δg . Then S(h; λ) has a simple pole at λ with residue Res n2 +N (S(h; λ)) = −cN P2N (h), where cN =

1 22N N !(N −1)!

(3.2.7)

.

We recall that our choice of the positive Laplacian is opposite to that in [128]. Let n be even. In view of Pn (h)(1) = 0, the formula (3.2.7) implies that S(h; λ)(1) is regular at λ = n. Graham and Zworski proved Theorem 3.2.2 ([128]). In the situation of Theorem 3.2.1, n

S(h; n)(1) = (−1) 2 c n2 Qn (h) for even n.

(3.2.8)

3.2. Scattering theory

95

Therefore, at λ = n the Laurent series of the scattering operator has the form

Pn (h) + S0 (h; λ) (3.2.9) λ−n such that S0 (h; n)(1) = c n2 Qn (h). This identity can be used to give the following family proof of (3.1.6). (3.2.6) applied to the function 1 gives S(h; λ) = −c n2

S(h; λ)(e(n−λ)ϕ ) = eλϕ S(e2ϕ h; λ)1. Now we use (3.2.8) and (3.2.9) to calculate the limit λ → n. We obtain (n−λ)ϕ e n n n c 2 lim Pn (h) + (−1) 2 c n2 Qn (h) = enϕ (−1) 2 c n2 Qn (e2ϕ h), λ→n n−λ i.e., using Pn (h)(1) = 0, we get

n

enϕ Qn (e2ϕ h) = (−1) 2 lim Pn (h) λ→n

e(n−λ)ϕ −1 n−λ

+ Qn (h) n

= (−1) 2 Pn (h)(ϕ) + Qn (h). The proof of (3.1.6) is complete. The pole of the scattering operator S(h; λ) at λ = n2 +N , N ∈ N0 corresponds to a speciﬁc asymptotic behaviour of the eigenfunctions for the corresponding eigenvalue. For these values of λ, a genuine eigenfunction u is of the form u = ρn−λ F + ρλ log ρG for F, G ∈ C ∞ (X), i.e., has a logarithmic singularity, and for λ = the relation G|∂X = 2 Res n2 +N (S(λ)f ), f = F |∂X .

(3.2.10) n 2

+ N , we have

The occurrence of a logarithmic singularity for a genuine eigenfunction is related to the simple poles of the rational coeﬃcients in the formal solutions aj (λ)ρλ+j and bj (λ)ρn−λ+j j≥0

j≥0

of the equation −Δg u = λ(n−λ)u. In fact, the set n2 + 12 Z is the set of all λ for which the two ladders λ + N0 and n − λ + N0 intersect non-trivially. At the pole n 2 + N both sums contribute to the asymptotics of a genuine eigenfunction and induce a logarithmic singularity. It is instructive to describe the details in the case of the hyperbolic upper half-space Hn+1 . The sums a2j (x ; λ)xλ+2j and b2j (x ; λ)xn−λ+2j n+1 n+1 j≥0

j≥0

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

96

are only formal eigenfunctions. Any genuine eigenfunction u is the HelgasonPoisson transformation of a hyperfunction vector of the corresponding principal series on the boundary ([152]). For the present discussion, we consider an eigenfunction with boundary values in C0∞ (Rn ), i.e., let u(x) = P (x, ζ)λ ω(ζ)dζ, ω ∈ C0∞ (Rn ), Rn

where P (x, ζ) =

xn+1 , x ∈ Hn+1 , ζ ∈ Rn |x − ζ|2

is the Poisson kernel of the upper half-space. Then −ΔHn+1 u = λ(n−λ)u and at least for λ ∈ n2 + iR, λ = n2 we have the asymptotics u(x , xn+1 ) ∼

a2j (x , λ)xλ+2j n+1 +

j≥0

b2j (x , λ)xn−λ+2j n+1

(3.2.11)

j≥0

for xn+1 → 0 with the leading terms a0 (λ) = Iλ (ω)

n

and b0 (λ) = π 2

Γ(λ− n2 ) def ω = c(λ)ω Γ(λ)

(3.2.12)

(see [195]). Here Iλ denotes the meromorphic continuation of the integral operator with the kernel 1 |x−y|2λ and c(λ) is the spherical Harish-Chandra c-function of Hn+1 . Iλ deﬁnes an intertwining operator for the (non-compact model of the) spherical principal series representations. It satisﬁes the intertwining relation nc nc (g) = π−λ (g) ◦ Iλ . Iλ ◦ πλ−n

The analog of Iλ in the compact model appears in (2.4.2). (3.2.12) shows that S(λ)b0 (λ) = a0 (λ), i.e., Iλ S(λ) = . c(λ) (5.2.52) shows that the coeﬃcients a2j (x , λ) are given by a2j (x ; λ) = and b2j (x ; λ) =

1 1 Δj a0 (x ; λ) 2j j! {(n−2j −2λ) · · · (n−2−2λ)}# n 1

1

2j j!

{(−n−2j +2λ) · · · (−n−2+2λ)}#

Δjn b0 (x ; λ),

3.2. Scattering theory

97

i.e., u∼

1 1 Δj Iλ (ω)xλ+2j n+1 2j j! {(n−2j −2λ) · · · (n−2−2λ)}# n j≥0 1 1 + c(λ) Δj (ω)xn−λ+2j . (3.2.13) n+1 2j j! {(−n−2j +2λ) · · · (−n−2+2λ)}# n j≥0

(3.2.13) holds true if (λ) > n2 , λ ∈ n2 = N0 ; then Iλ and c(λ) have no poles. Now we take a closer look at the asymptotics of u for λ ∈ n2 + N. Let n λ = 2 + N . In (3.2.13) both sums have a simple pole at λ = n2 +N with respective residues 1 n 1 2 +N +2j Δjn Res n2 +N (Iλ )(ω)xn+1 # j 2 j! {(−2j −2N ) · · · (−2−2N )} j≥0

and

1 n 1 2 −N +2j n Δjn (ω)xn+1 Res . +N # 2 2j j! {(−n−2j +2λ) · · · (−n−2+2λ)} j≥0

n

+N +2j

2 The latter sum contributes only for j ≥ N . The coeﬃcient of xn+1

is

1 1 μn Δj+N (ω) n 2j j! {(−2j −2N ) · · · (−2−2N )}#

n 1 +N j+N +c 2 2 (j +N )!

1 (ω) Δj+N × Res n2 +N n # {(−n−2j −2N +2λ) · · · (−n−2+2λ)} using Res n2 +N (Iλ ) = μN ΔN n.

(3.2.14)

We claim that these coeﬃcients vanish, i.e., there are no poles on the right-hand side of (3.2.13) at λ ∈ n2 + N. The assertion is equivalent to the identity μN = c

n 2

+N

1 1 n 1 1 1 = π2 n . (3.2.15) # 2N N 2 N !(N −1)! Γ( 2 ) {n · · · (n−2+2N )} 2 n!

Now it is well known ([105]) that |x|−λ is a meromorphic family of distributions on Rn with simple poles at λ ∈ n + 2N0 and residue vol(S n−1 )

1 ΔN δ0 2N N !n(n+2) · · · (n−2+2N ) n

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

98

at λ = n+ 2N . It follows that Iλ has a meromorphic continuation to C with simple poles at λ ∈ n2 + N0 and residues n

π2 1 vol(S n−1 ) ΔN = ΔN , n # # n n 2 2N N ! {n · · · (n−2+2N )} Γ( 2 )2N N ! {n · · · (n−2+2N )} i.e., we obtain the formula n

μN =

π2

#

Γ( n2 )2N N ! {n · · · (n−2+2N )}

.

This proves (3.2.15), i.e., the vanishing of the residues. It follows that at λ = n2 + N the asymptotics (3.2.13) takes the form 1 n 1 2 +N +2j Δj+N (ω)μN xn+1 log xn+1 n # j 2 j! {(−2j −2N ) · · · (−2−2N )} j≥0

1 n 1 n +N Res −c +N 2 2 2j j! {(−n−2j +2λ) · · · (−n−2+2λ)}# j≥0 n

−N +2j

2 × xn+1

n

−N

2 log xn+1 Δjn (ω) + F xn+1

with F |xn+1 = c( n2 + N )ω. The second sum contributes only for j ≥ N . Moreover, n

+N

2 log xn+1 is the coeﬃcient of xn+1

μN ΔN n (ω)

1 n 1 ΔN +N N Res n2 +N −c n (ω). # 2 2 N! {(−n−2N +2λ) · · · (−n−2+2λ)} But we have seen above that

1 n 1 = 0. μN + c +N N Res n2 +N 2 2 N! {(−n−2N +2λ) · · · (−n−2+2λ)}# n

+N

2 Hence the coeﬃcient of xn+1 log xn+1 is n 2μN ΔN n (ω) = 2 Res 2 +N (Iλ )(ω).

It follows that the asymptotics of u at λ = n2 + N takes the form λ xn+1 → 0 u ∼ xn−λ n+1 F + xn+1 log xn+1 G, with F |xn+1 =0 = c

n

+N ω

and G|xn+1 =0 = 2 Res n2 +N (Iλ )(ω). 2 This corresponds to the asymptotics (3.2.10).

3.2. Scattering theory

99

It is remarkable that analogous results hold true for general Poincar´e-Einstein metrics ([128]). Since the residues of the scattering operator yield the GJMSoperators, it follows that these can be seen in the coeﬃcients of the log-term in the asymptotics of genuine eigenfunctions. In contrast to the above perspective, in the present work it is more important that the relevant conformally covariant diﬀerential operators can be seen in the asymptotics of formal eigenfunctions. In fact, in the case of eigenfunctions on Hn+1 in the formal asymptotics b2j (x ; λ)xn−λ+2j n+1 j≥0

the coeﬃcient of xn−λ+2N is given by n+1 b2N (λ) = c(λ)

1 1 ΔN (ω). 2N N ! {(−n+2λ−2N ) · · · (−n−2+2λ)}# n

It has a simple pole at λ = n2 +N with residue given by a multiple of the GJMSoperator ΔN n. That picture extends to the case of a Poincar´e metric, too. In fact, Graham and Zworski ([128]) gave the following construction of GJMS-operators P2N in terms of Poincar´e metrics. Let M n be a closed manifold with metric h. We consider an associated Poincar´e metric g=

1 (dr2 +hr ), h0 = h r2

on (0, ε) × M and construct a formal solution u of the equation −Δg u = μ(n − μ)u by using a power series ansatz u ∼ j≥0 rμ+j aj . For odd n, all coeﬃcients aodd vanish and Γ( n2 −μ−j) P2j (h; μ)(a0 (h)) 22j j! Γ( n2 −μ) 1 P2j (h; μ)(a0 (h)) = 2j n 2 j! 2 −μ−1 · · · n2 −μ−j

a2j (h; μ) =

1

(3.2.16)

for certain polynomial families of natural diﬀerential operators P2j (h; μ) of order 2j on M so that P2j (h; μ) = Δjh + LOT . Then n

P2j h; −j = P2j (h), j ≥ 1. (3.2.17) 2 For even n, the situation is analogous. In particular, we have an (h; μ) =

22n

1 n n Pn (h; μ)(a0 (h)) 2 ! 2 −μ−1 · · · (−μ)

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

100

and the formula Pn (h) = Pn (h; 0)

(3.2.18)

n

for the critical GJMS-operator on M . The family P2j (h; μ) satisﬁes the identity P2j (h; μ)(1) = μQ2j (h; μ), j = 1, . . . ,

n 2

for certain polynomial families of scalar Riemannian invariants Q2j (h; μ). In particular, the operators P2j (h; 0) annihilate constants. Moreover, using (3.1.3) and (3.2.17), it follows that n

(3.2.19) Q2j (h) = (−1)j Q2j h; −j 2 and

n Qn (h) = (−1) 2 P˙ n (h; 0)(1).

(3.2.20)

For the details we refer to [128]. On the round sphere (S n , gc ) the notion of a natural conformally covariant diﬀerential operator and the notion of a diﬀerential intertwining operator are closely related. We brieﬂy describe that relation. On a general manifold M the naturality of D implies D(M ; f∗ (g)) = f∗ ◦ D(M ; g) ◦ f ∗

(3.2.21)

for a diﬀeomorphism f : M → M . Now for (S n , gc ) we have τ (S n , gc ) = n(n−1). The group Gn+1 = SO(1, n + 1)◦ operates by conformal diﬀeomorphims on S n , i.e., g∗ (gc ) = e2Φg gc , g ∈ Gn+1 , g∗ = (g −1 )∗ (3.2.22) for certain smooth functions 0 < Φg ∈ C ∞ (S n ). It follows that for a conformally covariant operator D of weight (a, b) on S n , e−bΦg ◦ D(S n ; gc ) ◦ eaΦg = D(S n ; e2Φg gc ) = D(S n ; g∗ (gc )) = g∗ ◦ D(S n ; gc ) ◦ g ∗ (3.2.23) i.e., (3.2.24) D(S n ; gc ) ◦ eaΦg g∗ = ebΦg g∗ ◦ D(S n ; gc ). Now (3.2.22) yields g∗ (vol(gc )) = enΦg vol(gc ) for the Riemannian volume form and the spherical principal series representation can be written in the form πλc (g)u = e−λΦg g∗ (u), u ∈ C ∞ (S n ). Thus the intertwining relation (3.2.24) reads c c D(S n ; gc ) ◦ π−a (g) = π−b (g) ◦ D(S n ; gc ), g ∈ Gn+1 .

For a generic parameter a, the spherical principal series πac is irreducible. Moreover, for generic parameters πac πbc iﬀ a + b = −n, where the equivalence is

3.2. Scattering theory

101

realized by constant multiples of the standard Knapp-Stein intertwining operator Iλ (see (2.4.2)). The meromorphic family Iλ of pseudo-diﬀerential operators has diﬀerential operator residues at λ ∈ − n2 + N0 . More precisely, up to a constant multiple, the residue of Iλ at λ = − n2 + N has leading part ΔN S n and is given by ([32], [23]) 2 2 n2 +N N −1 n−1 1 + j− (ΔS n −j(n−1−j)) . (3.2.25) ΔS n − = 2 2 n j=1 j= 2

Formula (3.2.25) is a special case of an analogous product formula for Einstein spaces. Theorem 3.2.3 ([110]). Let (M n , g) be Einstein and n even. Then n 2 +N −1

P2N (M, g) =

j= n 2

j(n−1−j) τg Δg − n(n−1)

and

Qn (g) = (n−1)!

τg n(n−1)

(3.2.26)

n2 .

(3.2.27)

For more information around Theorem 3.2.3 (including a proof) we refer to Section 6.16. Note that Theorem 3.2.3 yields the product formula n 2 +N −1

(Δ + j(n−1−j))

(3.2.28)

j= n 2

for the GJMS-operators of the compact hyperbolic manifolds M n = Γ\Hn of even dimension n ≥ 4. We close the present section with an outline of a derivation of formula (3.2.28) using scattering theory. For this purpose, we realize M n = Γ\Hn as one boundary component of a hyperbolic cylinder X n+1 = (0, π) × M n , ∂X n+1 = M n M n and apply an explicit formula for the scattering operator. X is deﬁned as the quotient X = Γ\Hn+1 , where the cocompact torsion-free Γ ⊂ Gn = SO(1, n)◦ is viewed as a subgroup of Gn+1 acting on Hn+1 . The map κ : Hn × (0, π) (x , xn , θ) → (x , xn cos θ, xn sin θ) ∈ Hn+1 π

= Hn+1 ∩ {xn = 0} Hn κ Hn , 2 and pulls back the hyperbolic metric gHn+1 on Hn+1 to the metric satisﬁes

g = (sin θ)−2 (dθ2 + gHn )

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

102

on the cylinder (0, π) × Hn . The conformal compactiﬁcation (xn sin θ)2 g restricts to gHn on κ(0) ∪ κ(π). Gn ⊂ Gn+1 operates trivially on (0, π), and for a cocompact Γ ⊂ Gn , we have X n+1 (0, π) × M n ,

∂X n+1 = κ(M n , 0) ∪ κ(M n , π),

where M n = Γ\Hn and X n+1 = Γ\Hn+1 . The scattering operator S(λ) : C ∞ (∂X) → C ∞ (∂X) can be identiﬁed with an operator on C ∞ (M ) ⊕ C ∞ (M ). The following result provides an explicit formula for the latter operator. Theorem 3.2.4. S(λ) decomposes into the direct sum of the operators S(λ, μ) =

1 n−2λ Γ( n2 −λ) sin π( n2 −μ) sin π( n2 −λ) 2 Γ(λ−μ)Γ(λ−(n−1−μ)) (3.2.29) sin π( n2 −λ) sin π( n2 −μ) π Γ(λ− n2 ) acting on the space E(μ) ⊕ E(μ), where E(μ) = ker(Δ+μ(n−1−μ)) ⊂ C ∞ (M ). For a brief outline of the proof of Theorem 3.2.4 we refer to [195]. Full details are given in [150]. A direct calculation shows that (3.2.29) implies the functional equation S(λ, μ)S(n−λ, μ) = id . We analyze the poles of S(λ). Let n be even and 1 ≤ N ≤ n2 . (3.2.29) is invariant n−1 under the substitution μ → n−1−μ and we choose μ in ( n−1 2 , n−1] ∪ 2 + iR; this is no restriction of generality since −Δ ≥ 0. If μ ∈ R, then S(λ, μ) has a simple pole at λ = n2 + N . Furthermore, if μ is real, but μ ∈ Z, then Γ(λ − μ) is regular at λ = n2 + N , i.e., S(λ, μ) has a simple pole at λ = n2 + N . We determine its residue π in these cases. In view of Γ(z)Γ(1−z) = sin(πz) , we ﬁnd Resλ= n2 +N (S(·, μ))

n

n

n

1 (−1)N 1 = − 2−2N Γ −μ+N Γ − −μ +1+N sin π −μ id π N ! Γ(N ) 2 2 2 1 =− N !(N −1)!22N

n 2 +N −1

(j −μ)(n−1−j −μ) id .

j= n 2

Hence the residue is given by 1 − N !(N −1)!22N

n 2 +N −1

(−μ(n−1−μ) + j(n−1−j)) id

j= n 2

1 =− N !(N −1)!22N

n 2 +N −1

j= n 2

(Δ + j(n−1−j)).

3.2. Scattering theory

103

If μ ∈ Z, then the matrix in (3.2.29) has the form 0 sin π( n2 −λ) sin π( n2 −λ) 0 and both Γ-factors Γ( n2 −λ) and Γ(λ−μ) contribute a simple pole at λ = n2 +N iﬀ n 2 +N −μ ∈ −N0 , i.e., iﬀ "n # μ∈ +N, . . . , n−1 . (3.2.30) 2 If (3.2.30) is not satisﬁed, i.e., iﬀ # "n n , . . . , +N −1 , μ∈ 2 2 then S(λ, μ) has no pole at λ = n2 + N . In fact, the latter values of μ correspond to the eigenspaces of Δ in the kernel of the residue. In the cases (3.2.30), S(λ, μ) has a simple pole at λ = n2 + N with a residue being proportional to 0 1 . 1 0 In particular, the residue is not local. Now the cases (3.2.30) are characterized by the existence of L2 -eigenfunctions of the Laplacian on the inﬁnite-volume cylinder X. This illustrates the role of the technical condition in Theorem 3.2.1 that λ(n−λ) is not an L2 -eigenvalue of −ΔX . In the present situation, it is actually possible to give explicit formulas for the L2 -eigenfunctions which have to be ruled out. Generalized eigenfunctions of −ΔX can be constructed by integration of the Eisenstein kernel 1 E(x, ζ; λ) = P (γ · x, ζ)λ , x ∈ Hn+1 , ζn = 0, (λ) > n γ∈Γ

against automorphic functions in C ∞ (M )⊕C ∞ (M ). For ω ∈ ker(ΔM +μ(n−1−μ)), we ﬁnd the generalized eigenfunction n

Γ(λ−μ)Γ(λ−(n−1−μ)) Γ(λ)Γ(λ− n2 +1) 1+cos θ n × 2 F 1 λ−μ, λ−(n−1−μ); λ− +1; ω(x , xn ) (3.2.31) 2 2

π 2 2n−2λ (sin θ)λ

in ker(ΔX + λ(n − λ)). (3.2.31) is holomorphic in (λ) > n and admits a meromorphic continuation in λ to C. It has a pole at λ = n2 + N if μ satisﬁes (3.2.30). 1 This

is an Eisenstein series of a convex-cocompact Kleinian group. For the details we refer to [173], [200], [51] and the last chapter of [151].

104

Chapter 3. Powers of the Laplacian, Q-curvature and Scattering

In that case, the residue of (3.2.31) is an eigenfunction for the eigenvalue λ(n−λ), λ = n2 + N . It is a constant multiple of the function 1+cos θ n (sin θ) 2 +N 2 F 1 −M, 2N +M +1; N +1; ω(x , xn ), 2 where we write n2 + N − μ = −M . Here the hypergeometric function degenerates to a polynomial. More precisely, 1−r N+ 1 ∼ CM 2 (r), 2 F 1 −M, 2N +M +1; N +1; 2 α where CM (r) is a Gegenbauer polynomial of degree M (see [21], 3.15, formula (3)). The latter formula implies that the eigenfunction is in L2 (X) with respect to the hyperbolic volume (sin θ)−n vol(Hn ) vol(0, π). Now the relation Resλ= n2 +N (S) = −cN P2N

(see (3.2.7)) implies that the GJMS-operators P2N on Γ\Hn are given by (3.2.28). Theorem 3.2.4 also implies that n

1 n−2λ Γ( n2 −λ) 0 1 Γ(λ)Γ(λ−(n−1)) sin π −λ . S(λ; 0) = 2 1 0 π Γ(λ− n2 ) 2 Hence

1 1 1 n−2λ Γ(λ)Γ(λ−(n−1)) S(λ) =2 1 Γ(λ− n2 ) Γ(1−( n2 −λ)) 1

so that for λ → n we obtain Γ(n) 1 1 1 −n S(n) =2 . = c n2 (n−1)! n n 1 1 1 Γ( 2 )Γ( 2 +1) Now (3.2.8) yields the formula n

Qn = (−1) 2 (n−1)! for the Q-curvature of the hyperbolic space Γ\Hn . The latter formula is a special case of (3.2.27).

Chapter 4

Paneitz Operator and Paneitz Curvature 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 . The early history of the pair (P4 , Q4 ) is as follows. The operator P4 was discovered in the middle of the 1980s independently by Eastwood-Singer ([91]), Paneitz ([192]) and Riegert ([208]) almost at the same time. Paneitz worked in general dimension n ≥ 3 and discovered the conformal covariance of the operator P4,n . Although the original paper of Paneitz remained unpublished until recently ([192]), it had a major inﬂuence. In [30], Branson supplied detailed proofs of the conformal covariance of the operator P4,n . The works [91] and [208] contain this result in dimension n = 4. Although the curvature quantity Q4,n in dimension n ≥ 3 appeared in [192] in the constant term of P4,n , Paneitz did not discuss the transformation properties of Q4 = Q4,4 under conformal changes of the metric. This aspect is important in Riegert’s independent work [208]. Riegert discovered that the transformation law of a certain linear combination of Q4 and |C|2 , under conformal changes of the metric, is governed by a linear diﬀerential operator: P4 . Later work ([12]) along these lines led to a formula for Q6 for conformally ﬂat metrics (see the discussion at the end of Section 6.14). In [30], Branson introduced the notation Q for Q4,n , and emphasized that this quantity (for n = 1, 2, 4) gives rise to a fourth-order analog of the Yamabe problem. In [48], Branson and Ørsted applied analytic continuation in dimension to derive the conformal transformation property of Q4 , i.e., the fundamental identity. The seminal work [31] embedded the material into the general context of GJMSoperators and introduced the hierarchy of Q-curvatures (as in Deﬁnition 3.1.1). For detailed information on the more recent role of Q4 in geometric analysis on four-manifolds we refer to [34], [73], [41], [43], [63], [64], [83], [170], [177] and

106

Chapter 4. Paneitz Operator and Paneitz Curvature

the bibliographies of these references. Analogous problems in higher dimension are discussed, for instance, in [49], [93], [187], [205].

4.1 P4 , Q4 and their transformation properties We introduce some notation. For a Riemannian manifold (M, g) of dimension n, let ∗g be the Hodge star operator on forms. Then the operators δg : Ωp (M ) → Ωp−1 (M ),

δg = (−1)np+n+1 ∗g d∗g

are the formal adjoints of the exterior diﬀerentials on forms with respect to the Hodge scalar product ω ∧ ∗g η M

([239]). Let Δ = δd denote the non-negative Laplacian (this sign convention is opposite to Chapter 3). A symmetric bilinear form T induces on Ω1 (M ) the linear operator T # : ω → (X → T (X, ω )). Here ω ∈ X (M ) denotes the vector ﬁeld dual to ω, i.e., g(ω , Y ) = ω, Y for all Y ∈ X (M ). In other words, T acts by the dual of the linear operator T on vector ﬁelds which corresponds to T by g(T (X), Y ) = T (X, Y ). Similarly, for the vector ﬁeld X, let X ∈ Ω1 (M ) be deﬁned by X , Y = g(X, Y ) for all Y ∈ X (M ). Note that g# acts as the identity on Ω1 (M ). The critical Q-curvature of a Riemannian manifold of dimension n = 4 is deﬁned by 1 1 1 (4.1.1) Q4 = 2J2 − 2|P|2 + ΔJ = τ 2 − | Ric |2 + Δτ. 6 2 6 We shall refer to this fourth-order curvature invariant also as to the Paneitz curvature or Paneitz quantity in order to distinguish it from Q-curvatures of orders = 4. The Paneitz curvature Q4 has the following conformal transformation law. Theorem 4.1.1 (The fundamental identity for Q4 ). For all ϕ ∈ C ∞ (M ), e4ϕ Q4 (e2ϕ g) = Q4 (g) + P4 (g)(ϕ),

(4.1.2)

where the Paneitz operator P4 is deﬁned by P4 = Δ2 + δ(2Jg − 4P)#d = Δ2 + δ

2 τ g − 2 Ric #d. 3

(4.1.3)

4.1. P4 , Q4 and their transformation properties

107

Although the individual terms in (4.1.1) transform with up to fourth-order powers of ϕ under conformal transformations, all non-linearities cancel in the given speciﬁc linear combination. This property distinguishes the quantity Q4 among other fourth-order scalar Riemannian curvature invariants. The fourth-order Paneitz-operator P4 is conformally covariant of weight (0, 4), i.e., Theorem 4.1.2. For all ϕ ∈ C ∞ (M ), e4ϕ ◦ P4 (e2ϕ g) = P4 (g).

(4.1.4)

Proof. The assertion is a consequence of Theorem 4.1.1. In fact, we ﬁnd

e4ϕ P4 (e2ϕ g)u = e4ϕ e4u Q4 (e2(u+ϕ) g) − Q4 (e2ϕ g) (by (4.1.2)) = e4(ϕ+u) Q4 (e2(u+ϕ) g) − e4ϕ Q4 (e2ϕ g) = Q4 (g) + P4 (g)(u + ϕ) − [Q4 (g) + P4 (g)ϕ] = P4 (g)u.

(by (4.1.2))

The proof is complete. Alternatively, (4.1.2) follows from the generalization n

n

e−( 2 +2)ϕ ◦ P4,n (e2ϕ g) = P4,n (g) ◦ e( 2 −2)ϕ

(4.1.5)

of (4.1.4) for the Paneitz-operator n−4 Q4,n (4.1.6) 2 on Riemannian manifolds of dimension n ≥ 3 (see Section 3.1). Here the constant term of P4,n is given by the fourth-order curvature invariant n Q4,n = J2 − 2|P|2 + ΔJ. (4.1.7) 2 Q4,n specializes to Q4 for n = 4. In terms of Ricci and scalar curvature, P4,n and Q4,n can be written in the less illuminating form P4,n = Δ2 + δ ((n−2)Jg − 4P) #d +

Δ2 + δ (an τ g + bn Ric) #d + with an =

(n−2)2 + 4 , 2(n−1)(n−2)

bn = −

and Q4,n = cn | Ric |2 + dn τ 2 + with cn = −

2 , (n−2)2

dn =

n−4 Q4,n 2 4 , n−2

1 Δτ 2(n−1)

n3 −4n2 +16n−16 . 8(n−1)2 (n−2)2

108

Chapter 4. Paneitz Operator and Paneitz Curvature

4.2 The fundamental identity for the Paneitz curvature In the present section we give a direct proof of Theorem 4.1.1. The proof will be interwoven with an informal description of how the basic principle of equivalence between conformal covariance and its inﬁnitesimal version acts in the given special case. Moreover, we use the instance to establish some important facts for later reference. We continue to use ˆ· to indicate the conformal change g → e2ϕ g. In particular, ˆ ˆ denote the respective operators for the metric e2ϕ g. We need the following δ and Δ simple result. Lemma 4.2.1. The operators eaϕ ◦ d ◦ e−aϕ : Ωp (M ) → Ωp+1 (M ),

e(a+2)ϕ ◦ δˆ ◦ e−aϕ : Ωp (M ) → Ωp−1 (M )

coincide with ω → dω − adϕ ∧ ω,

ω → δω − (n−2p−a)igrad ϕ (ω)

for all a ∈ C, respectively. Hence the operator ˆ ◦ e−aϕ : C ∞ (M ) → C ∞ (M ), e(a+2)ϕ ◦ Δ

Δ = δd

coincides with ω → Δω − aδ(dϕ ∧ ω) − (n−2−a)igrad ϕ (dω) + a(n−2−a)|dϕ|2ω. Proof. The assertion for d is obvious. Note that ˆ∗ = e(n−2p)ϕ ∗ : Ωp → Ωn−p . Hence ˆ =ˆ ∗dˆ ∗ω ( − 1)np+n+1 δω = e(n−2(n−p+1))ϕ ∗ d ∗ e(n−2p)ϕ ω = e−2ϕ (∗d ∗ ω + (n−2p) ∗ dϕ ∧ ∗ω) = e−2ϕ ∗d ∗ ω + (n−2p)(−1)n−p igrad(ϕ) (∗2 ω) = e−2ϕ ∗d ∗ ω + (−1)np+n (n−2p)igrad(ϕ) (ω) using ∗2 = (−1)np+p on Ωp . Hence ˆ = δω − (n−2p)igrad(ϕ) (ω). e2ϕ δω This proves the assertion on δ for a = 0. The proof for general a is analogous. In order to prove the assertion for the Laplacian, we write

ˆ −aϕ ω) = e(a+2)ϕ ◦ δˆ ◦ e−aϕ eaϕ d(e−aϕ ω) e(a+2)ϕ Δ(e and apply the identities for d and δ. The proof is complete.

4.2. The fundamental identity for the Paneitz curvature

109

Now δ = − ∗ d∗ in dimension n = 4. We get ˆ ˆJ 9 ˆJ = δd Δ = e−4ϕ δe2ϕ dˆJ

(by Lemma 4.2.1)

δd(J + Δϕ − |dϕ|2 ) − 2e−4ϕ δ(J + Δϕ − |dϕ|2 )dϕ = e−4ϕ ΔJ + Δ2 ϕ − 2JΔϕ + 2(dJ, dϕ) =e

−4ϕ

− Δ|dϕ|2 − 2(Δϕ)2 + 2(dΔϕ, dϕ) + 2|dϕ|2 Δϕ − 2(d|dϕ|2 , dϕ)

and e4ϕˆJ2 = J2 + 2JΔϕ + (Δϕ)2 − 2J|dϕ|2 − 2|dϕ|2 Δϕ + |dϕ|4 ˆ 2 is given by the product of by (2.5.7). In view of (2.5.9), the squared norm |P| −4ϕ e and the squared norm of 1 P − Hess(ϕ) + dϕ ⊗ dϕ − |dϕ|2 g. 2 A calculation yields

ˆ 4 = Q4 + Δ2 ϕ + 2JΔϕ + 2(dJ, dϕ) + 4 Pij Hessij (ϕ) e4ϕ Q i,j

+ − Δ|dϕ|2 + 2(dΔϕ, dϕ) − 2J|dϕ|2 − 2| Hess(ϕ)|2 − 4

Pij ϕi ϕj

i,j

+ − 2(d|dϕ|2 , dϕ) + 4

Hess (ϕ)ϕi ϕj . ij

(4.2.1) (4.2.2)

i,j

We rewrite this result in the form Lemma 4.2.2. ˆ 4 = Q4 + P4 (ϕ) + e4ϕ Q where

1 (4ϕP4 + P4• [ϕ]) (ϕ), 2

P4• (g)[ϕ] = (d/dt)|t=0 P4 (e2tϕ g) .

(4.2.3)

(4.2.4)

Now the relation (4.2.3) and the inﬁnitesimal conformal covariance 4ϕ ◦ P4 + P4• [ϕ] = 0 of P4 imply the fundamental identity (4.1.2).

(4.2.5)

110

Chapter 4. Paneitz Operator and Paneitz Curvature

Note that the conditions (4.1.4) (conformal covariance) and (4.2.5) (inﬁnitesimal conformal covariance) are equivalent. In fact, (4.2.5) implies 1 (d/dt) e4tϕ ◦ P4 (e2tϕ g) dt e4ϕ ◦ P4 (e2ϕ g) − P4 (g) = 0

1

=

(d/ds)|0 e4(t+s)ϕ ◦ P4 (e2(t+s)ϕ g) dt

0

=

1

e4tϕ ◦ 4ϕ ◦ P4 (e2tϕ g) + P4• (e2tϕ g)[ϕ] dt = 0.

0

Thus it remains to prove (4.2.3) and (4.2.5). For the proof of (4.2.3) we have to verify that P4 (u) = Δ2 u + 2JΔu + 2(dJ, du) + 4(P, Hess(u)),

(4.2.6)

and that (4.2.1) coincides with 1 1 (d/dt)|t=0 e4tϕ P4 (e2tϕ g) (ϕ) = (4ϕP4 (g) + P4• (g)[ϕ]) (ϕ); 2 2 the vanishing of (4.2.2) is obvious by

(4.2.7)

(d(dϕ, dϕ), dϕ) = ∇grad ϕ (dϕ, dϕ) = 2 (∇grad ϕ dϕ, dϕ) . We note that this proof of the fundamental identity is an explicit version of the following argument showing that (4.1.2) follows from its inﬁnitesimal version 4ϕQ4 + Q•4 [ϕ] = P4 (ϕ), where

(4.2.8)

Q•4 (g)[ϕ] = (d/dt)|t=0 (Q4 (e2tϕ g)),

and the conformal covariance (4.1.4). Lemma 4.2.3. (4.2.8) and (4.1.4) for all metrics in the conformal class of g imply (4.1.2) for g. Proof. We calculate

1

e4ϕ Q4 (e2ϕ g) − Q4 (g) = 0

1

=

(d/dt) e4tϕ Q4 (e2tϕ g) dt

(d/ds)|0 e4(t+s)ϕ Q4 (e2(t+s)ϕ g) dt

0

1

=

e4tϕ 4ϕQ4 (e2tϕ g) + Q•4 (e2tϕ g)[ϕ] dt

0

1

e4tϕ P4 (e2tϕ g)ϕdt

=

(by (4.2.8))

0

= P4 (g)ϕ. The proof is complete.

(by (4.1.4))

4.2. The fundamental identity for the Paneitz curvature

111

In particular, (4.2.6) (being equivalent to (4.2.8)) and (4.2.5) (being equivalent to (4.1.4)) imply (4.1.2). In order to prove (4.2.5), we calculate (4.2.9) (d/dt)|t=0 e4tϕ P4 (e2tϕ g)u . That calculation can be performed by using one of the equivalent formulas (4.1.3) or (4.2.6) for the operator P4 . It will be convenient to use the latter one. For the evaluation of (4.2.9) we apply Lemma 4.2.1 for p = 0 and a = n2 − 1. We ﬁnd

2 n ˆ ◦ e−( n2 −1)ϕ = Δ − n −1 δ(·dϕ) + igrad(ϕ) d + n −1 |dϕ|2 e( 2 +1)ϕ ◦ Δ 2 2

n

2 n 2 −1 Δ(ϕ) + −1 |dϕ| . = Δ− 2 2 Using e2ϕ ˆJ = J + Δ(ϕ) −

n 2

−1 |dϕ|2 ,

we conclude that

n ˆ + n −1 ˆJ ◦ e−( n2 −1)ϕ = Δ + n −1 J. e( 2 +1)ϕ ◦ Δ 2 2

This proves the conformal covariance of the Yamabe operator P2 . In view of

ˆ Hess(u)) ˆ 2 u + 2ˆJΔu ˆ + 2ˆ 7 e4ϕ P4 (e2ϕ g)u = e4ϕ Δ g(dˆJ, du) + 4ˆ g(P,

ˆ 2 = e4ϕ Δe ˆ −2ϕ e2ϕ Δ ˆ , (2.5.7) (see (4.2.6)), Lemma 4.2.1, the factorization e4ϕ Δ and (2.5.9) show that (4.2.9) equals − 2δ(digrad ϕ + dϕ ∧ δ)du + 2(−2Jigrad ϕ du + ΔϕΔu) + 2 +4

g rs ((Δϕ)r − 2Jϕr )us

r,s

P (−ϕi uj − ϕj ui + (du, dϕ)gij ) − 4(Hess(ϕ), Hess(u)) ij

i,j

= −2Δ(du, dϕ) + 2(dΔu, dϕ) + 2(du, dΔϕ) − 4J(du, dϕ) − 4(Hess(ϕ), Hess(u)) − 4 Pij (ϕi uj + ϕj ui ) .

(4.2.10)

i,j

For u = ϕ, (4.2.10) simpliﬁes to −2Δ|dϕ|2 + 4(dΔϕ, dϕ) − 4J|dϕ|2 − 4| Hess(ϕ)|2 − 8

Pij ϕi ϕj ,

i,j

and this coincides with (4.2.1), up to the factor 2. Thus (4.2.1) coincides with (4.2.7). But Lemma 4.2.4 shows that (4.2.10) vanishes. This completes the proof of (4.2.5).

112

Chapter 4. Paneitz Operator and Paneitz Curvature

Lemma 4.2.4. Δ(dϕ, dψ) = (dΔϕ, dψ) − 2(Hess(ϕ), Hess(ψ)) + (dϕ, dΔψ) − 2(Ric, dϕ ⊗ dψ) for ϕ, ψ ∈ C ∞ (M ). Proof. For α, β ∈ Ω1 (M ) we have Δ(α, β) = −(ΔB α, β) − 2(∇α, ∇β) − (α, ΔB β) ock formula ΔB = with the Bochner Laplacian ΔB . But the Bochner-Weitzenb¨ −Δ + Ric ([24], 1.155) implies Δ(dϕ, dψ) = (Δdϕ, dψ) − 2(Hess(ϕ), Hess(ψ)) − 2(Ric, dϕ ⊗ dψ) + (dϕ, Δdψ).

This proves the assertion.

It only remains to verify that P4 , deﬁned by (4.1.3), can be written in the form (4.2.6). That observation is the special case n = 4 of the fact that P4,n (see (4.1.6)) coincides with the operator Δ2 u + (n−2)JΔu + (6−n)(dJ, du) + 4(P, Hess(u)) +

n−4 Q4,n u. 2

(4.2.11)

That assertion is obviously equivalent to Lemma 4.2.5. −δ(P#du) = (dJ, du) + (P, Hess(u)).

(4.2.12)

This result follows from Lemma 4.2.6. δ(P#du) = (δP, du) − (P, Hess(u)), where the divergence δ acts on a symmetric bilinear form h by δh(X) = − i ∇i (h)(ei , X) using an orthonormal frame. and the relation −δP = dJ, which is a consequence of the second Bianchi identity. For the convenience of later reference and for the sake of completeness, these consequences are discussed in the following lemma. Lemma 4.2.7 (Bianchi identities). ∇Z (R)(U, V, X, Y ) + ∇Y (R)(U, V, Z, X) + ∇X (R)(U, V, Y, Z) = 0 and div1 (C)(X, Y, Z) =

(4.2.13)

∇i (C)(ei , X, Y, Z)

i

= (n−3) (∇Z (P)(X, Y ) − ∇Y (P)(X, Z)) , −δP = dJ. The tensor on the right-hand side of (4.2.14) is called the Cotton tensor.

(4.2.14) (4.2.15)

4.2. The fundamental identity for the Paneitz curvature

113

Proof of Lemma 4.2.6. We observe that for α ∈ Ω1 (M ), ∇X (P#α)(Y ) = X ((P#α)(Y )) − (P#α)(∇X (Y )) = X(P(Y, α )) − P(∇X (Y ), α ) = ∇X (P)(Y, α ) + P(Y, ∇X (α )). Hence δ(P#du) = −

∇i (P#du)(ei )

i

=−

∇i (P)(ei , (du) ) −

i

P(ei , ∇i ((du) ))

i

= δ(P), (du) − (P, Hess(u)) = (δ(P), du) − (P, Hess(u)).

This completes the proof.

Proof of Lemma 4.2.7. The ﬁrst identity is the diﬀerential Bianchi identity. Since both sides of (4.2.14) are linear in X, Y and Z, it suﬃces to prove the assertion for a geodesic frame ei at m. We contract (4.2.13) in the arguments U and X. Then ∇k (R)(es , ej , es , ei ) + ∇i (R)(es , ej , ek , es ) + ∇s (R)(es , sj , ei , ek ) = 0. Now using that C is trace-free and R = C − P g, we ﬁnd " ∇s (C)(es , ej , ei , ek ) = ek (P g)(es , ej , es , ei ) s

s

# + ei (P g)(es , ej , ek , es ) + es (P g)(es , ej , ei , ek ) .

The sums on the right-hand side simplify to [es (Pis )δjk − es (Pks )δji ] + ek (J)δji − ei (J)δjk . (n−3) (ek (Pji ) − ei (Pjk )) +

s

Since, by (4.2.15), s es (Pis ) = Ji , we ﬁnd ∇s (C)(es , ej , ei , ek ) = (n−3) (∇k (P)(ej , ei ) − ∇i (P)(ej , ek )) , s

i.e.,

∇i (C)(ei , X, Y, Z) = (n−3) (∇Z (P)(X, Y ) − ∇Y (P)(X, Z)) .

i

In order to prove (4.2.15), we ﬁrst contract (4.2.13) in U and X. Then −∇k (Ric)ij + ∇i (Ric)jk + ∇s (R)(es , ej , ei , ek ) = 0. s

114

Chapter 4. Paneitz Operator and Paneitz Curvature

A second contraction yields −∇k (τ ) +

∇i (Ric)ik +

2 The latter identity implies

∇s (Ric)sk = 0,

s

i

i.e.,

∇i (Ric)ik = ∇k (τ ).

i

i

∇i (P)ik = ∇k (J). In fact,

1 (∇i (Ric)ik − ∇i (J)δik ) n−2 i 1 1 ∇k (τ ) − ∇k (J) = n−2 2 1 1 1 = − ∇k (τ ) n−2 2 2(n−1) 1 = ∇k (τ ) = ∇k (J). 2(n−1)

∇i (P)ik =

i

The proof is complete.

An alternative proof of Theorem 4.1.1 will be given in Chapter 6. In Section 6.6 (see Remark 6.6.5), we shall derive the formula (4.1.6) from the asymptotics of eigenfunctions of the Laplacian of an associated Poincar´e-Einstein metric. For n = 4, this result will later imply that the residue family D4res (h; λ) specializes to P4 (h)i∗ at λ = 0. The conformal covariance of D4res (h; λ) then yields the conformal covariance of P4 (h). Moreover, we will prove that the quantity D˙ 4res (h; 0)(1) coincides with −Q4 (h). From this perspective, the fundamental identity for Q4 (h) is just a consequence of the conformal covariance of the family D4res (h; λ). Note that the latter argument does not require going to dimensions = 4. Section 6.22 will contain a further proof of the fundamental identity for Q4 . It generalizes the latter argument but replaces the residue family D4res (h; λ) by the tractor family D4T (g; λ).

4.3 Q4 and v4 The discussion in Section 4.2 clearly shows that the fundamental identity for Q4 = 2(J2 − |P|2 ) + ΔJ

(4.3.1)

is a consequence of a subtle interplay between the quantities v4 = J2 − |P|2 and ΔJ. The behaviour of v4 in a conformal class is not governed by a linear operator.

4.3. Q4 and v4

115

Its inﬁnitesimal conformal transformation law is v4• [ϕ] + 4v4 ϕ = 2JΔϕ + 2(P, Hess(ϕ)) = 3JΔϕ + (Ric, Hess(ϕ)) = (S, Hess(ϕ)) ,

(4.3.2)

where S = Ric − τ2 g is the Einstein tensor on M 4 . In addition to its relation to Q4 , the quantity J2 − |P|2 appears naturally in a number of other contexts. The multiple 18 (J2 − |P|2 ) is the holographic anomaly of the renormalized volume of Poincar´e-Einstein metrics in dimension 5 ([141], [119]). In this connection, the relation (4.3.2) appears, e.g., in [147]. The same relation plays an important role in the study ([66], [63]) of the fully non-linear partial diﬀerential equation for ϕ so that v4 (e2ϕ g) = f ∈ C ∞ (M ). The quantity J2 − |P|2 is also of interest in dimensions = 4. In fact, for a closed manifold M n of dimension n ≥ 3, we notice that the functional (J2 −|P|2 ) vol F2 = M

satisﬁes (d/dt)|t=0 (F2 (e2tϕ g)) = (n−4) ϕ(J2 −|P|2 ) vol(g) + 2 (JΔϕ + (Hess(ϕ), P)) vol M M = (n−4) ϕ(J2 −|P|2 ) vol(g) − δ(P#dϕ) vol M M = (n−4) ϕv4 vol M

(Lemma 4.2.5). It follows that F2 is conformally invariant for n = 4, and that for n > 4 the critical points of the functional

2 (J −|P| ) vol 2

M

n−4 n vol

2

M

on a conformal class are solutions of the equation J2 − |P|2 = constant. This observation generalizes the description of the equation J = constant as the EulerLagrange equation of the Yamabe functional ([165]). More generally, Viaclovsky ([235]) discovered that for 2k < n (k ≥ 3) and C = 0, the critical points of the functional σk (P) vol, σk (P) = tr(∧k P) Fk = M

are metrics with constant σk . In Lemma 6.14.1, we will see that the symmetric polynomials σk (P) of the eigenvalues of the Schouten tensor are related to the

116

Chapter 4. Paneitz Operator and Paneitz Curvature

holographic coeﬃcients v2k of conformally ﬂat metrics. For general metrics, the equations v2k = constant are the Euler-Lagrange equations of the respective functionals 2 n−2k n v2k vol vol M

M

([65], [117]). Finally, we emphasize the relation between the quantity v4 and the GaußBonnet integrand of M 4 . The observation is that the Euler form of M is proportional to 8J2 − 8|P|2 + |C|2 (see (6.14.15)). Hence for closed M the integral (J2 − |P|2 ) vol M

depends only on the conformal class of g. This re-proves the conformal invariance of F2 . The latter integral is a multiple of the total Q-curvature M Q4 vol. The theory of residue families in Chapter 6 will recognize the relations between P4 , Q4 and v4 as special cases of general facts. In particular, the holographic formula will give the two summands 2(J2 − |P|2 ) and ΔJ in (4.3.1) a natural interpretation.

Chapter 5

Intertwining Families c In the present chapter, we construct two sequences of polynomial families DN (λ) : nc C ∞ (S n ) → C ∞ (S n−1 ) and DN (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ) of diﬀerential intertwining operators for spherical principal series representations. These families are induced by families DN (λ) of homomorphisms of Verma modules. We show how nc (λ) naturally arises from the asymptotics of eigenfunctions of the Laplacian of DN the hyperbolic metric on the upper half-space. This is the simplest special case of the construction of residue families in Section 6.6. In Chapter 6, the coincidence of nc (λ) will be interpreted as the simplest two constructions of diﬀerent nature for DN case of a relation between residue families and tractor families (holographic duality). The induction and the mutual relations between the families are discussed in Section 5.2. We prove that all families DN (λ) satisfy a system of factorization identities. These relations imply corresponding identities for the induced families of diﬀerential operators. In turn, these give rise to a recursive algorithm which c (λ). In Section 6.11, allows determination of explicit formulas for the families DN analogs of these factorization identities for residue families will shed light on the recursive structure of Q-curvatures and GJM S-operators.

5.1 The algebraic theory In this section, we construct the families DN (λ), describe them as homomorphisms of generalized Verma modules, and relate them to the asymptotics of eigenfunctions of the hyperbolic Laplacian on the upper half-space.

5.1.1 Even order families D2N (λ) We introduce the notation Δ− n−1 =

n−1

(Yj− )2 ∈ U(n− n ) ⊂ U(gn ),

j=1

Δ− n =

n j=1

(Yj− )2 ∈ U(n− n+1 ) ⊂ U(gn+1 ).

118

Chapter 5. Intertwining Families

Theorem 5.1.1. For any non-negative integer N and any λ ∈ C, the element D2N =

N

j − 2N −2j aj (Δ− ∈ U(n− n−1 ) (Yn ) n+1 )

j=0

satisﬁes

Yi+ , D2N ∈ U(n− n+1 )(mn+1 ⊕ C(H0 −λ)), i = 1, . . . , n − 1

(5.1.1)

iﬀ the coeﬃcients aj are subject to the recursive relations (N −j +1)(2N −2j +1)aj−1 + j(n−1+2λ−4N +2j)aj = 0, j = 1, . . . , N. (5.1.2) Since (N − j + 1)(2N − 2j + 1) = 0 for all j = 0, . . . , N , the recursive relations (5.1.2) imply that the coeﬃcients aj are determined by aN . In particular, for all λ the element D2N ∈ U(n− n+1 ) is uniquely determined by (5.1.1), up to a multiple. More precisely, we have Corollary 5.1.1. For 0 ≤ j ≤ N −1, we have aj (λ) =

N −1 N! (−2)N −j (2λ−4N +2k+n+1)aN (λ). j!(2N −2j)! k=j

0 (λ) ∈ U(n− Let D2N n+1 ) be the unique element satisfying (5.1.1) which is 0 (λ) is given by normalized by the condition aN (λ) = 1, i.e., D2N 0 − 2N −2 N D2N (λ) = a0 (λ)(Yn− )2N + a1 (λ)Δ− + · · · + (Δ− n−1 (Yn ) n−1 )

(5.1.3)

with the polynomial coeﬃcients aj (λ) =

N −1 N! (−2)N −j (2λ−4N +2k+n+1) j!(2N −2j)!

(5.1.4)

k=j

(N )

for j = 0, . . . , N −1. We shall write aj (λ) if it is appropriate to emphasize the order of the corresponding operator. (N ) The coeﬃcients aj (λ) are coeﬃcients of a Gegenbauer polynomial of degree 2N . The Gegenbauer polynomial Cnλ (r) is deﬁned by the power series 1 = Cnλ (r)xn . (5.1.5) 2 λ (1 − 2rx + x ) n≥0

(5.1.5) implies Cnλ (−r) = (−1)n Cnλ (r). On the other hand, we have Cnλ (r) =

Γ(2λ+n) 1 F Γ(2λ) n!

1 1−r −n, 2λ+n; λ+ ; , 2 2

(5.1.6)

(5.1.7)

i.e., Cnλ (r) is a hypergeometric polynomial. For details see [21], Section 3.15.

5.1. The algebraic theory

119

Theorem 5.1.2. a0 (λ)x2N − a1 (λ)x2N −2 ± · · · + (−1)N x0 = (N )

(N )

N! −λ− n−1 C2N 2 (x). n−1 (−λ− 2 )N

Proof. We recall the explicit formula ([21], 3.15, formula (9)) λ (x) = C2N

N (−1)j (λ)2N −j j=0

j!(2N −2j)!

(2x)2N −2j ,

where (λ)m = λ(λ+1) · · · (λ+m−1). Now, by (5.1.4), we have N −1 N! n+1 (N ) N −j N −j (−2) aj (λ) = 2 λ−2N +k+ , j!(2N −2j)! 2 k=j

i.e., (N ) aj (−λ)

n−1 N! n−1 2N −2j 2 +2N −j −1 · · · λ− +N . = λ− j!(2N −2j)! 2 2

Hence we obtain a0 (−λ)x2N − a1 (−λ)x2N −2 ± · · · + (−1)N x0 =

N n−1 N! j (λ− 2 )2N −j (2x)2N −2j (−1) j!(2N − 2j)! (λ− n−1 ) N 2 j=0

=

N! λ− n−1 C2N 2 (x). n−1 (λ− 2 )N

The proof is complete.

Corollary 5.1.2. For all N ∈ N0 , n

n−1 0 N 0 N − +N = (Δ− +N = (Δ− ) and D D2N − n 2N n−1 ) . 2 2 0 Proof. The coeﬃcients ar = ar − n2 +N , 0 ≤ r ≤ N of D2N (− n2 +N ) are characterized by (N −r+1)(2N −2r+1)ar−1 + r(−1−2N +2r)ar = 0, aN = 1, i.e., aN −1 + N (−1)aN = 0, 6aN −2 + (N −1)(−3)aN −1 = 0, ··· (N −1)(2N −3)a1 + 2(−2N +3)a2 = 0, N (2N −1)a0 + (−2N +1)a1 = 0.

120

Chapter 5. Intertwining Families

Hence a0 = i.e., ar =

1 2! N! a1 = a2 = · · · = aN = aN = 1, N N (N −1) N!

N . Therefore, r

N n

N 0 j − 2N −2j − 2 N − N − +N = D2N = (Δ− (Δ− n−1 ) (Yn ) n−1 + (Yn ) ) = (Δn ) . j 2 j=0

For the proof of the second assertion we note that the coeﬃcients n−1 +N , 0 ≤ r ≤ N ar = ar − 2 are characterized by (N −r+1)(2N −2r+1)ar−1 + r(−2N +2r)ar = 0, aN = 1, i.e., aN −1 = 0, 6aN −2 + (N −1)(−2)aN −1 = 0, ··· N (2N −1)a0 + (−2N +2)a1 = 0. Hence aN −1 = · · · = a0 = 0, aN = 1.

We split the proof of Theorem 5.1.1 into a series of partial results on commutators. Lemma 5.1.1. H0 , (Yr− )j = −j(Yr− )j for 1 ≤ r ≤ n and j ≥ 1. Proof. The assertion is obvious for j = 1. We argue by induction and calculate H0 (Yr− )j+1 = H0 , (Yr− )j Yr− + (Yr− )j H0 Yr− = −j(Yr− )j+1 + (Yr− )j H0 , Yr− + (Yr− )j+1 H0 (by assumption) = −(j + 1)(Yr− )j+1 + (Yr− )j+1 H0 . The proof is complete. Lemma 5.1.2. M1r , (Yr− )2j = 2jY1− (Yr− )2j−1 for 1 ≤ r ≤ n and j ≥ 1.

Proof. The calculation M1r (Yr− )2 = M1r , Yr− Yr− + Yr− M1r Yr− = Y1− Yr− + Yr− M1r , Yr− + (Yr− )2 M1r = Y1− Yr− + Yr− Y1− + (Yr− )2 M1r

(5.1.8)

5.1. The algebraic theory

121

proves the assertion for j = 1. Now in sense of an induction we have M1r (Yr− )2(j+1) = M1r , (Yr− )2j (Yr− )2 + (Yr− )2j M1r (Yr− )2 = 2jY1− (Yr− )2j+1 + (Yr− )2j M1r , (Yr− )2 + (Yr− )2(j+1) M1r by assumption and thus M1r (Yr− )2(j+1) = 2(j +1)Yr− (Yr− )2(j+1) + (Yr− )2(j+1) M1r

by (5.1.8). The proof is complete. Lemma 5.1.3. + Y1 , (Yr− )2N =

8

−2N (2N −1)(Y1− )2N −1 + 4N (Y1− )2N −1 H0 , 2N (2N −1)Y1− (Yr− )2N −2 + 4N (Yr− )2N −1 M1r ,

r = 1, 2 ≤ r ≤ n.

Proof. We ﬁrst prove the assertion for r = 1. We ﬁnd Y1+ (Y1− )2 = Y1+ , Y1− Y1− + Y1− Y1+ Y1− = 2H0 Y1− + Y1− Y1+ , Y1− + (Y1− )2 Y1+ = 2 H0 , Y1− + 2Y1− H0 + 2Y1− H0 + (Y1− )2 Y1+ = 4Y1− H0 − 2Y1− + (Y1− )2 Y1+ , i.e.,

+ Y1 , (Y1− )2 = −2Y1− + 4Y1− H0 .

(5.1.9)

Now we proceed by induction. We calculate Y1+ (Y1− )2N +2 = Y1+ , (Y1− )2N (Y1− )2 + (Y1− )2N Y1+ (Y1− )2 = Y1+ , (Y1− )2N (Y1− )2 + (Y1− )2N Y1+ , (Y1− )2 + (Y1− )2N +2 Y1+ = −2N (2N −1)(Y1− )2N +1 + 4N (Y1− )2N −1 H0 (Y1− )2 + 4(Y1− )2N +1 H0 − 2(Y1− )2N +1 + (Y1− )2N +2 Y1+ , by assumption and by (5.1.9), i.e., we get + Y1 , (Y1− )2N +2 = (−2N (2N −1) − 2 − 8N )(Y1− )2N +1 + (4N +4)(Y1− )2N +1 H0 = −2(N +1)(2N +1)(Y1− )2N +1 + 4(N +1)(Y1− )2N +1 H0 using Lemma 5.1.1. This completes the proof of the claim for r = 1. For 2 ≤ r ≤ n and N = 1, the assertion follows from Y1+ (Yr− )2 = Y1+ , Yr− Yr− + Yr− Y1+ Yr− = 2M1r Yr− + Yr− Y1+ , Yr− + (Yr− )2 Y1+ = 2 M1r , Yr− + 2Yr− M1r + 2Yr− M1r + (Yr− )2 Y1+ = 2Y1− + 4Yr− M1r + (Yr− )2 Y1+ .

(5.1.10)

122

Chapter 5. Intertwining Families

We proceed by induction. We obtain Y1+ (Yr− )2N +2 = Y1+ , (Yr− )2N (Yr− )2 + (Yr− )2N Y1+ (Yr− )2 = Y1+ , (Yr− )2N (Yr− )2 + (Yr− )2N Y1+ , (Yr− )2 + (Yr− )2N +2 Y1+ = 2N (2N −1)Y1− (Yr− )2N + 4N (Yr− )2N −1 M1r (Yr− )2 + (Yr− )2N (2Y1− + 4Yr− M1r ) + (Yr− )2N +2 Y1+ by using the assumption and (5.1.10). Now M1r , (Yr− )2 = 2Y1− Yr− (Lemma 5.1.2) implies

Y1+ , (Yr− )2N +2

= (2N (2N −1)+2)Y1− (Yr− )2N + 8N Y1− (Yr− )2N + 4(N +1)(Yr− )2N +1 M1r = 2(N +1)(2N +1)Y1− (Yr− )2N + 4(N +1)(Yr− )2N +1 M1r . The proof is complete.

Lemma 5.1.4. For j ≥ 1, + j j−1 j−1 Y1 , (Δ− − 2j(n−1−2j)Y1− (Δ− − 4jY1− (Δ− H0 ∈ U(n− n )mn . n−1 ) n−1 ) n−1 ) j Proof. We write (Δ− n−1 ) in the form j (Δ− n−1 ) =

j! − )2an−1 , a = (a1 , . . . , an−1 ), (Y − )2a1 · · · (Yn−1 a! 1

|a|=j

where |a| = a1 + · · · + an−1 and apply Lemma 5.1.3 successively. The ﬁrst step yields j Y1+ (Δ− n−1 ) =

j! − T1 (a) + (Y1− )2a1 Y1+ (Y2− )2a2 · · · (Yn−1 )2an−1 , a!

|a|=j

where T1 (a)

− )2an−1 . = −2a1 (2a1 −1)(Y1− )2a1 −1 + 4a1 (Y1− )2a1 −1 H0 (Y2− )2a2 · · · (Yn−1 (5.1.11)

In the next step, we use Lemma 5.1.3 for r = 2. We get j Y1+ (Δ− n−1 ) j! − = )2an−1 , T1 (a) + T2 (a) + (Y1− )2a1 (Y2− )2a2 Y1+ (Y3− )2a3 · · · (Yn−1 a! |a|=j

5.1. The algebraic theory

123

where − T2 (a) = 2a2 (2a2 −1)(Y1− )2a1 +1 (Y2− )2a2 −1 (Y3− )2a3 · · · (Yn−1 )2an−1 − + 4a2 (Y1− )2a1 (Y2− )2a2 −1 M12 (Y3− )2a3 · · · (Yn−1 )2an−1 .

The continuation of this method ﬁnally gives the formula j! + j (T1 (a) + T2 (a) + · · · + Tn−1 (a)) = Y1 , (Δ− n−1 ) a! |a|=j

with T1 (a) given by (5.1.11) and − Tk (a) = 2ak (2ak −1)(Y1− )2a1 +1 (Y2− )2a2 · · · (Yk− )2ak −2 · · · (Yn−1 )2an−1 − + 4ak (Y1− )2a1 · · · (Yk− )2ak −1 M1k · · · (Yn−1 )2an−1 ,

k = 2, . . . , n − 1. Here we use that n− is abelian to get the powers of Y1− left. It follows that j! + − j 2a1 (2a1 −1)Y1− (Y1− )2a1 −2 (Y2− )2a2 · · · (Yn−1 =− )2an−1 Y1 , (Δ− n−1 ) a! |a|=j

+

j! − 2a2 (2a2 −1)Y1− (Y1− )2a1 (Y2− )2a2 −2 · · · (Yn−1 )2an−1 a!

|a|=j

+ ···+ j! − 2an−1 (2an−1 −1)Y1− (Y1− )2a1 (Y2− )2a2 · · · (Yn−1 )2an−1 −2 + a! |a|=j

+2

j! − 2a1 (Y1− )2a1 −1 H0 (Y2− )2a2 · · · (Yn−1 )2an−1 a!

|a|=j

+2

j! − 2a2 (Y1− )2a1 (Y2− )2a2 −1 M12 · · · (Yn−1 )2an−1 a!

|a|=j

+ ···+ j! − 2an−1 (Y1− )2a1 · · · (Yn−1 )2an−1 −1 M1n−1 . +2 a!

(5.1.12)

|a|=j

Next, we sum up the ﬁrst n − 1 sums in (5.1.12). Since n− is abelian, it is enough to apply the following result. Lemma 5.1.5. ∂2 ∂2 ∂2 (|y|2j ) = 2j(n−5+2j)|y|2(j−1) − 8j(j−1)y12|y|2(j−2) , − 2 + 2 + ··· + 2 ∂y1 ∂y2 ∂yn−1 2 where |y|2 = y12 + · · · + yn−1 .

124

Chapter 5. Intertwining Families

Proof. We use the identities ∂ 2 2j ∂ |y| = 2j y1 |y|2j−2 = 2j|y|2j−2 + 4j(j −1)y12 |y|2j−4 ∂y12 ∂y1

(5.1.13)

and n−1 ∂ ΔRn−1 |y|2j = 2j (yr |y|2j−2 ) ∂y r r=1

= 2j(2j −2)|y|2j−2 + 2j(n−1)|y|2j−2 = 2j(n−3+2j)|y|2j−2

(5.1.14)

to calculate ∂2 ∂2 ∂2 ∂2 (|y|2j ) = −2 2 |y|2j + Δ |y|2j − 2 + 2 + ···+ 2 ∂y1 ∂y2 ∂yn−1 ∂y1 = −4j|y|2j−2 − 8j(j −1)y12 |y|2j−4 + 2j(n−3+2j)|y|2j−2 = 2j(n−5+2j)|y|2j−2 − 8j(j −1)y12 |y|2j−4 . The proof is complete.

Now Lemma 5.1.5 implies that the ﬁrst n−1 sums in (5.1.12) can be simpliﬁed to

j−1 j−2 − 8j(j −1)(Y1− )3 (Δ− . 2j(n−5+2j)Y1− (Δ− n−1 ) n−1 )

(5.1.15)

Next, successive application of Lemma 5.1.1 gives 2

j! − 2a1 (Y1− )2a1 −1 H0 (Y2− )2a2 · · · (Yn−1 )2an−1 a!

|a|=j

=2

(5.1.16)

j! − 2a1 (Y1− )2a1 −1 (Y2− )2a2 · · · (Yn−1 )2an−1 H0 a!

|a|=j

−2

j! − 2a1 2a2 (Y1− )2a1 −1 (Y2− )2a2 · · · (Yn−1 )2an−1 a!

|a|=j

− ···− j! − 2a1 2an−1 (Y1− )2a1 −1 (Y2− )2a2 · · · (Yn−1 )2an−1 . −2 a! |a|=j

In order to simplify the sums in (5.1.16), we apply the identities ∂ 2j |y| = 2jy1 |y|2(j−1) ∂y1 n−1

∂ ∂ 2j |y| = 4j(j −1) y1 |y|2(j−1) − y13 |y|2(j−2) . yr ∂y1 r=2 ∂yr

(5.1.17) (5.1.18)

5.1. The algebraic theory

125

For the proof of (5.1.18) we calculate ∂ ∂y1

n−1

n−1 ∂ 2j ∂ 2j ∂ 2j ∂ ∂ yr yr y1 |y| = |y| − |y| ∂yr ∂y1 r=1 ∂yr ∂y1 ∂y1 r=2 = 2j

∂2 ∂ 2j ∂ 2j |y| − y1 2 |y|2j − |y| . ∂y1 ∂y1 ∂y1

The latter sum simpliﬁes to (2j −1)2jy1|y|2(j−1) − 2jy1 |y|2(j−1) − 4j(j −1)y13 |y|2(j−2)

= 4j(j −1) y1 |y|2(j−1) − y13 |y|2(j−1) using (5.1.13) and (5.1.17). Now (5.1.16) and (5.1.18) imply 2

j! − 2a1 (Y1− )2a1 −1 H0 (Y2− )2a2 · · · (Yn−1 )2an−1 a!

(5.1.19)

|a|=j

j−1 j−1 j−2 = 4jY1− (Δ− H0 − 8j(j −1)Y1− (Δ− + 8j(j −1)(Y1− )3 (Δ− . n−1 ) n−1 ) n−1 )

Finally, we notice that [M1r , Ys− ] = 0 for r < s. Hence the last n − 2 sums in (5.1.12) deﬁne an element in U(n− n )mn . Thus we see that by (5.1.15) and (5.1.19)

j j−1 = 2j(2j +n−5)Y1− (Δ− Y1+ , (Δ− n−1 ) n−1 ) j−2 j−1 − 8j(j −1)(Y1− )3 (Δ− + 4jY1− (Δ− H0 n−1 ) n−1 ) j−1 j−2 − 8j(j −1)Y1− (Δ− + 8j(j −1)(Y1− )3 (Δ− n−1 ) n−1 )

(mod U(n− n )mm ),

i.e.,

j Y1+ , (Δ− n−1 )

j−1 j−1 = 2j(n−1−2j)Y1− (Δ− + 4jY1− (Δ− H0 n−1 ) n−1 )

(mod U(n− n )mm ).

The proof is complete. Now since M n ﬁxes Yn− Lemma 5.1.4 implies − 2k j − 2k j j + − 2k Y1+ (Δ− = Y1+ , (Δ− + (Δ− n−1 ) (Yn ) n−1 ) (Yn ) n−1 ) Y1 (Yn ) j−1 j−1 = 2j(n−1−2j)Y1− (Δ− (Yn− )2k + 4jY1− (Δ− H0 (Yn− )2k n−1 ) n−1 ) j + − 2k + (Δ− n−1 ) Y1 (Yn )

(mod U(n− n+1 )mn ).

126

Chapter 5. Intertwining Families

Hence using Lemma 5.1.1 and Lemma 5.1.3 (for r = n), we obtain + j − 2k j−1 Y1 , (Δ− = 2j(n−1−2j)Y1− (Δ− (Yn− )2k n−1 ) (Yn ) n−1 ) j−1 j−1 + 4jY1− (Δ− (Yn− )2k H0 − 8jkY1− (Δ− (Yn− )2k n−1 ) n−1 ) j − 2k−2 + 2k(2k−1)Y1− (Δ− n−1 ) (Yn ) j − 2k−1 + 4k(Δ− M1n n−1 ) (Yn )

(mod U(n− n+1 )mn )

j−1 = {2j(n−1−2j) + 4jλ − 8jk} Y1− (Δ− (Yn− )2k n−1 ) j − 2k−2 + 2k(2k−1)Y1− (Δ− n−1 ) (Yn )

(5.1.20)

(mod U(n− n+1 )(mn+1 ⊕ C(H0 −λ))). Now we are ready to complete the

Proof of Theorem 5.1.1. We use (5.1.20) to determine the commutator Y1+ , D2N , where N j − 2N −2j aj (Δ− . D2N = n−1 ) (Yn ) j=0

We obtain

N j−1 Y1+ , D2N = aj (2j(n−1−2j) + 4jλ − 8j(N −j)) Y1− (Δ− (Yn− )2N −2j n−1 ) j=0 j − 2N −2j−2 +2(N −j)(2N −2j −1)Y1− (Δ− n−1 ) (Yn )

up to an element in U(n− n+1 )(mn+1 ⊕ C(H0 −λ)). It follows that + Y1 , D2N ≡ 0 (mod U(n− n+1 )(mn+1 ⊕ C(H0 −λ))) is equivalent to the set of conditions {2r(n−1−2r) + 4rλ − 8r(N −r)} ar + 2(N −r+1)(2N −2r+1)ar−1 = 0 for r = 1, . . . , N . The latter conditions are equivalent to those given in Theorem 5.1.1. Now for each 2 ≤ j ≤ n − 1 there exists mj ∈ M n = SO(n−1) such that Ad(mj )Y1+ = Yj+ . Then

Yj+ , D2N = Ad(mj )Y1+ , D2N

+ = Ad(mj ) Y1+ , Ad(m−1 j )D2N = Ad(mj ) Y1 , D2N

− n since Δ− n−1 and Yn are ﬁxed under Ad(M ). It follows that the set of conditions + Yj , D2N ≡ 0 (mod U(n− n+1 )(mn+1 ⊕ C(H0 −λ)))

for 1 ≤ j ≤ n−1, is equivalent to the condition for j = 1. The proof is complete.

5.1. The algebraic theory

127

5.1.2 Odd order families D2N +1 (λ) We start with the following analog of Theorem 5.1.1 for families in U(n− n+1 ) of odd order. It is the algebraic core of the theory of odd order families. Theorem 5.1.3. For any non-negative integer N and any λ ∈ C, the element D2N +1 =

N

j − 2N −2j+1 bj (Δ− ∈ U(n− n−1 ) (Yn ) n+1 )

j=0

satisﬁes + Yi , D2N +1 ∈ U(n− n+1 )(mn+1 ⊕ C(H0 −λ)), i = 1, . . . , n − 1

(5.1.21)

iﬀ the coeﬃcients bj are subject to the recursive relations (N −j +1) (2N −2j +3) bj−1 + j (n−3+2λ−4N +2j)bj = 0

(5.1.22)

for j = 1, . . . , N . Proof. The proof is similar to that of Theorem 5.1.1. Lemma 5.1.4 gives − 2k+1 j − 2k+1 j j + − 2k+1 Y1+ (Δ− = Y1+ , (Δ− + (Δ− n−1 ) (Yn ) n−1 ) (Yn ) n−1 ) Y1 (Yn ) j−1 j−1 = 2j(n−1−2j)Y1− (Δ− (Yn− )2k+1 + 4jY1− (Δ− H0 (Yn− )2k+1 n−1 ) n−1 ) j + − 2k+1 + (Δ− n−1 ) Y1 (Yn )

(mod U(n− n+1 )mn ).

Hence by Lemma 5.1.1 we get

j − 2k+1 j−1 = 2j(n−1−2j)Y1− (Δ− (Yn− )2k+1 Y1+ , (Δ− n−1 ) (Yn ) n−1 )

j−1 j−1 + 4jY1− (Δ− (Yn− )2k+1 H0 − 4j(2k+1)Y1− (Δ− (Yn− )2k+1 n−1 ) n−1 ) + j − 2k+1 (mod U(n− + (Δ− n−1 ) Y1 , (Yn ) n+1 )(mn+1 )). + Now we use Lemma 5.1.3 in order to determine the commutator Y1 , (Yn− )2k+1 . We have

Y1+ , (Yn− )2N +1 = Y1+ , (Yn− )2N Yn− + (Yn− )2N Y1+ Yn− − (Yn− )2N +1 Y1+ = 2N (2N −1)Y1− (Yn− )2N −1 + 4N (Yn− )2N −1 M1n Yn− + (Yn− )2N Y1+ , Yn− = 2N (2N −1)Y1− (Yn− )2N −1 + 4N (Yn− )2N −1 Y1− + (4N +2)(Yn− )2N M1n .

Hence

j − 2k+1 j−1 = (2j(n−1−2j)−4j(2k+1))Y1− (Δ− (Yn− )2k+1 Y1+ , (Δ− n−1 ) (Yn ) n−1 ) j−1 + 4jY1− (Δ− (Yn− )2k+1 H0 n−1 ) j − − 2k−1 + (2k(2k−1)+4k) (Δ− n−1 ) Y1 (Yn )

(mod U(n− n+1 )mn+1 ),

128

Chapter 5. Intertwining Families

i.e.,

j − 2k+1 Y1+ , (Δ− n−1 ) (Yn )

j−1 = {2j(n−1−2j)−4j(2k+1)+4jλ}Y1− (Δ− (Yn− )2k+1 n−1 ) j − − 2k−1 + 2k(2k+1)(Δ− n−1 ) Y1 (Yn )

(mod U(n− n+1 )(mn+1 ⊕ C(H0 −λ))),

i.e., (5.1.20) remains true if we replace 2k on both sides by 2k+1. − j − 2N −2j+1 Now for D2N +1 = N ∈ U(n− n+1 ), it follows that j=0 bj (Δn−1 ) (Yn )

Y1+ , D2N +1 N " j−1 = bj 2j (n−1−2j −2(2N −2j +1)+2λ) Y1− (Δ− (Yn− )2N −2j+1 n−1 ) j=0

j − 2N −2j−1 + 2(N −j)(2N −2j +1)Y1− (Δ− n−1 ) (Yn )

#

up to an element in U(n− n+1 )(mn+1 ⊕ C(H0 −λ)). Hence, for i = 1, the condition in Theorem 5.1.3 is equivalent to the set of conditions {2r(n−1−2r) + 4rλ − 8r(N −r) − 4r} br + 2(N −r+1)(2N −2r+3)br−1 = 0 for r = 1, . . . , N . These conditions for the coeﬃcients are equivalent to those formulated in the theorem. As for D2N , it follows that the remaining conditions for i = 2, . . . , n − 1 do not imply additional conditions. The proof is complete. Corollary 5.1.3. For 0 ≤ j ≤ N −1, bj (λ) =

N −1 N! (−2)N −j q (2λ−4N +2k+n−1)bN (λ). j!(2N −2j +1)! k=j

− 0 Let D2N +1 (λ) ∈ U(nn+1 ) be the unique element satisfying (5.1.22) which is 0 normalized by the condition bN (λ) = 1, i.e., D2N +1 (λ) is given by 0 − 2N +1 − 2N −1 N − + b1 (λ)Δ− + · · ·+ (Δ− D2N +1 (λ) = b0 (λ)(Yn ) n−1 (Yn ) n−1 ) Yn , (5.1.23)

with the polynomial coeﬃcients bj (λ) =

N −1 N! (−2)N −j (2λ−4N +2k+n−1) j!(2N −2j +1)!

(5.1.24)

k=j

for j = 0, . . . , N − 1. Note that D10 (λ) = Yn− . Instead of bj (λ) we shall also write (N ) bj (λ) if it is appropriate to emphasize the order of the operator. The following theorem extends Theorem 5.1.2 to the odd order case.

5.1. The algebraic theory

129

Theorem 5.1.4. b0 (λ)x2N +1 − b1 (λ)x2N −1 ± · · · + (−1)N x = (N )

(N )

N! −λ− n−1 2 C (x). 2N +1 2 −λ− n−1 2 N +1

Proof. We recall the explicit formula λ C2N +1 (x) =

N (−1)j (λ)2N +1−j j=0

j!(2N +1−2j)!

2N +1−2j

(2x)

.

Now, by (5.1.24), we have (N ) bj (λ)

N −1 N! n−1 N −j N −j (−2) = 2 λ−2N +k+ , j!(2N +1−2j)! 2 k=j

i.e., (N ) bj (−λ)

n−1 N! n−1 2N −2j λ− = 2 +2N −j · · · λ− +N +1 j!(2N +1−2j)! 2 2 λ− n−1 N! 2 2N +1−j 2N −2j 2 . = j!(2N +1−2j)! λ− n−1 2 N +1

Hence b0 (λ)x2N +1 − b1 (λ)x2N −2 ± · · · + (−1)N x1 N λ− n−1 N! 1 2 2N +1−j 2N +1−2j j (2x) = (−1) 2 λ− n−1 j!(2N +1−2j)! 2 N +1 j=0 (N )

=

(N )

N! 1 λ− n−1 C2N +12 (x). n−1 2 (λ− 2 )N +1

The proof is complete.

5.1.3 DN (λ) as homomorphism of Verma modules We prove that both families D2N (λ) (Theorem 5.1.1) and D2N +1 (λ) (Theorem 5.1.3) are naturally interpreted as families of homomorphisms of generalized Verma 0 (λ) modules. We abuse notation by writing D2N (λ) for the normalized family D2N with polynomial coeﬃcients. The same convention applies to the odd order families. Let p m = mm ⊕ a ⊕ n+ m ⊂ gm be a parabolic subalgebra. pm is the Lie algebra of the minimal parabolic subgroup P m = M m A(N m )+ ⊂ Gm . For λ ∈ C, we consider the character ξλ : A at → eλt ∈ Aut(C).

130

Chapter 5. Intertwining Families

ξλ gives rise to a character of P m by ξλ (man+ ) = ξλ (a). Let ξλ also denote the corresponding character of pm . Let Iλ (gm ) ⊂ U(gm ) ⊗ C(λ) be the left U(gm )-ideal which is generated by the elements X ⊗ 1 − 1 ⊗ ξλ (X)1 ∈ U(gm ) ⊗ C(λ), X ∈ pm . We consider the left U(gm )-module Mλ (gm ) = (U(gm ) ⊗ C(λ))/Iλ (gm ). Mλ (gm ) is a generalized Verma module for gm . Theorem 5.1.5. Let DN (λ) ∈ U(n− n+1 ) be as in Theorem 5.1.1 (even N ) and Theorem 5.1.3 (odd N ). Then for all λ ∈ C the map U(gn ) ⊗ C(λ−N ) T ⊗ 1 → i(T )DN (λ) ⊗ 1 ∈ U(gn+1 ) ⊗ C(λ) induces a homomorphism Mλ−N (gn ) → Mλ (gn+1 ) of U(gn )-modules. Here i : gn → gn+1 is the canonical inclusion. Proof. It is enough to prove that DN (λ) induces a map Iλ−N (gn ) → Iλ (gn+1 ). For X = H0 ∈ pn , we have Iλ−N (gn ) (H0 ⊗ 1 − 1 ⊗ ξλ−N (H0 )1) = (H0 ⊗ 1 − 1 ⊗ (λ−N )) DN (λ)

−−−−→ H0 DN (λ) ⊗ 1 − DN (λ) ⊗ (λ−N ) = (DN (λ)H0 − N DN (λ)) ⊗ 1 − DN (λ) ⊗ (λ−N ) = DN (λ) (H0 ⊗ 1 − 1 ⊗ λ) = DN (λ) (H0 ⊗ 1 − 1 ⊗ ξλ (H0 )1) ∈ Iλ (gn+1 ) using [H0 , DN (λ)] = −N DN (λ). Next, for X ∈ mn , we obtain Iλ−N (gn ) (X ⊗ 1 − 1 ⊗ ξλ−N (X)1) = X ⊗ 1 DN (λ)

−−−−→ XDN (λ)⊗1 = DN (λ)X ⊗1 = DN (λ) (X ⊗ 1 − 1 ⊗ ξλ (X)1) ∈ Iλ (gn+1 ). Here we have used that [mn , DN (λ)] = 0 which is a consequence of the deﬁnition of DN (λ) and the commutator relations [X, Yn− ] = 0 and [X, Δ− n−1 ] = 0 for X ∈ mn . + , i.e., for X = Y , j = 1, . . . , n−1 we ﬁnd that DN (λ) maps Finally, for X ∈ n+ n j Iλ−N (gn ) (X ⊗ 1 − 1 ⊗ ξλ−N (X)1) = X ⊗ 1

5.2. Induced families

131

into XDN (λ) ⊗ 1 = DN (λ)X ⊗ 1 + RN (λ)X ⊗ 1 = DN (λ) (X ⊗ 1 − 1 ⊗ ξλ (X)1) + RN (λ)X ⊗ 1, where − RN (λ)X ⊗ 1 ∈ U(n− n+1 )(mn+1 ⊗ 1) ⊕ U(nn+1 )(H0 ⊗ 1 − 1 ⊗ ξλ (H0 )1) ∈ Iλ (gn+1 )

according to Theorem 5.1.1 and Theorem 5.1.3. We notice that there is no analog of the latter calculation for X = Yn+ . The proof is complete.

5.2 Induced families 5.2.1 Induction 0 In the present section, we use the polynomial families D2N (λ) ∈ U(n− n+1 ) to induce n G -equivariant families of diﬀerential operators. First of all, we have a model on induced representation spaces. Restriction of that picture to K and N − yields the respective compact and non-compact models. In addition, we consider restrictions to totally umbilic subspheres Σ with a conformal group isomorphic to Gn . For λ ∈ C, let ξλ be the (non-unitary) A-character ξλ (at ) = eλt acting on the space C(λ). Let m

∞ m P IndG P m (ξλ ) = C (G , C(λ))

m

= {u ∈ C ∞ (Gm ) | u(gman) = ξλ (a)u(g), man ∈ P m } . m

Gm operates by left translation on the space IndG P m (ξλ ). Note that the latter deﬁnition slightly diﬀers from the conventions in representation theory ([156]) in that the above space usually would be considered to be induced by the character ξλ−1 . The following result is a consequence of Theorem 5.1.1. 0 (λ) induces a polyTheorem 5.2.1. For any non-negative integer N , the family DN n+1 nomial family of G -equivariant diﬀerential operators n+1

n+1

G 0 (λ) : IndG DN P n+1 (ξλ ) → IndP n (ξλ−N ).

Hence the composition

(5.2.1)

0 (λ) DN (λ) = i∗ ◦ DN

0 of DN (λ) with the Gn -equivariant restriction map i∗ : C ∞ (Gn+1 ) → C ∞ (Gn ) deﬁnes a polynomial family of Gn -equivariant operators n+1

n

G DN (λ) : IndG P n+1 (ξλ ) → IndP n (ξλ−N ).

(5.2.2)

132

Chapter 5. Intertwining Families

Proof. We let U(g) operate from the right on C ∞ (G), i.e., the action of U(g) extends the representation R(X) : u(g) → (d/dt)|t=0 (u(g exp(tX))), X ∈ g n+1

0 ∞ of g. Now let DN (λ) operate on u ∈ IndG P n+1 (ξλ ) ⊂ C (G). We have to prove that 0 0 DN (λ)u(gmat n) = e(λ−N )t DN (λ)u(g)

(5.2.3)

− − for mat n ∈ P n = M n AN n , g ∈ Gn+1 . Now Ad(m)Δ− n−1 = Δn−1 and Ad(m)Yn = − n Yn for m ∈ M immediately imply 0 0 DN (λ)u(gm) = DN (λ)u(g), m ∈ M n . n+1

Next, we observe that for 1 ≤ j ≤ n − 1 and u ∈ IndG P n+1 (ξλ ), 0 0 0 Yj+ DN (λ)u(g) = Yj+ , DN (λ) u(g) + DN (λ)Yj+ u(g) + 0 = Yj , DN (λ) u(g) (by right N n -invariance of u) =0

(by Theorem 5.1.1 or Theorem 5.1.3).

0 0 Hence DN (λ)u(gn) = DN (λ)u(g), n ∈ N n . Finally, we note that 0 0 0 H0 DN (λ)u(g) = H0 , DN (λ) u(g) + DN (λ)H0 u(g) 0 0 (λ)u(g) + λDN (λ)u(g) = −N DN

=

(by Lemma 5.1.1)

0 (λ)u(g), (λ−N )DN

0 0 i.e., DN (λ)u(ga) = ξλ−N (a)DN (λ)u(g). The proof of (5.2.3) is complete. The remaining assertions are obvious.

Restriction from G = KAN + to K deﬁnes a Gm -isomorphism C ∞ (Gm , C(λ))P where Gm acts on C ∞ (K m )M λ

m

m

m

→ C ∞ (K m )M , λ

by

ϕ(k) → ϕ(κ(g −1 k))eλH(g

−1

k)

= ϕ(κ(g −1 k))e−λg·O,kM .

We compose this isomorphism with C ∞ (K m )M and obtain an isomorphism

m

C ∞ (K m /M m ) C ∞ (S m−1 )

m

∞ m−1 α : IndG )λ P m (ξλ ) → C (S

(5.2.4)

of Gm -modules. Hence DN (λ) induces a Gn -equivariant operator DN (S n , S n−1 ; λ) : C ∞ (S n )λ → C ∞ (S n−1 )λ−N .

(5.2.5)

5.2. Induced families

133

The renormalized operator c (λ) = 2−N DN (S n , S n−1 ; λ) DN

(5.2.6)

is called the compact model of DN (λ). The renormalization in (5.2.6) is introduced in order to recover the GJMS-operators for the round metric of S n as values of c (λ) at λ = − n2 +N (compare Lemma 5.2.2 with Lemma 5.2.8). the families D2N Similarly, restriction from G to N − ⊂ N − M AN + = N − P + ⊂ G induces a map m ∞ − ∞ m−1 ) (5.2.7) β : IndG P m (ξλ ) → C (N ) C (R and a non-compact model nc (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ) DN

(5.2.8)

of DN (λ). By Lemma 2.2.1, the Gm -action on Rm−1 which is induced by transport from S m−1 using κS coincides with the Gm -action which is induced by restriction to N − Rm−1 . The following result is obvious. Lemma 5.2.1. nc (λ) D2N

=

N

2N −2j

aj (λ)ΔjRn−1 i∗ (∂/∂xn )

,

j=0

where i : Rn−1 x → (x , 0) ∈ Rn , ΔRn−1 = satisfy (5.1.2) and aN (λ) = 1. Similarly, nc D2N +1 (λ)

=

N

n−1 j=1

∂ 2 /∂x2i , and the coeﬃcients

bj (λ)ΔjRn−1 i∗ (∂/∂xn )

2N +1−2j

,

j=0

where the coeﬃcients satisfy (5.1.22) and bN (λ) = 1. Now we establish an explicit relation between the induced picture on the one hand, and the compact and the non-compact picture on the other hand. m

2 m Lemma 5.2.2. Φλm ◦ κ∗S ◦ α = β on IndG P m (ξλ ), where Φm (x) = 1 + |x| , x ∈ R . + + Proof. Let n− be the Iwasawa decomposition of n− s = ks as ns ∈ KAN s . Then − m−1 ns N = ks N ∈ S and for u ∈ IndG P (ξλ ) we get − + β(u)(n− s ) = u(ns ) = u(ks as ns )

= ξλ (as )u(ks ) = ξλ (as )α(u)(ks N ) = ξλ (as )α(u)(n− s N ). Hence Lemma 2.2.1 gives ∗ β(u)(n− s ) = ξλ (as )α(u)(κS (s)) = ξλ (as ) (κS ◦ α) (u)(s),

134

Chapter 5. Intertwining Families

and it only remains to prove that ξλ (as ) = (1+|s|2 )λ , s ∈ Rm−1 .

(5.2.9)

In order to prove (5.2.9), we note that, by M -invariance, it is clearly enough to verify it in the case n = 2, i.e., G = SO(1, 2)◦ . Then the poles are N = (1, 0) and S = (−1, 0) in ∂B2 = S 1 ⊂ R2 C. The element ⎞ ⎛ s2 /2 s 1 + s2 /2 ⎝ −s2 /2 1 − s2 /2 −s⎠ , s ∈ R n− s = s s 1 ﬁxes S, and for its action on B2 we have −s2 2s −s − ∈ B2 . ns (O) = , = 4 + s2 4 + s2 s + 2i Now

−1

ξλ (as ) = e−λas

(O),1

.

−1 −1 (O) = (n+ (as )−1 (O), we get Since (n− s ) s ) −1 (O), 1 = (as )−1 (O), 1, (n− s )

i.e., − −1

ξλ (as ) = e−λ(ns ) Now

−1 (O) = n− (n− s ) −s (O) =

(O),1

.

s ∈ B2 −s + 2i

is on the circle |z −a| = (1−a)2 , where a = 1/s2 +2. It intersects the real axis in −1 < x = −s2 /s2 +2 ≤ 0. Now the hyperbolic distance of x and O is − log

1 + |x| = − log(1 + s2 ) 1 − |x|

(O is inside the horosphere through x with normal 1). Hence − −1

ξλ (as ) = e−λ(ns ) and the proof is complete.

(O),1

= (1 + s2 )λ

Lemma 5.2.2 and (2.3.17) show that β transports the induced representation m nc IndG m P (ξλ ) to the representation πλ . Moreover, Lemma 5.2.2 implies the relation λ−N c nc (λ) = DN (λ) ◦ Ψλn ◦ κ∗S , Ψm = Φm /2 (5.2.10) Ψn−1 ◦ κ∗S ◦ DN

5.2. Induced families

135

c nc between the models DN (λ) and DN (λ). In fact, using

β −1 ◦ (Φλm ◦ κ∗S ) ◦ α = id for m = n and m = n−1, we get ∗ −1 ◦ (Φλn−1 ◦ κ∗S ) ◦ α, β −1 ◦ (Φλ−N n−1 ◦ κS ) ◦ α ◦ DN (λ) = DN (λ) ◦ β

i.e., ∗ −1 ) = β ◦ DN (λ) ◦ β −1 ◦ (Φλn−1 ◦ κ∗S ). Φλ−N n−1 ◦ κS ◦ (α ◦ DN (λ) ◦ α

(5.2.10) then follows from the deﬁnitions c (λ) = α ◦ DN (λ) ◦ α−1 2 N DN

nc and DN (λ) = β ◦ DN (λ) ◦ β −1 .

(5.2.10) implies Lemma 5.2.3.

c nc κ∗S ◦ DN (λ) ◦ (κS )∗ = Ψ−λ+N ◦ DN (λ) ◦ Ψλn . n−1

(5.2.11)

In particular, the equivariance nc nc nc (g) ◦ DN (λ) = DN (λ) ◦ πλnc (g), g ∈ Gn πλ−N

of the non-compact model is a consequence of the equivariance of the compact model and (2.3.17). Finally, we induce operators from S n to totally umbilic hypersurfaces Σ as in Section 2.3. As before, let Gn ⊂ Gn+1 so that, if Gn+1 is considered as the conformal group of (S n , gc ), then Gn is the subgroup which leaves the subsphere S n−1 = S n ∩ {xn+1 = 0} invariant. We have seen above that the family DN (λ) ∈ n+1 U(n− -equivariant family n+1 ) induces the G 0 (λ) : C ∞ (Gn+1 , C(λ))P DN

n+1

→ C ∞ (Gn+1 , C(λ−N ))P

n

by diﬀerentiation from the right. Its composition with the restriction i∗ to Gn ⊂ Gn+1 then deﬁnes the left Gn -equivariant family DN (λ) (see Theorem 5.2.1). Using respective restrictions to K n+1 ⊂ Gn+1 and K n ⊂ Gn , it can be identiﬁed with the Gn -equivariant family DN (S n , S n−1 ; λ) : C ∞ (S n )λ → C ∞ (S n−1 )λ−N c and we set DN (S n ; λ) = 2−N DN (S n , S n−1 ; λ). Then c c c (λ) ◦ πλc (S n )(g) = πλ−N (S n−1 )(g) ◦ DN (λ), g ∈ Gn . DN

This is a consequence of the fact that the restriction to K m deﬁnes the Gm -module isomorphism α. Now the families DN (λ) (N even) also induces equivariant families DN (S n , Σ; λ) : C ∞ (S n ) → C ∞ (Σ)

136

Chapter 5. Intertwining Families

for all totally umbilic hypersurfaces Σ of S n . We describe the corresponding induction mechanism and the equivariance of such families. We write Σ = σ(S n−1 ) ⊂ S n for some σ ∈ Gn+1 . The group σGn σ −1 leaves Σ invariant. We consider the submanifold Gn [σ] = σGn ⊂ Gn+1 . The left-action of σGn σ −1 on Gn+1 restricts to a left-action of this group on the manifold Gn [σ]: Lσg0 σ−1 : Gn [σ] → Gn [σ], σg → σg0 g. It induces representations Lσg0 σ−1 : C ∞ (Gn [σ]) → C ∞ (Gn [σ]), u(σg) → u(σg0−1 g). 0 Then the composition of DN (λ) with the restriction i∗σ to Gn [σ] is left σGn σ −1 equivariant. The map n

βσ : C ∞ (Gn [σ], C(λ))P → C ∞ (σ(S n−1 )) = C ∞ (Σ) u(σg) → (σ(kM ) → u(σk)) is an isomorphism. Hence we can use the commutative diagram C ∞ (Gn+1 , C(λ))P ⏐ ⏐α : C ∞ (S n )

i∗ ◦D0 (λ)

n+1

σ −− −−N −−→ C ∞ (Gn [σ], C(λ−N ))P ⏐ ⏐α : σ

DN (S n ,Σ;λ)

−−−−−−−−→

n

C ∞ (Σ)

to deﬁne the family DN (S n , Σ; λ). The following result describes its equivariance. Lemma 5.2.4. πλ−N (Σ, gc )(g) ◦ DN (S n , Σ; λ) = DN (S n , Σ; λ) ◦ πλ (S n , gc )(g) for all g ∈ σGn σ −1 ⊂ Gn+1 , i.e., for all g ∈ Gn+1 which leave Σ invariant. Proof. We recall that πλ (Σ, gc )(g) =

g∗ (vol(Σ, gc )) vol(Σ, gc )

λ − n−1

◦ g∗ ,

on functions on Σ (see (2.3.18)). Here g ∈ Gn+1 preserves Σ and vol(Σ, gc ) is the Riemannian volume form on Σ for the restriction of the round metric. It suﬃces to prove that the maps α and ασ satisfy the intertwining identities α ◦ Lg0 = πλ (S n , gc )(g0 ) ◦ α and ασ ◦ Lσg0 σ−1 = πλ (Σ, gc )(σg0 σ −1 ) ◦ ασ , g0 ∈ Gn .

5.2. Induced families

137

The equivariance of α has been established above. We prove the second assertion by identifying πλ (Σ, gc )(σg0 σ −1 ) as the bottom line in the commutative diagram C ∞ (Gn [σ], C(λ))P ⏐ ⏐α : σ C ∞ (Σ)

n

Lσgσ−1

−−−−−→ C ∞ (Gn [σ], C(λ))P ⏐ ⏐α : σ

n

(5.2.12)

C ∞ (Σ).

−−−−→

n

Let u ∈ C ∞ (Gn [σ], C(λ))P . Then ασ (u) is the function Σ σ(kM ) → u(σk) = u(κ(σk))eλH(σk) . On the other hand, we get Lσgσ−1 (u)(σg) = u(σg0−1 g). Hence the composition ασ Lσgσ−1 (u) is the function −1

σ(kM ) → u(σg0−1 k) = u(κ(σg0−1 k))eλH(σg0

k)

.

It follows that the bottom line in (5.2.12) is given by the operator −1

eλH(σg0

k)−λH(σk)

(σg0 σ −1 )∗ = e−λg0 σ

−1

= e−λσg0 σ

·O,b+λσ−1 ·O,b

−1

·O,σ(b)

(σg0 σ −1 )∗

(σg0 σ −1 )∗

−1 = Φ−λ )∗ , b = kM, σg0 σ−1 (σ(b))(σg0 σ

where Φg (b) = eg·O,b . Here we have used the identity g2 g1 · O, b = g2 · O, b + g1 · O, g2−1 (b)

(5.2.13)

and its consequence g · O, b = −g −1 · O, g −1 (b). (5.2.13) is equivalent to the cocycle identity Φg2 g1 = Φg2 · (g2 )∗ (Φg1 ). (5.2.14) Now it only remains to interpret the exponential term as

g∗ (vol(Σ, gc )) vol(Σ, gc )

λ − n−1

at σ(b) for g = σg0 σ −1 , i.e., we have to prove (σgσ −1 )∗ (vol(Σ, gc )) = Φn−1 σg0 σ−1 (σ(b)). vol(Σ, gc ) We ﬁrst note that σ ∗ (gc |Σ ) = σ ∗ (gc )|S n−1 = (Φ2σ−1 gc )|S n−1 . Hence gc |Σ = σ∗ (Φ2σ−1 |S n−1 )σ∗ (gc |S n−1 ) = (Φ−2 σ )|S n−1 σ∗ (gc |S n−1 ),

(5.2.15)

138

Chapter 5. Intertwining Families

and therefore )|S n−1 . vol(Σ, gc ) = σ∗ (vol(S n−1 , gc ))/(Φn−1 σ Now we calculate (σg0 σ −1 )∗ (vol(Σ, gc )) = σ∗ (g0 )∗ σ ∗ (vol(Σ, gc )) = σ∗ (g0 )∗ vol(S n−1 , gc )Φn−1 σ−1 n−1 = σ∗ Φn−1 , gc )(g0 )∗ (Φn−1 g0 vol(S σ−1 ) n−1 = σ∗ (Φn−1 , gc ))(σg0 )∗ (Φn−1 g0 )σ∗ (vol(S σ−1 ) n−1 n−1 = σ∗ (Φn−1 vol(Σ, gc ) g0 )(σg0 )∗ (Φσ−1 )Φσ n−1 n−1 Φσg0 σ−1 Φσg0 Φn−1 vol(Σ, gc ) = σ Φσ Φσg0

= Φn−1 σg0 σ−1 vol(Σ, gc ) using (5.2.14). The proof is complete.

The deﬁnitions immediately imply that the families DN (S n , S n−1 ; λ) and DN (S n , Σ; λ) are conjugate. More precisely, we have Lemma 5.2.5. Let Σ = σ(S n−1 ). Then

σ∗ (vol(S n−1 , gc )) vol(Σ, gc )

− λ−N n−1

◦ σ∗ ◦ DN (S n , S n−1 ; λ) = DN (S n , Σ; λ) ◦ πλ (S n )(σ). (5.2.16)

We omit the proof. Lemma 5.2.5 can be used to derive the equivariance of the family DN (S n , Σ; λ) (see Lemma 5.2.4) from the equivariance of the family DN (S n , S n−1 ; λ). Lemma 5.2.5 can also be used as an equivalent alternative definition of DN (S n , Σ; λ). In fact, DN (S n , Σ; λ) is well deﬁned by (5.2.16) since by the Gn -equivariance of the family DN (S n , S n−1 ; λ) (see (5.2.5)) it only depends on the class of σ in Gn+1 /Gn . As noticed above, using respective restrictions to (N n+1 )− ⊂ Gn+1 and n − (N ) ⊂ Gn , the Gn -equivariant family DN (λ) induces a Gn -equivariant family nc (λ) = DN (Rn , Rn−1 ; λ) : C ∞ (Rn ) → C ∞ (Rn−1 ), DN i.e., we have nc nc nc (λ) ◦ πλnc (Rn )(g) = πλ−N (Rn−1 )(g) ◦ DN (λ), g ∈ Gn . DN

Moreover, for a (generalized) sphere Σ = σ(Rn−1 ) ⊂ Rn , σ ∈ Gn+1 we can use DN (λ) to induce an equivariant family DN (Rn , Σ; λ) : C ∞ (Rn ) → C ∞ (Σ). We

5.2. Induced families

139

have no reason to describe that construction in more detail for general Σ. However, for later purposes, we formulate the basic properties of such families in the special case Σ = S n−1 = ∂Bn ⊂ Rn . Lemma 5.2.6. πλ−N (S n−1 , gc )(g) ◦ DN (Rn , S n−1 ; λ) = DN (Rn , S n−1 ; λ) ◦ πλ (Rn , gc )(g) for all g ∈ Gn+1 which leave S n−1 ⊂ Rn invariant. Of course, in Lemma 5.2.6 we use on the left-hand side the action on S n−1 ⊂ R which is induced by that on Rn . The families DN (Rn , Rn−1 ; λ) and DN (Rn , S n−1 ; λ) are conjugate. More precisely, n

Lemma 5.2.7. Let S n−1 = ∂Bn ⊂ Rn and let σ0 : Rn−1 → S n−1 be as in (2.2.9). Then

(σ0 )∗ (vol(Rn−1 , gc )) vol(S n−1 , gc )

− λ−N n−1

◦ (σ0 )∗ ◦ DN (Rn , Rn−1 ; λ) = DN (Rn , S n−1 ; λ) ◦ πλ (Rn , gc )(σ0 ).

In Section 5.4, the last result will be used to ﬁnd explicit formulas for the induced families DN (Rn , S n−1 ; λ) for N ≤ 3. In particular, it will be shown that the Chang-Qing boundary operator P3 on B4 is an induced one. nc c 5.2.2 Even order families: D2N (λ) and D2N (λ)

The main results of the present section are the factorization identities for the compact and the non-compact models of DN (λ). nc (− n2 +j) and Theorem 5.2.2 (Factorization). For j = 1, . . . , N , the operators D2N n−1 nc D2N (2N −j − 2 ) factorize as maps

C ∞ (Rn )− n2 +j → C ∞ (Rn )− n2 −j → C ∞ (Rn−1 )− n2 +j−2N , C ∞ (Rn )− n−1 +2N −j → C ∞ (Rn−1 )− n−1 +j → C ∞ (Rn−1 )− n−1 −j 2

2

2

in the form

n

n

nc nc n − +j = D2N D2N −2j − −j ◦ P2j (R ), 2 2 n−1 n−1 nc nc D2N 2N −j − 2N −j − = P2j (Rn−1 ) ◦ D2N . −2j 2 2

(5.2.17) (5.2.18)

We note that for j = N , the factorizations (5.2.17) and (5.2.18) state that n

n−1 nc nc − +N = i∗ ◦ (ΔRn )N and D2N +N = (ΔRn−1 )N ◦ i∗ . − D2N 2 2

140

Chapter 5. Intertwining Families

Proof. It is enough to prove the factorizations n

n

0 0 − j D2N − +j = D2N −2j − −j (Δn ) , 2 2 n−1 n−1 − 0 j 0 D2N 2N −j − = (Δn−1 ) D2N −2j 2N −j − 2 2 in U(n− n+1 ). For both sets of identities we use diﬀerent types of arguments. We begin with the proof of the ﬁrst set of identities. We ﬁrst notice that the righthand side is of the form j N −j j − r − 2N −2j−2r s − 2(j−s) ar (Δn−1 ) (Yn ) (Δ− n−1 ) (Yn ) s r=0

=

s=0

N −j

j

r=0 s=0

=

N

j r+s ar (Yn− )2N −2r−2s (Δ− n−1 ) s

k − 2N −2k sk (Δ− n−1 ) (Yn )

k=0 N and the coeﬃcient of (Δ− is equal to 1. In order to apply the uniqueness n−1 ) consequence of Theorem 5.1.1, it is enough to verify that the product n

− 0 − j T(j,N ) = D2N −2j − −j (Δn ) ∈ U(nn+1 ) 2 n 0 satisﬁes the same commutator relations as D2N − 2 +j , i.e.,

n

+ −j . Yi , T(j,N ) ∈ U(n− n+1 ) mn+1 ⊕ C H0 + 2

(5.2.19)

Now we write n n + 0 j 0 j (Δ− Yi+ , (Δ− Yi , T(j,N ) = Yi+ , D2N −2j − −j n ) + D2N −2j − −j n) 2 2 n

j +j (Δ− ) m ⊕ C H + ∈ U(n− n+1 0 n) n+1 2 n

+ U(n− −j n+1 ) mn+1 ⊕ C H0 + 2 n j 0 n+1 leaves Δ− using (Δ− n ) = D2j (− 2 + j) (Corollary 5.1.2). But M n invariant and − j − j H0 , (Δn ) = −2j(Δn ) (Lemma 5.1.1). Hence

n

j +j (Δ− U(n− n) n+1 ) mn+1 ⊕ C H0 + 2 n

−j . ⊆ U(n− ) m ⊕ C H + n+1 0 n+1 2

5.2. Induced families

141

0 This proves (5.2.19), i.e., T(j,N ) and D2N (− n2 + j) satisfy the same commutator relations and are normalized both by the condition that the coeﬃcient of Δ− n−1 is 1. Theorem 5.1.1 implies that these two elements coincide. A proof of the second set of identities along the above lines does not work 0 (2N−j−n−1 since M n+1 does not leave D2N 2 ) invariant. However, here the assertion follows from Corollary 5.1.1. In view of

0 D2N

N n−1 n−1 ) r − 2N −2r a(N 2N −j − 2N −j − = (Δ− r n−1 ) (Yn ) 2 2 r=0

and n−1 j 0 (Δ− ) D 2N −j − 2N −2j n−1 2 N −j n−1 r+j ar(N −j) 2N −j − (Yn− )2N −2j−2r = (Δ− n−1 ) 2 r=0 N n−1 (N −j) r − 2N −2r = ar−j , 2N −j − (Δ− n−1 ) (Yn ) 2 r=j the assertion is equivalent to n−1 ) a(N 2N −j − =0 r 2 and ) a(N r

for r = 0, . . . , j − 1

n−1 n−1 (N −j) 2N −j − 2N −j − = ar−j 2 2

for r = j, . . . , N.

(5.2.20)

(5.2.21)

Now Corollary 5.1.1 gives N −1 n−1 N! ) N −r (−2) (−2j + 2k + 2) 2N −j − = a(N r 2 r!(2N −2r)!

(5.2.22)

k=r

and thus (5.2.20) is obvious. For the proof of (5.2.21), we have to compare (5.2.22) with N −j−1 n−1 (N −j)! (N −j) (−2)N −r (2j +2k +2) (5.2.23) 2N −j − ar−j = 2 (r−j)!(2N −2r)! k=r−j

(Corollary 5.1.1). But (5.2.22) simpliﬁes to (N −j)! N! (−2)N −r 2N −r , r!(2N −2r)! (r−j)!

142

Chapter 5. Intertwining Families

whereas (5.2.23) coincides with N! (N −j)! (−2)N −r 2N −r . (r−j)!(2N −2r)! r! The proof is complete. Notice that the direct type of arguments used in the proof of the second set of identities do not give an analogous proof of the ﬁrst set since these lead to a set of non-trivial identities for binomial coeﬃcients. Now we prove analogous factorizations for the compact models. c Theorem 5.2.3 (Factorization). For j = 1, . . . , N , the operators D2N (− n2 +j) and n−1 c D2N (2N −j − 2 ) factorize as equivariant maps

C ∞ (S n )− n2 +j → C ∞ (S n )− n2 −j → C ∞ (S n−1 )− n2 +j−2N , C ∞ (S n )− n−1 +2N −j → C ∞ (S n−1 )− n−1 +j → C ∞ (S n−1 )− n−1 −j 2

2

2

in the form n

n

c c n − +j = D2N D2N −2j − −j ◦ P2j (S ), 2 2 n−1 n−1 c c D2N 2N −j − 2N −j − = P2j (S n−1 ) ◦ D2N . −2j 2 2

(5.2.24) (5.2.25)

Proof. We prove the ﬁrst set of identities. The identity n

n

nc nc n − +j = D2N D2N −2j − −j ◦ P2j (R , gc ) 2 2 (see (5.2.17)) and Lemma 5.2.3 imply n

c − +j ◦ (κS )∗ κ∗S ◦ D2N 2 n

n − n +j nc 2 −j−2N − +j ◦ Ψn 2 = Ψn−1 ◦ D2N 2 n

n

n −n −n nc n 2 −j−2N 2 −j 2 +j 2 +j − ◦ Ψ −j ◦ Ψ = Ψn−1 ◦ D2N ◦ P (R , g ) ◦ Ψ n n n 2j c −2j n 2

c n = κ∗S ◦ D2N −2j − −j ◦ (κS )∗ ◦ P2j (R , gs ) (by Lemma 5.2.8) 2 n

c n = κ∗S ◦ D2N −2j − −j ◦ ((κS )∗ ◦ P2j (R , gs )) 2 n

c n = κ∗S ◦ D2N −2j − −j ◦ P2j (S , gc ) ◦ (κS )∗ (by (2.2.1)). 2 Now Lemma 5.2.3 implies the ﬁrst set of identities. The proof of the second set is similar.

5.2. Induced families

143

Note that for eϕ(x) = 2/(1+|x|2), the identity (3.1.1) implies Lemma 5.2.8. N + n2 N − n2 1+|x|2 1+|x|2 ◦ P2N (Rn , gc ) ◦ = P2N (Rn , gs ), 2 2 where gs is given by (2.2.2). Now we use Theorem 5.2.3 to deﬁne a recursive method which can be used to c (λ) : C ∞ (S n ) → C ∞ (S n−1 ) in terms of P2M (S N ) deduce explicit formulas for D2N n−1 c and P2M (S ) for 1 ≤ M ≤ N − 1. For that purpose, we prove that λ → D2N (λ) is a polynomial operator family of degree 2N with a scalar leading coeﬃcient, i.e., c D2N (λ) = A2N i∗ λ2N + A2N −1 λ2N −1 + · · · + A1 λ + A0

(5.2.26)

for some A2N ∈ C. Then the 2N unknown operator-valued coeﬃcients Aj , j = 0, . . . , 2N − 1 can be determined by solving the system n

n

c c n − +j = D2N D2N −2j − −j ◦ P2j (S ), 2 2 n−1 n−1 c c D2N 2N −j − 2N −j − = P2j (S n−1 ) ◦ D2N −2j 2 2 c (λ), 0 ≤ M ≤ N −1. Note that the for j = 1, . . . , N using the known families D2M determinant of the system is the Vandermonde determinant ⎞ ⎛ 2N −1 ... ε01 ε1 ⎜ .. ⎟ = (ε − ε ) = 0 det ⎝ ... j i . ⎠ −1 j>i 0 ε2N . . . ε 2N 2N

for εj = − n2 + j and εj+N = 2N − j − method is well deﬁned.

n−1 2

for j = 1, . . . , N . Hence the recursive

c Lemma 5.2.9. λ → D2N (λ) is a polynomial of degree 2N . The leading coeﬃcient is scalar. More precisely, A2N = (−1)N .

Proof. The proof rests on the formula c nc κ∗S ◦ D2N (λ) ◦ (κS )∗ = Ψ−λ+2N ◦ D2N (λ) ◦ Ψλn n−1

(Lemma 5.2.3) and the following result. Lemma 5.2.10. Let a, b ∈ N0 . Then −λ+2(a+b)

λ → Ψn−1

◦ ΔbRn−1 ◦ i∗ ◦

∂2 ∂x2n

a ◦ Ψλn

(5.2.27)

144

Chapter 5. Intertwining Families

is an operator-valued polynomial of degree 2b + a with leading term 2−2a

(2a)! 2b |x | (1+|x |2 )a λ2b+a i∗ . a!

Proof. We write the operator polynomial as the composition

−λ+2(a+b) Ψn−1

◦

ΔbRn−1

◦

Ψλ−2a n−1

2 a

∂ −λ+2a ∗ λ ◦ Ψn . ◦ Ψn−1 ◦ i ◦ ∂x2n

(5.2.28)

Now the second factor ◦ i∗ ◦ λ → Ψ−λ+2a n−1

∂2 ∂x2n

a ◦ Ψλn

is a polynomial of degree a the leading coeﬃcient of which is the operator 2−2a

(2a)! (2a)! a (1+|x |2 )a i∗ = 2−a Ψn−1 (x )i∗ . a! a!

In fact, this follows from c−λ+2a

2

d dx2

a

1 1

(c + x2 )λ = c2a x=0

2

d dx2

a

1 1

⎛ x=0

⎝ 1+

x √ c

2 λ

⎞ ⎠

a λ x 2n 1 d2 1 √ x=0 n dx2 c n≥0 λ = ca (2a)! . a

= c2a

−λ+2(a+b)

Now it is easy to see that the leading term of the polynomial λ → Ψn−1 ΔbRn−1 ◦ Ψλ−a n−1 is the operator

◦

|x |2b Ψan−1 (x )λ2b . It follows that the composition (5.2.28) is a polynomial of degree 2b + a with a leading term as claimed. Now Lemma 5.2.10 and the deﬁnition nc D2N (λ) =

N

(N )

aj

j=0 (N )

with polynomial coeﬃcients aj

(λ)ΔjRn−1 ◦ i∗ ◦

∂2 ∂x2n

N −j

(λ) of degree N −j (see (5.1.4)) imply that the

5.2. Induced families

145

c leading term of D2N (λ) is given by the composition of the sum N j=0

2

2N −2j

N −j

(−1)

N! j!(2N −2j)!

(2N −2j)! 2j |x | (1+|x |2 )N −j λN −j λ2j+N −j (N −j)! N N = (−1)N −j |x |2j (1+|x |2 )N −j λ2N j

× 2−2(N −j)

j=0

= (|x |2 − (1+|x |2 ))N λ2N = (−1)N λ2N with i∗ . This completes the proof of Lemma 5.2.9.

We use these results for the c (λ) is of the form Proof of Theorem 1.4.1. In fact, Lemma 5.2.9 shows that D2N

(−1)N λ2N i∗ + A2N −1 λ2N −1 + · · · + A1 λ + A0 . By the above recursive method, the 2N coeﬃcients A2N −1 , . . . , A0 are determined by 2N factorization identities. More precisely, they are given as compositions of the operators P2 (S m ), . . . , P2N (S m ) for m = n and m = n − 1 and the values n

n−1 c c 2N −j − − D2N −j , D −2j 2N −2j 2 2 for j = 1, . . . , N of families of lower order. Now the factorization identities applied to the latter families imply formulas for these in terms of P2 (S m ), . . . , P2N −2 (S m ) c for m = n and m = n − 1 and certain values of the families D≤2N −4 (λ). A continuation of the procedure proves the assertion in Theorem 1.4.1 concerning the structure of the coeﬃcients Aj . c (λ). The behaviour of the normal We determine the normal order of D2N nc (λ). The explicit formula order can be read oﬀ from the non-compact model D2N 0 nc for D2N (λ) (see (5.1.3)) shows that the normal order of D2N (λ) degenerates, i.e., is smaller than the order 2N of the family, iﬀ N −1

(2λ−4N +2k+n+1) = 0,

k=0

i.e., iﬀ λ∈

n−1 n−1 − +2N −1, . . . , − +N . 2 2

More precisely, for λ = − n−1 2 + N + M with 0 ≤ M ≤ N − 1, precisely the coeﬃcients a0 (λ), . . . , aN −M−1 (λ) vanish, i.e., the normal order is 2N −2(N −M ) =

146

Chapter 5. Intertwining Families

2M . In the most extreme case M = 0, the normal order is 0, i.e., for λ = − n−1 2 +N the family is tangential. In fact, this operator coincides with P2N (S n−1 )i∗ . This completes the proof of Theorem 1.4.1/(i). c The SO(n)-equivariance of D2N (λ) : C ∞ (S n ) → C ∞ (S n−1 ) implies that the c n−1 , i.e., function D2N (λ)(1) is constant on S c (λ)(1) Q2N (λ) = D2N

(5.2.29)

for a polynomial Q2N (λ) of degree 2N . We determine an explicit formula for this polynomial. It is a special case of the tractor Q-polynomial which generalizes Qcurvature (Deﬁnition 6.21.5). The identity (5.2.27) implies c nc (λ)(1) = Ψ−λ+2N D2N (λ)(Ψλn ). Q2N (λ) = D2N n−1

(5.2.30)

We use this identity to prove that Q2N (2N ) = (−1)N (2N )!.

(5.2.31)

In order to verify (5.2.31), it is enough to determine the value of nc (2N )(Ψ2N D2N n )(x ) at x = 0

(since it does not depend on x ), i.e., to calculate the sum 2

−2N

N

(N ) aj (2N )ΔjRn−1

j=0

∂2 ∂x2n

N −j

(1+|x |2 +x2n )2N

at (x , xn ) = 0. We obtain

2

−2N

N

(N ) aj (2N )

2N 2N r=0

j=0

r

1 × ΔjRn−1 |x |2r 1x =0 But

∂2 ∂x2n

N −j

1 (1+x2n )2N −2r 1xn =0 . (5.2.32)

1 # ΔkRn−1 |x |2r 1x =0 = δrk r!2r {(n−3+2r) · · · (n−1)}

(see (5.1.14)) and

∂2 ∂x2n

k

1 (1+x2n )M 1xn =0 =

M (2k)!. k

5.2. Induced families

147

Hence (5.2.32) reads N

(N ) aj (2N )

j=0

=

N

(N )

aj

2N 2N −j # j−2N 2 j! (2N −2j)! {(n−3+2j) · · · (n−1)} j N −j

(2N )2j−2N

j=0

(2N )!(2N −2j)! # {(n−3+2j) · · · (n−1)} . N !(N −j)!

Now using Corollary 5.1.1, it follows that the last expression coincides with (−1)N 2−N (2N )!

N j=0

(−1)j {(n−1+2N ) · · · (n−1)}# . j!(N −j)! n−1 +2j

Therefore, for the proof of (5.2.31) it only remains to verify that the latter sum is equal to 2N . However, this follows from the identity N j=0

(−1)j {x · · · (x+2N )}# = 2N . j!(N −j)! x+2j

(5.2.33)

In order to prove (5.2.33), we notice that the left-hand side is a polynomial qN (x) of degree N in x. But qN (−2r) =

(−1)r # # {(−2r) · · · (−2)} {2 · · · (−2r+2N )} = 2N r!(N −r)!

for 0 ≤ r ≤ N . Hence qN (x) = 2N and the proof of (5.2.31) is complete. In addition to the identity (5.2.31) we have a series of recursive relations for the polynomials Q2N (λ). These follow from the identities (5.2.24) and (5.2.25). We get n

n

Q2N − +j = Q2N −2j − −j P2j (S n )(1), (5.2.34) 2 2 n−1 n−1 Q2N 2N −j − = P2j (S n−1 )(1)Q2N −2j 2N −j − , (5.2.35) 2 2 for j = 1, . . . , N . Note that n 2 +j−1

n

P2j (S )(1) =

(−r)(n − 1 − r)

(for all n).

(5.2.36)

r= n 2

The following lemma gives the unique polynomials which satisfy (5.2.31), (5.2.34) and (5.2.35). Lemma 5.2.11. Q2N (λ) = (−1)N λ(λ−1) . . . (λ−2N +1).

148

Chapter 5. Intertwining Families

Proof. (5.2.31) is obviously satisﬁed. In order to prove that the given polynomial satisﬁes the identities (5.2.34) and (5.2.35) for j = 1, . . . , N , we use induction over N . Moreover, we use the formula n 2 +N −1

n

P2N (S )(1) =

r= n 2

n

n

(−r(n−1−r)) = (−1)N − −N +1 · · · − +N . 2 2

Assuming that Q2M (λ) = (−1)M (λ− 1) · · · (λ− 2M + 1) for 1 ≤ M ≤ N − 1, we obtain for the right-hand side of (5.2.34)

n

n n (−1)N −j − −j − −j −1 · · · − +j −2N +1 2 n2 2 n n n

j × (−1) − +j · · · − −j +1 = (−1)N − +j · · · − +j −2N +1 2 2 2 2 n

= Q2N − +j . 2

This proves (5.2.34). The proof of (5.2.35) is analogous. nc c 5.2.3 Odd order families: D2N +1 (λ) and D2N +1 (λ)

In the present section, we extend the results of Section 5.2.2 to the odd order nc c 0 families D2N +1 (λ) and D2N +1 (λ) which are induced by the families D2N +1 (λ). n c Theorem 5.2.4 (Factorization). For j = 1, . . . , N , the operators D2N +1 (− 2 + j) n−1 c and D2N +1 (2N +1−j − 2 ) factorize as equivariant maps

C ∞ (S n )− n2 +j → C ∞ (S n )− n2 −j → C ∞ (S n−1 )− n2 +j−2N −1 , C ∞ (S n )− n−1 +2N +1−j → C ∞ (S n−1 )− n−1 +j → C ∞ (S n−1 )− n−1 −j 2

2

2

in the form n

n

c c n − +j = D D2N +1 2N −2j+1 − −j ◦ P2j (S ), 2 2 n−1 n−1 c n−1 c D2N +1 2N +1−j − ) ◦ D2N +1−2j 2N +1−j − = P2j (S . 2 2 nc There are analogous factorizations for the non-compact models D2N +1 (λ). nc Proof. For the proof of the factorization identities for D2N +1 (λ), it suﬃces to prove the 2N identities n

n

0 0 − j (5.2.37) D2N +1 − +j = D2N +1−2j − −j (Δn ) , 2 2 n−1 n−1 0 j 0 D2N = (Δ− (5.2.38) +1 2N +1−j − n−1 ) D2N +1−2j 2N +1−j − 2 2

5.2. Induced families

149

in U(n− n+1 ). For the proof of the ﬁrst set of identities we note that the right-hand sides of (5.2.37) is of the form N −j N − r − 2N +1−2j−2r − 2 s − 2N +1−2s br (Δn−1 ) (Yn ) +(Y ) ) = ds (Δ− (Δ− n n−1 n−1 ) (Yn ) r=0

s=0

n N 0 and that the coeﬃcient of (Δ− in the product is 1. D2N n−1 ) +1 (− 2 +j) has the same properties. Moreover, the product on the right-hand side of (5.2.37) satisﬁes n 0 the same commutator relations as D2N +1 (− 2 +j). This can be seen as in the proof of Theorem 5.2.3; we recall that the main point of the argument is that mn+1 commutes with Δ− n . Thus the asserted identity follows from Theorem 5.1.3. For the proof of (5.2.38) we have to compare N n−1 (N ) r − 2N +1−2r br 2N +1−j − (Δ− n−1 ) (Yn ) 2 r=1

with

N −j

br(N −j)

r=1

n−1 r+j (Yn− )2N +1−2j−2r . 2N +1−j − (Δ− n−1 ) 2

In view of Corollary 5.1.3, that means to compare N r=0

with

N −j r=0

N −1 N! (−2)N −r (2k−2j +2) r!(2N −2r+1)!

(5.2.39)

k=r

N −j−1 (N −j)! (−2)N −j−r (2k+2j +2). r!(2N −2j −2r+1)!

(5.2.40)

k=r

Now (5.2.39) coincides with N −j r=0

N −1−j N! N −r−j (−2) (2k+2). (r+j)!(2N −2j −2r+1)!

But since (N −j)! r!

k=r

N −j−1 k=r

(2k+2j +2) =

N! (r+j)!

N −1−j

(2k+2),

k=r

the proof of (5.2.38) is complete. The assertions for the compact model follow as in the proof of Theorem 5.2.3. As for the even order operator families, we now deﬁne a recursive method c c which allows us to determine the family D2N +1 (λ) by using the families D2M+1 (λ) for 0 ≤ M ≤ N − 1. The method rests on the 2N factorization identities for the c family D2N +1 (λ) (Theorem 5.2.4) and the following result (compare with (5.2.26) and Lemma 5.2.9).

150

Chapter 5. Intertwining Families

c Lemma 5.2.12. The operator families D2N +1 (λ) are polynomials of degree 2N of the form (5.2.41) B2N λ2N + B2N −1 λ2N −1 + · · · + B1 λ + B0

with B2N = (−1)N D1c = (−1)N i∗ grad(H0 ). Proof. The proof rests on the identity −λ+2N +1 c nc λ ◦ D2N D2N +1 (λ) = Ψn−1 +1 (λ) ◦ Ψn

(5.2.42)

(Lemma 5.2.3) and the following result. Lemma 5.2.13. For a, b ∈ N0 , λ →

−λ+2(a+b)+1 Ψn−1

◦

ΔbRn−1

∗

◦i ◦

∂ ∂xn

2a+1 ◦ Ψλn

is an operator-valued polynomial of degree 2b+a with leading term −2a−1 (2a+1)! 2b 2 a+1 ∗ ∂ |x | (1+|x | ) 2 i λ2b+a . a! ∂xn Proof. We write the polynomial as the composition 2a+1

∂ −λ+2(a+b)+1 λ−2a −λ+2a b ∗ λ Ψn−1 ◦ ΔRn−1 ◦ Ψn−1 ◦ Ψn−1 ◦ i ◦ ◦ Ψn . ∂xn Similarly as in the proof of Lemma 5.2.10, it can be shown that the second factor is a polynomial with leading term −2a (2a+1)! 2 a ∗ ∂ (1+|x | ) i 2 λa . a! ∂xn

From here the assertion follows easily. Now Lemma 5.2.13 and the deﬁnition nc D2N +1 (λ) =

N

(N )

bj

(λ)ΔjRn−1 ◦ i∗ ◦

j=0

∂ ∂xn

2N +1−2j

(N )

with polynomial coeﬃcients bj (λ) of degree N −j (see (5.1.24)) imply that the c leading term of D2N +1 (λ) is given by the composition of the sum N j=0

22N −2j (−1)N −j

N! j!(2N −2j +1)!

× 2−2(N −j)−1

(2N −2j +1)! 2j |x | (1+|x |2 )N −j+1 λN −j λ2j+(N −j) (N −j)!

5.2. Induced families

151

with i∗ ∂/∂xn . Since the latter sum simpliﬁes to N 1 1 N (−1)N −j |x |2j (1+|x |2 )N −j+1 λ2N = (−1)N (1+|x |2 )λ2N , j 2 j=0 2 it is enough to notice that i∗ grad(H0 ) =

1 ∂ (1+|x |2 ) |x =0 2 ∂xn n

(see (2.2.4)) to complete the proof of Lemma 5.2.12.

We complete the Proof of Theorem 1.4.1. The proof of Theorem 1.4.1/(ii) is analogous to that of c part (i) on page 145. Lemma 5.2.12 shows that D2N +1 (λ) is of the form (−1)N i∗ ∇N λ2N + B2N −1 λ2N −1 + · · · + B1 λ + B0 . By the recursive method, the 2N coeﬃcients B2N −1 , . . . , B0 are determined by 2N factorization identities (Theorem 5.2.4). More precisely, these are given as compositions of the operators P2 (S m ), . . . , P2N (S m ) for m = n and m = n − 1 and the values n

n−1 c c D2N −j , D − 2N +1−j − +1−2j 2N +1−2j 2 2 for j = 1, . . . , N of lower order families. This proves the assertion in Theorem 1.4.1/(ii) concerning the structure of the coeﬃcients Bj . c We determine the normal order of D2N +1 (λ). In order to calculate the nornc mal order, we use the non-compact model D2N +1 (λ). By the explicit formula for 0 nc D2N (λ) (see (5.1.23)), the normal order of D +1 2N +1 (λ) degenerates iﬀ N −1

(2λ−4N +2k+n−1) = 0,

k=0

i.e., iﬀ λ∈

n−1 n−1 − +2N, . . . , − +N +1 . 2 2

More precisely, for λ = − n−1 2 + N + M with 1 ≤ M ≤ N precisely the coeﬃcients b0 (λ), . . . , bN −M (λ) vanish, i.e., the normal order is 2N +1−2(N −M +1) = 2M −1. n−3 c In the most extreme case M = 1, the normal order of D2N +1 (− 2 + N ) is 1. n−1 c In fact, this operator equals P2N (S )D1 . This completes the proof of Theorem 1.4.1/(ii).

152

Chapter 5. Intertwining Families

c ∞ n ∞ n−1 The K n -equivariance of D2N ) and the K n +1 (λ) : C (S ) → C (S c ∞ n−1 ) is SO(n)-invariant, i.e., invariance of H0 imply that D2N +1 (λ)(H0 ) ∈ C (S a constant function. We deﬁne c Q2N +1 (λ) = D2N +1 (λ)(H0 ).

(5.2.43)

Note that since D1c (H0 ) = 1 the leading coeﬃcient of Q2N +1 (λ) coincides with the coeﬃcient of D1c in B2N . We continue with the determination of the degree 2N polynomial Q2N +1 (λ). Lemma 5.2.14. Q2N +1 (λ) = (−1)N (λ−1) · · · (λ−2N ). In order to prove Lemma 5.2.14, we use induction over N . The idea is to ﬁnd 2N +1 conditions for the polynomial family Q2N +1 (λ) which uniquely determine it using the families Q2M+1 (λ) for 0 ≤ M ≤ N −1. The following lemma provides the ﬁrst condition. Lemma 5.2.15. Q2N +1 (2N +1) = (−1)N (2N )!. Proof. We evaluate the right-hand side of the identity c Q2N +1 (2N +1) = D2N +1 (2N +1)(H0 )

=2

−2N

(by deﬁnition)

nc 2 2 2N D2N xn ) +1 (2N +1)((1+|x | +xn )

at x = 0. The result coincides with 2

−2N

N

(N ) bj (2N +1)ΔjRn−1

j=0

∂ ∂xn

2N +1−2j

1 (1+|x |2 +x2n )2N xn 1x =xn =0

i.e., 2−2N

N

(N )

bj

(2N +1)

j=0

×

2N 2N r=0

r

ΔjRn−1

2r |x | |x =0

∂ ∂xn

2N +1−2j

1 (1+x2n )2N −r xn 1x

But using 1 # ΔkRn−1 |x |2r 1x =0 = δrk r!2r {(n−3+2r) · · · (n−1)} and

∂ ∂xn

2k+1

1 (1+x2n )M xn 1xn =0 =

M (2k+1)!, k

n =0

.

5.2. Induced families

153

we obtain the following formula for Q2N +1 (2N +1). 2−2N

N

(N )

bj

(2N +1)

j=0

×

2N 2N r=0

= 2−2N

r

N

δrj j!2r {(n−3+2r) · · · (n−1)}#

(N )

bj

(2N +1)2j−2N

j=0

2N −r (2N −2j +1)! N −j

(2N )!(2N −2j +1)! {(n−3+2r) · · · (n−1)}#. N !(N −j)!

Now we apply Corollary 5.1.3 and (5.2.33). It follows that Q2N +1 (2N +1) is given by (−1)N 2−N (2N )! ×

N j=0

(−1)j {{(n−1) · · · (n−3+2j)}#{(n+1+2j) · · · (n−1 +2N )}# j!(N −j)!

= (−1)N 2−N (2N )!

N j=0

(−1)j {(n−1) · · · (n−1+2N )}# = (−1)N (2N )!. j!(N −j)! n−1 +2j

The proof is complete.

Now we apply the identities in Theorem 5.2.4 to the function H0 . We obtain n

n

c n Q2N +1 − +j = D2N (5.2.44) +1−2j − −j P2j (S )(H0 ), 2 2 n−1 n−1 Q2N +1 2N −j − = Q2N +1−2j 2N +1−j − P2j (S n−1 )(1) (5.2.45) 2 2 for j = 1, . . . , N . H0 is the restriction of the degree 1 harmonic polynomial xn on Rn+1 to S n . It follows that H0 is an eigenfunction of ΔS n : ΔS n (H0 ) = −nH0 .

(5.2.46)

Hence n 2 +j−1

n

P2j (S )(H0 ) =

n 2 +j−1

(−n−r(n−1−r)) =

r= n 2

(r−n)(r+1).

r= n 2

Together with (5.2.36) we see that (5.2.44) and (5.2.45) are equivalent to n

n

n 2 +j−1 Q2N +1 − +j = Q2N +1−2j − −j (r−n)(r+1) 2 2 n r= 2

(5.2.47)

154

Chapter 5. Intertwining Families

and n−1 +j−1 2 n−1 n−1 Q2N +1 2N −j − (−r)(n−1−r) = Q2N +1−2j 2N +1−j − 2 2 n−1 r=

2

(5.2.48) for j = 1, . . . , N . Now we are able to complete the proof of Lemma 5.2.14. We use induction over N to prove (5.2.49) Q2N +1 (2N +1) = (−1)N (2N )!, (5.2.47) and (5.2.48). (5.2.49) is obvious. For the right-hand side of (5.2.47) we obtain " n

n

# − −j −1 · · · − +j −2N (−1)N −j 2 " 2

n

n # n n · · · − +j −1 +1 · · · +j × − 2 2 2 2 "

# n n − +j −1 · · · − = (−1)N 2

n # " n

2 n

# " n − −j −1 · · · − +j −2N × − −1 · · · − −j 2 2n n

2 n

2 N − +j −1 · · · − +j −2N = Q2N +1 − +j . = (−1) 2 2 2 The proof of (5.2.48) is analogous. The proof is complete.

nc 5.2.4 Eigenfunctions of ΔHn and the families DN (λ)

In the present section, we relate the families DN (λ) to the asymptotics of eigenfunctions of the Laplacian on the upper half-space with respect to the hyperbolic metric of curvature −1. That relation leads to an alternative proof of a system of nc (λ). We also show how these families factorization identities for the families D2N appear in the residues of a family of distributions, and that the latter observation implies their equivariance. In Section 6.6, these results will be generalized within a much wider framework. Let aj (x )xλ+j (5.2.50) u(x , xn ) ∼ n , xn > 0, aodd ≡ 0 j≥0

be a formal solution of the eigenequation −ΔHn u = λ(n−1−λ)u, where ΔHn = x2n

n 2 ∂2 ∂ ∂ 2 2 ∂ n−1 + x − (n−2)x = x Δ − (n−2)xn . n R n n 2 2 ∂x ∂x ∂x ∂x n n n i i=1

The coeﬃcients aj are determined recursively by the ﬁrst coeﬃcient a0 . More precisely, the maps T2j (λ) : a0 (·) → a2j (λ, ·)

5.2. Induced families

155

are diﬀerential operators (with rational coeﬃcients in λ) of order 2j on Rn−1 with T0 (λ) = id. We use these data to deﬁne for any N ∈ N0 the family S2N (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ) by S2N (λ) =

N j=0

1 2N −2j T2j (λ)i∗ (∂/∂xn ) , (2N −2j)!

(5.2.51)

where i : Rn−1 → Rn , x → (x , 0). 0 On the other hand, we have seen in Section 5.2.1 that D2N (λ) induces a nc ∞ n ∞ n−1 family D2N (λ) : C (R ) → C (R ). The following result establishes a relation between both families. nc (λ) are proportional by Theorem 5.2.5. The families S2N (λ+n−1−2N ) and D2N a rational function in λ.

Proof. The series in (5.2.50) corresponds to an eigenfunction iﬀ

λ+j−2 x2n xλ+j (λ+j)(λ+j −1)aj − (n−2)(λ + j)xλ+j n ΔRn−1 aj + xn n aj

j≥0

= −λ(n−1−λ)

xλ+j n aj .

j≥0

Comparing the coeﬃcients of xλ+j we obtain n ((λ+j)(λ+j −1) − (λ+j)(n−2)) aj + ΔRn−1 aj−2 = −λ(n−1−λ)aj . The latter relations are equivalent to (λ+j)(λ+j −(n−1))aj + ΔRn−1 aj−2 = −λ(n−1−λ)aj , i.e., ΔRn−1 aj−2 = j(n−1−2λ−j)aj , j ≥ 2.

(5.2.52)

It follows that we can write S2N (λ) in the form 1 ∗ i A0 (λ) (2N )!

∂2 ∂x2n

N

1 ΔRn−1 i∗ + A2 (λ) (2N −2)!

∂2 ∂x2n

N −1

∗ + · · · + A2N (λ)ΔN Rn−1 i

with coeﬃcients A2j (λ) which are determined recursively by 2j(n−1−2λ−2j)A2j (λ) = A2j−2 (λ), A0 (λ) = 1.

(5.2.53)

156

Chapter 5. Intertwining Families

Equivalently, we can write S2N (λ) as B0 (λ)i∗

∂2 ∂x2n

N

+ B2 (λ)ΔRn−1 i∗

∂2 ∂x2n

N −1

∗ + · · · + B2N (λ)ΔN Rn−1 i

with coeﬃcients B2j (λ) which are determined recursively by 2j(n−1−2λ−2j) B2j (λ) = B2j−2 (λ) (2N −2j +2)(2N −2j +1)

and B0 (λ) =

1 . (2N )!

(5.2.54)

nc According to Lemma 5.2.1, the family D2N (λ) has the form

a0 (λ)i

∗

∂2 ∂x2n

N + a1 (λ)ΔRn−1 i

∗

∂2 ∂x2n

N −1

∗ + · · · + aN (λ)ΔN Rn−1 i ,

where the coeﬃcients aj (λ) are determined recursively by aj−1 (λ) = −

j(n−1+2λ−4N +2j) aj (λ) (N −j +1)(2N −2j +1)

and aN (λ) = 1.

(5.2.55)

But (5.2.54) implies B2j (λ+n−1−2N )

j(−(n−1)−2λ+4N −2j) = B2j−2 (λ+n−1−2N ). (N −j +1)(2N −2j +1)

Comparing these relations with (5.2.55), it follows that the quotients B2j (λ+n−1−2N ) aj (λ)

(5.2.56)

do not depend on j. The proof is complete.

It is easy to determine the proportionality coeﬃcient in Theorem 5.2.5. In fact, it is enough to ﬁnd the quotient (5.2.56) for j = N . Then aN (λ) = 1 and we calculate, using (5.2.54), 1 (2N )! B0 (λ) # {2N · · · 2} {(n−1−2λ−2N ) · · ·(n−1−2λ−2)}# 1 = N . 2 N !{(n−1−2λ−2N ) · · · (n−3−2λ)}#

B2N (λ) =

Here # indicates that each second term in the corresponding product is deleted. Hence S2N (λ+n−1−2N ) =

1 Dnc (λ), 2N N !{(−(n−1)−2λ+2N ) · · · (−(n−1)−2λ+4N −2)}# 2N

(5.2.57)

5.2. Induced families

157

nc i.e., D2N (λ) is of the form

2N N ! {(−(n−1)−2λ+2N ) · · · (−(n−1)−2λ+4N−2)}#i∗ (2N )!

∂2 ∂x2n

N

∗ + · · · + ΔN Rn−1 i .

Theorem 5.2.5 extends to odd order families. For any N ∈ N0 , we deﬁne the family S2N +1 (λ) : C ∞ (Rn ) → C ∞ (Rn−1 ) by S2N +1 (λ) =

N j=0

1 T2j (λ)i∗ (∂/∂xn )2N +1−2j . (2N +1−2j)!

(5.2.58)

The following result is the analog of Theorem 5.2.5. nc Theorem 5.2.6. The families S2N +1 (λ+n−1−(2N +1)) and D2N +1 (λ) are proportional by a rational function in λ.

Proof. By the arguments in the proof of Theorem 5.2.5, we can write S2N +1 (λ) in the form A0 (λ)

1 i∗ (2N +1)!

∂ ∂xn

2N +1

1 ΔRn−1 i∗ + A2 (λ) (2N −1)!

∂ ∂xn

2N −1 + ···+

∗ A2N (λ)ΔN Rn−1 i

∂ ∂xn

with coeﬃcients A2j (λ), j = 0, . . . , N which are determined recursively by (5.2.53). Equivalently, we can write S2N +1 (λ) as 2N +1 2N −1 ∂ ∂ ∂ ∗ + C2 (λ)ΔRn−1 i∗ + · · · + C2N (λ)ΔN i C0 (λ)i∗ Rn−1 ∂xn ∂xn ∂xn with coeﬃcients C2j (λ) which are determined recursively by 2j(n−1−2λ−2j) C2j (λ) = C2j−2 (λ) (2N −2j +3)(2N −2j +2)

and C0 (λ) =

1 . (5.2.59) (2N +1)!

Comparing (5.2.59) with (5.1.22) gives the desired result.

In order to determine the proportionality factor in Theorem 5.2.6, it is enough to ﬁnd the quotient C2N (λ+(n−1)−(2N +1) bN (λ) with bN (λ) = 1. Now (5.2.59) gives C2N (λ) =

(2N +1)! C0 (λ). 2N N !{(n−1−2λ−2N ) · · ·(n−3−2λ)}#

158

Chapter 5. Intertwining Families

Hence S2N +1 (λ+(n−1)−(2N +1)) =

1 2N N !{(−(n−1)−2λ+2(N +1)) · · ·(−(n−1)−2λ+4N )}#

nc D2N +1 (λ),

(5.2.60) nc i.e., D2N +1 (λ) is of the form

2N N ! {(−(n−1)−2λ+2(N +1)) · · ·(−(n−1)−2λ+4N )}# (2N +1)! ∗ × i∗ (∂/∂xn )2N +1 + · · · + ΔN Rn−1 i (∂/∂xn ) . nc (λ) in terms of eigenfunctions of the Laplacian on The interpretation of DN the hyperbolic upper half-space allows us to give a conceptually alternative proof of one half of the system of factorization identities for these families. The following result restates the corresponding part of Theorem 5.2.2. For convenience of later use we shift the dimension by 1. nc (λ) : C ∞ (Rn+1 ) → C ∞ (Rn ) factorize in the Theorem 5.2.7. The families D2N form n

n

nc nc 2N −k− = ΔkRn ◦ D2N D2N −2k 2N −k− 2 2 for k = 0, . . . , N .

Proof. We recall that nc D2N (λ) =

N

(N )

aj

(λ)ΔjRn i∗ (∂/∂xn+1 )2N −2j

j=0

(Lemma 5.2.1), where (N ) aj (λ)

N −1 N! N −j (−2) = (2λ−4N +2k+n+2). j!(2N −2j)! k=j

Since

(N )

a0

2N −k−

n

n

(N ) = · · · = ak−1 2N −k− = 0, 2 2

it is enough to prove that n

n k (N ) (N −k) 2N −k− 2N −k− aj = aj−k ΔRn 2 2 for all j = k, . . . , N . By Theorem 5.2.5, this is equivalent to the proportionality of the operators n

+k ΔkRn Res n2 −k (T2j (·)) and T2j−2k 2

5.2. Induced families

159

for all j = k, . . . , N . But for λ = n2 + k, the two ladders in the asymptotics u∼ T2j (λ)(a0 )xλ+2j T2j (n−λ)(b0 )xn−λ+2j , xn+1 → 0 n+1 + n+1 j≥0

j≥0

overlap, and we ﬁnd the relation n

+k Res n2 +k (S(·)) = Res n2 −k (T2j (·)) T2j−2k 2 for all j ≥ k, where a0 = S(λ)b0 with S(λ) = Iλ /c(λ) (see Section 3.2). Here the residue Res n2 +k (S) of the scattering operator is a multiple of ΔkRn = P2k (Rn ). This completes the proof. We close the present section with the outline of an alternative argument for the equivariance of the families SN (λ) which are deﬁned by (5.2.51) and (5.2.58). We ﬁrst prove that, up to a meromorphic coeﬃcient, SN (μ) coincides with the operator RN (μ) given by Resλ=−μ−1−N

Rn +

xλn uϕdx

= Rn−1

RN (μ)(ϕ)ωdζ,

where the eigenfunction u is the Poisson transformation (see Section 2.3) u(x) = Pn−1−μ (ω)(x) = P (x, ζ)n−1−μ ω(ζ)dζ Rn−1

C0∞ (Rn−1 ).

of ω ∈ Then the equivariance of SN (λ) will be a direct consequence. In order to determine the operator RN (μ), we use repeated partial integration. n−1 n−1 Assume that (μ) = n−1 2 , μ = 2 . Then for (λ) > − 2 , ∞ xλn uϕdx = xλn u(x , xn )ϕ(x , xn )dx dxn Rn +

Rn−1

0

=

0

∞

Rn−1 j≥0

xλ+μ+2j T2j (μ)(a0 )(x )ϕ(x , xn )dx dxn n

up to an analogous integral with powers xnλ+n−1−μ+2j ; in the following we ignore the second integral. It is enough to establish the meromorphic continuation of the integrals ∞ n−1 xλ+μ+2j ϕ(x , xn )dxn , (λ) > − n 2 0 in λ. The latter integral is holomorphic on (λ) > −(μ)− 1 − 2j. We integrate M ≥ 1 times by parts and ﬁnd ∞ xλ+μ+2j ϕ(x , xn )dx dxn n 0 ∞ (−1)M = xλ+μ+2j+M ϕ(M) (x , xn )dx dxn , n (λ+μ+2j +1) · · · (λ+μ+2j +M ) 0

160

Chapter 5. Intertwining Families

where ϕ(M) denotes the M th derivative in the variable xn . Hence the integral has a meromorphic continuation to (λ) > −(μ)− 2j − M − 1 with simple poles at λ = −μ−2j −1, . . . , λ = −μ−2j −M and ∞ xλ+μ+2j T (μ)(a )(x )ϕ(x , x )dx dx Resλ=−μ−2j−M 2j 0 n n n 0 Rn−1 1 = T2j (μ)(a0 )(x )ϕ(M−1) (x , 0)dx . (M −1)! Rn−1 Hence we ﬁnd the residues Resλ=−μ−1−L

Rn +

xλn uϕdx

=

[ L2 ]

1 (L−2j)!

j=0

Rn−1

T2j (μ)(a0 )ϕ(L−2j) (x , 0)dx

for L ≥ 0. In other words,

RL (μ)(ϕ)(x ) = c(n−1−μ)

[ L2 ] j=0

1 T ∗ (μ) (L−2j)! 2j

∂ ∂xn

L−2j

(ϕ)(x , 0).

∗ Now (5.2.51) and (5.2.58) follow from the self-adjointness of T2j . Here we have used that the leading coeﬃcient a0 and ω are related by

a0 = c(n−1−μ)ω, since

Rn−1

(|x

c(λ) = π

n−1 2

Γ(λ− n−1 2 ) Γ(λ)

xλn dζ = xnn−1−λ c(λ). − ζ|2 + x2n )λ

Now the equivariance of the family RN (μ) follows from the equivariance of the Poisson transformation. The Gn -equivariance of Pλ states that nc Pλ ◦ πλ−(n−1) (g) = g∗ ◦ Pλ n

(Corollary 2.3.2). Now observe that xn ∈ C ∞ (Rn )G 1 . In fact, π1nc (g)(xn ) Hence

=

g∗ (vol(gc )) vol(gc )

− n1 g∗ (xn ) =

g∗ (vol(x−2 n gc )) −2 vol(xn gc )

− n1 xn = xn .

C0∞ (Rn−1 ) ω → xλn Pn−1−μ (ω) ∈ C −∞ (Rn )

(5.2.61)

on functions on R with on distributions on R . In particintertwines ular, the residue of the meromorphic continuation of (5.2.61) at λ = −μ−1−N nc nc and π−μ−1−N , i.e., intertwines π−μ nc nc (g)ω = π−μ−1−N (g)R∗N (μ)ω, (5.2.62) R∗N (μ) π−μ nc π−μ

n−1

πλnc

n

5.3. Some low order examples

161

where R∗N (μ)ω, ϕ

= ω, RN (μ)ϕ =

Therefore, (5.2.62) yields nc ωπμ−(n−1) (g −1 )RN (μ)(ϕ)dζ = Rn−1

Rn

Rn−1

ωRN (μ)ϕdζ.

nc ωRN (μ) πμ−(n−1)+N (g −1 )ϕ dζ.

This proves the equivariance nc (g) ◦ RN (λ+n−1−N ), g ∈ Gn . RN (λ+n−1 −N ) ◦ πλnc (g) = πλ−N

(5.2.63)

5.3 Some low order examples c Here we derive explicit formulas for the intertwining families DN (λ) : C ∞ (S n ) → ∞ n−1 n C (S ) of order N ≤ 4 in terms of GJMS-operators on S and S n−1 . These results serve as illustration and as convenient future reference. The ﬁrst-order family is induced by D10 (λ) = Yn− , and we ﬁnd D1c (λ) = ∗ i grad(H0 ). The second-order family is induced by − D20 (λ) = −(2λ+n−3)(Yn− )2 + Δ− n−1 ∈ U(nn+1 ).

(5.3.1)

Lemma 5.3.1. D2c (λ)

5 ∗ n−1 ∗ ∗ n = −λ i + λ 2P2 (S )i − 2i P2 (S ) − n− i 2 (n−2)(n−3) ∗ i , + (n−2)P2 (S n−1 )i∗ − (n−3)i∗P2 (S n ) − 4 2 ∗

i.e., D2c (λ) = −(2λ+n−3)i∗P2 (S n ) + (2λ+n−2)P2(S n−1 )i∗ n−2 n−3 ∗ − λ+ λ+ i . 2 2

(5.3.2)

These formulas are equivalent to D2c (λ) = −(2λ+n−3)i∗ΔS n + (2λ+n−2)ΔS n−1 i∗ − λ(λ−1)i∗ = −(2λ+n−3)i∗(grad H0 )2 + (ΔS n−1 −λ(λ−1)) i∗ .

(5.3.3)

Proof. We determine D2c (λ) = −λ2 i∗ + aλ + b (recall that by Lemma 5.2.9 the leading term of D2c (λ) is −i∗ ) so that the factorization identities n

n−3 D2c − +1 = i∗ P2 (S n ), D2c − (5.3.4) = P2 (S n−1 )i∗ 2 2

162

Chapter 5. Intertwining Families

are satisﬁed. This yields the ﬁrst two formulas for D2c (λ). For the proof of the remaining two formulas we use the identity i∗ (grad(H0 ))2 = i∗ ΔS n − ΔS n−1 i∗

(5.3.5)

and the formulas

n n −1 , P2 (S ) = ΔS n − 2 2 n

P2 (S

n−1

n−1 ) = ΔS n−1 − 2

n−1 −1 . 2

For the proof of (5.3.5) we use the coordinates κ : S n−1 × (0, 1) (y, ρ) → ( 1−ρ2y, ρ) ∈ S n on S n . Then κ∗ (gS n ) = (1−ρ2 )gS n−1 +

1 dρ2 1−ρ2

and the Laplacian on S n−1 × (0, 1) is given by ΔS n−1 ×(0,1) = (1−ρ2 )

1 ∂2 ∂ + − nρ ΔS n−1 . ∂ρ2 ∂ρ 1−ρ2

(5.3.6)

Finally, we have the following formula for the gradient of the deﬁning function ρ = κ∗ (H0 ) with respect to the metric κ∗ (gS n ). grad(ρ) = (1−ρ2 )

∂ . ∂ρ

(5.3.7)

In these terms, (5.3.5) reads 2 1 ∂ ∗ 2 ∂ i (1−ρ ) 2 − nρ + ΔS n−1 − ΔS n−1 i∗ = i∗ (grad(ρ))2 , ∂ρ ∂ρ 1−ρ2 where now i : y → (y, 0). But the latter identity is equivalent to i∗

∂2 = i∗ (grad(ρ))2 , ∂ρ2

which is a direct consequence of (5.3.7). The proof is complete.

(5.3.8)

The proof of Lemma 5.3.1 rests on the relations (5.3.4). Next, we illustrate how the residue method (see Section 1.5) works in that special case. The calculation of the residue of the pole of Mu (λ) at λ = −μ−3 gives the distribution 1 ∂2 n−2 a i a (f ) + i (f ) − (f ) a ∗ 0 ∗ 2 0 2! ∂ρ2 2

5.3. Some low order examples

163

(we ignore the contributions of the coeﬃcients bN ). Here a2 (f ) is determined by a0 (f ) via 2(2μ−(n−3))a2(f ) = μ(μ+1)a0 (f ) − ΔS n−1 a0 (f ) (see (1.5.1)), i.e., a2 (f ) =

μ(μ+1) 1 a0 (f ) − Δ n−1 a0 (f ). 2(2μ−(n−3)) 2(2μ−(n−3)) S

Hence we arrive at the operator μ(μ+1) 1 n−2 ∗ 1 ∗ ∂2 i i∗ − ΔS n−1 i∗ − i + 2 ∂ρ2 2(2μ−(n−3)) 2(2μ−(n−3)) 2 (compare with the deﬁnition of S2 (μ)). In order to ﬁnd D2c (μ), we have to replace μ by μ+n−3 and to renormalize. We obtain ∂2 + [(μ+n−3)(μ+n−2) − (n−2)(2μ+n−3)] i∗ − ΔS n−1 i∗ ∂ρ2 ∂2 = (2μ+n−3)i∗ 2 − [ΔS n−1 −μ(μ−1)] i∗ . ∂ρ

(2μ+n−3)i∗

Now (5.3.8) shows that this operator coincides with −D2c (μ). We continue with the discussion of the third-order family. It is induced by 1 − − D30 (λ) = − (2λ+n−5)(Yn− )3 + Δ− n−1 Yn ∈ U(nn+1 ). 3 Now D3c (λ) satisﬁes the factorization identities n

n−5 c c n c D3 − +1 = D1 P2 (S ), D3 − = P2 (S n−1 )D1c . 2 2 Similarly as for D2c (λ), we apply these two identities to ﬁnd an explicit formula for D3c (λ) by using that it has the form D3c (λ) = −λ2 D1c + λA + B with coeﬃcients A and B to be determined (Lemma 5.2.12). We ﬁnd Lemma 5.3.2.

2 7 P2 (S n−1 )D1c − D1c P2 (S n ) − n− D1c D3c (λ) = −λ2 D1c + λ 3 2 n−2 (n−2)(n−5) c n−5 c + − D1 P2 (S n ) + P2 (S n−1 )D1c − D1 , 3 3 4

i.e., 1 1 D3c (λ) = − (2λ+n−5)D1c P2 (S n ) + (2λ+n−2)P2(S n−1 )D1c 3 3 n−5 n−2 − λ+ λ+ D1c . 2 2

(5.3.9)

164

Chapter 5. Intertwining Families

Similarly, we get Lemma 5.3.3. 1 1 D4c (λ) = (2λ+n−5)(2λ+n−7)i∗ P4 (S n ) + (2λ+n−4)P2(S n ) 3 2 1 1 + (2λ+n−2)(2λ+n−4) P4 (S n−1 ) − (2λ+n−5)P2(S n−1 ) i∗ 3 2 2 − (2λ+n−4)(2λ+n−5)P2(S n−1 )i∗ P2 (S n ) 3 1 + (2λ+n−2)(2λ+n−4)(2λ+n−5)(2λ+n−7)i∗. 16 The zeros of the four monomials (2λ + n − 2), (2λ + n − 4), (2λ + n − 5), (2λ + n − 7) coincide with the special values of λ for which the family satisﬁes a factorization identity. Proof. The family is of the form λ4 i∗ + a3 λ2 + a2 λ2 + a1 λ + a0 and satisﬁes the characterizing identities

n n n D4c − +2 = i∗ P4 (S n ), D4c − +1 = D2c − −1 P2 (S n ) 2 2 2 and n−1 D4c 2− = P4 (S n−1 )i∗ , 2

n−1 n−1 D4c 3− = P2 (S n−1 )D2c 3− . 2 2

It suﬃces to prove that the given family satisﬁes these identities. The respective ﬁrst two identities are obvious. We prove the factorization identity for λ = − n2 + 1. The deﬁnition of D4c (λ) and the explicit formulas (see (3.2.25))

n

n +1 −2 , P4 (S n ) = P2 (S n ) ΔS n − 2 2

P2 (S n ) = ΔS n −

n n −1 2 2

imply n

D4c − +1 = 5P4 (S n ) − 15P2 (S n ) − 4P2 (S n−1 )P2 (S n ) 2

n

n

+1 −2 − 4P2 (S n−1 ) − 15 P2 (S n ) = 5 ΔS n − 2 2 = 5P2 (S n ) − 4P2 (S n−1 ) − 5 P2 (S n ) n

= D2c − −1 P2 (S n ); 2 here we omit i∗ . For the latter identity we have used Lemma 5.3.1. The proof of

5.4. Families for (Rn , S n−1 )

165

the fourth identity is symmetric to the latter argument. In fact, we have n−1 D4c 3− = 5P4 (S n−1 ) − 15P2 (S n−1 ) − 4P2 (S n−1 )P2 (S n ) 2 n+1 n−5 = P2 (S n−1 ) 5 ΔS n−1 − − 4P2 (S n ) − 15 2 2 n−1 n n−1 = P2 (S ) −4P2 (S ) + 5P2 (S )−5 n−1 = P2 (S n−1 )D2c 3− . 2 The proof is complete.

5.4 Families for (Rn , S n−1) In the present section, we take a closer look at the intertwining families b DN (λ) = DN (Rn , S n−1 ; λ) : C ∞ (Rn ) → C ∞ (S n−1 )

which are induced by DN (λ). The fact that they are induced by homomorphisms of Verma modules implies that they are special cases of tractor families (Section 6.21). Therefore, they serve as an illustration of the general theory of tractor families. b (λ) demonstrate that, in general, tractor families In particular, the examples DN are not determined by the factorization identities they satisfy. In Section 5.4.1, we b (λ) in terms of the asymptotics of eigenfunctions describe the induced families DN of the Poincar´e metric on the unit ball. From the perspective of Chapter 6 that relation is interpreted as follows. The residue construction in Section 5.4.1 is a special case of the general construction in Section 6.5. The families constructed res (λ) (deﬁned in Section 6.6) for the here are conjugate to the residue families DN n−1 . Both arise from two diﬀerent conformal compactiﬁcations round metric on S of the Poincar´e metric on the unit ball (see also the discussion at the end of Section 6.5 and Example 6.6.2). The relation between two types of constructions (via induction and via Poincar´e metric) is a prototype of general results concerning the relation between residue families and tractor families (Section 6.21). Here we b (λ) for N ≤ 3. Last not least, the family D3b (λ) use that relation to determine DN (for n = 4) deserves special interest. In Section 5.4.3, we apply an explicit formula for that family to relate its value at λ = 0 to an operator of Chang and Qing. b 5.4.1 The families DN (λ) b The families DN (λ) : C ∞ (Rn ) → C ∞ (S n−1 ) are equivariant with respect to principal series representations of the subgroup Gn0 = σ0 Gn σ0−1 ⊂ Gn+1 (acting on Rn ) which leaves the sphere S n−1 invariant. More precisely, b b πλ−N (S n−1 , gc )(g) ◦ DN (λ) = DN (λ) ◦ πλ (Rn , gc )(g)

166

Chapter 5. Intertwining Families

for all g ∈ Gn0 (see Lemma 5.2.6). In order to derive explicit formulas for the b (λ), we identify them as residues. For that purpose, we regard the families DN families as boundary operators C ∞ (Bn ) → C ∞ (S n−1 ) on the unit ball Bn = {x ∈ Rn | |x| < 1}. We consider Bn as a hyperbolic manifold with the metric 4 (dx21 + · · · + dx2n ). (1−|x|2 )2 Let ΔBn be the associated Laplacian and u ∈ C ∞ (Bn ) an eigenfunction −ΔBn u = μ(n−1−μ)u, where (μ) = (n−1)/2. We associate to u a family of measures on Rn by Mu (λ), ϕ = (1−|x|2 )λ u(x)ϕ(x)dx = (1−|x|2 )λ u(x)ϕ(x)dx Bn

Rn

for test functions ϕ ∈ C0∞ (Rn ). In the second integral, u denotes the extension of u by 0 in the complement of Bn . The integral converges if (λ) > − n−1 2 . We construct a meromorphic continuation of the family Mu (λ). For test functions ϕ with compact support in Bn , the family Mu (λ), ϕ is holomorphic on C. Thus in order to see the poles it is enough to choose ϕ with support in a neighborhood of the boundary. The existence of the continuation follows from the existence of an asymptotic expansion of the eigenfunction u. We assume that e(n−1−μ)x,b ω(b)db u(x) = S n−1

for some ω ∈ C ∞ (S n−1 ); here the distance x, b is deﬁned using the metric of curvature −1 ([140]). In order to write the asymptotic expansion of u, we use polar coordinates (0, 1) × S n−1 → Bn . In these coordinates, the metric is given by 4 (dr2 + r2 gc ) (1−r2 )2 with the round metric gc on S n−1 , and for the hyperbolic Laplacian we have the formula 2 ∂ 1 n−1 ∂ ∂ n−1 4ΔBn = (1−r2 )2 + + Δ + 2(n−2)(1−r2)r . S 2 2 ∂r r ∂r r ∂r Now we write the asymptotics of u in the form u(r, b) ∼ (1−r2 )μ+N aN (μ, b) + (1−r2 )n−1−μ+N bN (μ, b). N ≥0

(5.4.1)

N ≥0

u is a spherical function if ω is constant. In that case, it can be identiﬁed with a hypergeometric function in the argument |x|2 . a0 (μ, b) coincides with ω(b), up to a coeﬃcient which only depends on μ (c-function).

5.4. Families for (Rn , S n−1 )

167

A routine calculation shows that the coeﬃcients aN (μ, b), N ≥ 1 are determined by a0 (b) via the recursion formula 4(k+1)(n−2−2μ−k)ak+1 = 2(μ+k)(n−2− 2(μ+k))ak + ΔS n−1 (ak−1 + · · · + a0 ).

(5.4.2)

A similar relation holds true for bN (μ, b), N ≥ 1. Now let N ∈ N0 . We construct an equivariant diﬀerential operator of order N in terms of the operators Tj (μ) : a0 (μ, b) → aj (μ, b), j = 0, . . . , N . We ﬁrst rewrite the integral (1−|x|2 )λ u(x)ϕ(x)dx Bn

in polar coordinates as 1 (1−r2 )λ u(r, b)ϕ(r, b)rn−1 drdb. 0

S n−1

In view of (5.4.1), we are led to study the continuation of the function ⎞ ⎛ 1 λ → (1−r2 )λ ⎝ (1−r2 )μ+N aN (μ, b)⎠ rn−1 ϕ(r, b)drdb. 0

S n−1

N ≥0

For convenience we change to the coordinates (ρ, b) ∈ (0, 1) × S n−1 , ρ = 1 − r2 . Let ψ be deﬁned by ϕ(r, b) = ψ(1 − r2 , b). Then we rewrite the integral as the sum of n−2 1 1 ρλ+μ+N aN (μ, b)(1−ρ) 2 ψ(ρ, b)dρdb 2 0 S n−1 N ≥0 ⎞ ⎛ (− n−2 )m 1 1 2 ρm ⎠ ψ(ρ, b)dρdb = ρλ+μ+N aN (μ, b) ⎝ 2 0 S n−1 m! N ≥0 m≥0 ⎛ ⎞ N n−2 (− ) 1 1 N −j 2 = aj (μ, b)⎠ ψ(ρ, b)dρdb ρλ+μ+N ⎝ 2 0 S n−1 (N −j)! j=0 N ≥0 ⎛ ⎞ N n−2 (− ) 1 1 N −j 2 = Tj (μ)(a0 (μ, b))⎠ ψ(ρ, b)dρdb, ρλ+μ+N ⎝ 2 0 S n−1 (N −j)! j=0 N ≥0

and a similar term using b0 (μ, b). We introduce the abbreviation AN (μ) =

N (− n−2 )N −j 2

j=0

(N −j)!

Tj (μ).

168

Chapter 5. Intertwining Families

In the integrals

0

1

S n−1

ρλ+μ+N AN (μ)(a0 )ψ(ρ, b)dρdb

we use M ≥ 1 partial integrations in the variable ρ to get 1 1 AN (μ)(a0 ) ρλ+μ+N +1 ψ(ρ, b) 0 db λ+μ+N +1 S n−1 1 1 − AN (μ)(a0 ) ρλ+μ+N +2 ψ (ρ, b) 0 db (λ+μ+N +1)(λ+μ+N +2) S n−1 ± ...

1 (−1)M−1 + AN (μ)(a0 ) ρλ+μ+N +M ψ (M−1) (ρ, b) db (λ+μ+N +1) · · · (λ+μ+N +M ) S n−1 0 1 M (−1) + ρλ+μ+N +M AN (μ)(a0 )ψ (M) (ρ, b)dρdb (λ+μ+N +1) · · · (λ+μ+N +M ) 0 S n−1 1 (−1)M = ρλ+μ+N +M AN (μ)(a0 )ψ (M) (ρ, b)dρdb (λ+μ+N +1) · · · (λ+μ+N +M ) 0 S n−1 for (λ) > −μ − 1 − N . We use this formula for meromorphic continuation to (λ) > −μ−1−N −M with simple poles at λ = −μ−1−N, . . . , −μ−N −M. The residue at λ = −μ−N −M is given by −1 (M −1)!

0

1

S n−1

AN (μ)(a0 )ψ (M) (ρ, b)dρdb =

1 (M −1)!

S n−1

AN (μ)(a0 )ψ (M−1) (0, b)db.

It follows that Mu (λ), ϕ has a meromorphic continuation to C with simple poles in the ladder −μ−1−N0 . More precisely, only the integrals which involve A0 , . . . , AP contribute to the residue of the pole in λ = −μ−1−P , P ≥ 0 and Res−μ−1−P (Mu (λ), ϕ) 1 1 = AP (μ)(a0 )(b)ψ(0, b)db + · · · + A0 (μ)(a0 )(b)ψ (P ) (0, b)db . 2 P ! S n−1 S n−1 But since the operators Aj are self-adjoint we obtain Res−μ−1−N (Mu (λ), ϕ) =

N j=0

1 (N −j)!

S n−1

a0 (μ, b)Aj (μ)ψ (N −j) (0, b)db.

5.4. Families for (Rn , S n−1 )

169

In order to formulate the result in a compact form, we introduce an operator family RN (μ) by ψ(ρ, b) →

N j=0

1 (N −j)!

(− n−2 (− n−2 ∂ N −j 2 )j 2 )1 T0 (μ) + · · · + Tj−1 (μ) + Tj (μ) i∗ N −j ψ, j! 1! ∂ρ (5.4.3)

where i∗ restricts functions in ρ to ρ = 0. We regard RN (μ) also as an operator family RN (μ) : C ∞ (Bn ) → C ∞ (S n−1 ). The above result can be formulated in the following form. Lemma 5.4.1. Let u be the Poisson-transform e(n−1−μ)x,b ω(b)db S n−1

of ω ∈ C ∞ (S n−1 ). Assume that (μ) = (n−1)/2 and μ = (n−1)/2. Then u is an eigenfunction −ΔBn u = μ(n−1−μ)u. The family Mu (λ), ϕ has simple poles at λ = −μ− 1 − N for all N ∈ N0 . The residues are Res−μ−1−N (Mu (λ), ϕ) = a0 (μ, b)RN (μ)(ψ)(b)db, S n−1

where a0 is one of the leading terms in the asymptotics of u. Next, we relate the family RN (μ) to the induced family DN (Rn , S n−1 ; μ). The method is as follows. We use σ0 to establish a connection between the distribution Mu (λ) with support in Bn and a corresponding distribution with support in Rn+ . The residues of the latter one are induced families. The main identity for that argument is given in the following result. Lemma 5.4.2.

Bn

1−|x|2 2

where

Iλ =

λ

u(x)ϕ(x)dx =

σ0∗ (vol(Rn , gc )) vol(Rn , gc )

Rn +

λ/n

xλn σ0∗ (u)Iλ+n (ϕ)dx,

σ0∗ , dx = vol(Rn , gc ).

(5.4.4)

170

Chapter 5. Intertwining Families

Proof. By the identity (2.2.10), the left-hand side equals ∗ λ/n λ/n dx σ0 (dx) λ λ (σ0 )∗ (xn ) uϕdx = xn σ0∗ (uϕ)σ0∗ (dx) (σ0 )∗ (dx) dx Bn Rn + ∗ λ/n+1 σ0 (dx) = xλn σ0∗ (u) σ0∗ (ϕ)dx. n dx R+

The proof is complete. Now we calculate residues at λ = −μ−1−N on both sides of (5.4.4). a0 (μ, b)RN (μ)(ϕ)(b)db 2μ+N S n−1

=2

μ Rn−1

σ0∗ (a0 )

σ0∗ (vol(S n−1 , gc )) vol(Rn−1 , gc )

μ n−1

SN (μ)I−μ−N +n−1 (ϕ)dx.

(5.4.5)

Here we use the fact that the residue at λ = −μ−1−N of the family λ → xλn u(x)ϕ(x)dx, u ∈ ker(ΔRn +μ(n−1−μ)), ϕ ∈ Cc∞ (Rn ) Rn +

is given by

Rn−1 n−1

a0 (μ, x )SN (μ)(ϕ)(x )dx ,

) is its leading term in the asymptotics u(x , xn ) ∼ where a0 (μ, x ) ∈ C ∞ (R xμn a0 (μ, x ) + · · · for xn → 0. The proof of that result is analogous to the above one for Mu (λ). We omit the details. Moreover, we have used that an asymptotics u(x) ∼ (1−|x|2 )μ a0 (μ, b) + · · ·

for |x| → 1

implies an asymptotics σ0∗ (u)

∼

xμn 2μ σ0∗ (a0 )

σ0∗ (vol(S n−1 , gc )) vol(Rn−1 , gc )

μ n−1

+ ···

for xn → 0.

In fact, (2.2.10) implies ∗ ∗ μ μ σ0∗ (1−|x|2 )μ σ0 (dx) n σ0 (vol(S n−1 , gc )) n−1 μ μ =2 =2 . xμn dx vol(Rn−1 , gc ) Now the integral on the right-hand side of (5.4.5) can be rewritten as μ n−1 db (σ0 )∗ (dx) db a0 (μ, b) (σ0 )∗ (SN (μ)I−μ−N +n−1 (ϕ)) (σ0 )∗ (dx) db S n−1 μ−(n−1) n−1 db = a0 (μ, b) (σ0 )∗ (SN (μ)I−μ−N +n−1 (ϕ)) db. (σ0 )∗ (dx) S n−1

5.4. Families for (Rn , S n−1 )

171

Hence 2

N

σ0∗ (db) dx

−μ+n−1 n−1

◦

σ0∗

◦ RN (μ) = SN (μ) ◦

σ0∗ (dx) dx

−μ+n−1−N n

◦ σ0∗ .

Now we shift μ into μ + n − 1 − N . Then the last identity states 2N

σ0∗ (db) dx

− μ−N n−1

◦ σ0∗ ◦ RN (μ+n−1−N ) = SN (μ+n−1+N ) ◦

σ0∗ (dx) dx

− μn

◦ σ0∗ ,

which, in turn, is equivalent to

db (σ0 )∗ (dx)

μ−N n−1

◦ (σ0 )∗ ◦ SN (μ+n−1+N ) = 2 RN (μ+n−1−N ) ◦ N

dx (σ0 )∗ (dx)

nμ

◦ (σ0 )∗ ,

(5.4.6)

i.e.,

(σ0 )∗ (vol(Rn−1 , gc )) vol(S n−1 , gc )

− μ−N n−1

◦ (σ0 )∗ ◦ SN (μ+n−1+N ) = 2N RN (μ+n−1−N ) ◦ πμ (Rn , gc )(σ0 ).

nc Now Theorem 5.2.5 and Theorem 5.2.6 identify SN (λ+n−1+N ) with DN (λ) (up to a normalizing coeﬃcient which only depends on λ). Thus, by Lemma 5.2.7, we can identify RN (λ+n−1−N ) with the induced family DN (Rn , S n−1 ; λ) (up to a polynomial normalizing coeﬃcient). The following theorem summarizes these results.

Theorem 5.4.1. The family RN (λ + n − 1 − N ) coincides with the induced family b DN (λ) = DN (Rn , S n−1 ; λ), up to a rational coeﬃcient in λ. More precisely, DN (Rn , S n−1 ; λ) = 2N kN (λ)RN (λ+n−1−N ),

(5.4.7)

where kN (λ)SN (λ+n−1−N ) = DN (Rn , Rn−1 ; λ). Here k2N (λ) = 2N N !{(−(n−1)−2λ+2N ) · · · (−(n−1)−2λ+4N −2)}# and k2N +1 (λ) = 2N N !{(−(n−1)−2λ+2N +2) · · ·(−(n−1)−2λ+4N )}#. For the formula for kN (λ) we refer to (5.2.57) and (5.2.60).

172

Chapter 5. Intertwining Families

5.4.2 D1b (λ), D2b (λ) and D3b (λ) In the present section, we determine explicit formulas for the induced families b DN (λ) for N ≤ 3. For N = 3, the result allows us to recognize the Chang-Qing operator P3 as a member of the family D3b (λ) in dimension 4. The method is to apply Theorem 5.4.1 and to determine the families RN (λ) of order N ≤ 3 via a calculation of the relevant residues of the extension of the family Mu (λ). Explicit b (λ), at least in principle, can be derived also directly from the formulas for DN identity in Lemma 5.2.7. The method used here, however, is easier to handle. For k = 0, (5.4.2) shows that T1 (λ)(a0 (λ)) = a1 (λ) = 12 λa0 (λ). Therefore, R1 (λ) is given by n−2 ∂ψ λ R1 (λ)(ψ(ρ, b)) = (0, b) + − + ψ(0, b). ∂ρ 2 2 In terms of (r, b)-coordinates, we obtain n−2−λ ∗ 1 ∂ − i , R1 (λ) = − i∗ 2 ∂r 2 where i∗ restricts functions in r to r = 1. Now, by Theorem 5.4.1, D1b (λ) = 2R1 (λ+n−2) = −i∗

∂ + λi∗ . ∂r

(5.4.8)

(5.4.9)

Thus D1b (λ) coincides with D1 (Bn , S n−1 ; λ) (Theorem 6.2.1) for the inner normal ﬁeld −∂/∂r; we recall that in our convention the mean curvature for the inner normal is −1. In particular, n

D1b − +1 : C ∞ (Bn )− n2 +1 → C ∞ (S n−1 )− n2 2 is the boundary operator −i∗

n−2 ∗ n−2 ∗ ∂ ∂ − i = −i∗ − Hi , ∂r 2 ∂r 2

i.e., D1b (− n2 +1) is a Robin boundary condition in the conformal Neumann problem n

P2 (Bn ), D1b − +1 : C ∞ (Bn )− n2 +1 → C ∞ (Bn )− n2 −1 ⊕ C ∞ (S n−1 )− n2 . 2 We continue with the discussion of the second-order family. Theorem 5.4.1 yields the ﬁrst assertion of the following result. Lemma 5.4.3. D2b (λ) = −(2λ+n−3)8R2 (λ+n−3) is induced by D20 (λ) (see (5.3.1)). Moreover, D2b (λ) = −(2λ+n−3)i∗

∂2 ∂r2

+ (2λ+n−3)(2λ−1)i∗

∂ + (ΔS n−1 + λ(1−λ)(2λ+n−2)) i∗ . (5.4.10) ∂r

5.4. Families for (Rn , S n−1 )

173

Proof. It only remains to establish the given formula for D2b (λ). (5.4.3) shows that R2 (λ) is given by n (− 2 +1)(− n2 +2) (− n2 +1) 1 ∗ ∂2 n−2−λ ∗ ∂ i i + T0 (λ) + T1 (λ) + T2 (λ) i∗ . − 2 ∂ρ2 2 ∂ρ 2! 1! But T0 (λ) = 1, T1 (λ) = T2 (λ) =

λ 2

and

1 (ΔS n−1 + λ(λ + 1)(n−4−2λ)) 8(n−3−2λ)

(5.4.11)

by (5.4.2). Now the result follows by a direct calculation. (5.4.10) re-proves the factorization identities n−3 n−1 n 3 D2b − + = ΔS n−1 − i∗ = P2 (S n−1 )i∗ 2 2 2 2 and

2 n

n−1 ∂ ∂ n−1 D2b − +1 = i∗ + Δ + = i∗ ΔRn = i∗ P2 (Rn ). S 2 ∂r2 r ∂r In terms of Laplacians ΔRn , ΔS n−1 and ∂/∂r, we ﬁnd

D2b (λ) = −(2λ+n−3)i∗ΔRn + (2λ+n−2) (ΔS n−1 + λ(1−λ)) i∗ + (2λ+n−3)(2λ+n−2)i∗

∂ . (5.4.12) ∂r

In Section 6.3, we shall recognize the family D2b (λ) as a special case of a general construction of a conformally covariant second-order family. Notice that ∂/∂r is the (exterior) unit normal geodesic vector ﬁeld. We continue with the discussion of D3b (λ). Theorem 5.4.1 implies the ﬁrst part of the following result. Lemma 5.4.4. D3b (λ) = −16(2λ+n−5)R3 (λ+n−4) is induced by D30 (λ). Moreover, 1 ∂3 ∂2 −D3b (λ) = − (2λ+n−5)i∗ 3 + (2λ+n−5)(λ−1)i∗ 2 3 ∂r ∂r ∂ ∗ ∂ − (λ−1) ((λ−1)(2λ+n−5) + (λ−2)) i + ΔS n−1 i∗ ∂r ∂r 1 ∗ ∗ − λΔS n−1 i + (2λ+n−2)λ(λ−1)(λ−2)i . 3 D3b (λ) is a polynomial family of degree 4 with leading term 23 λ4 . Since there are only two factorization identities at disposal, these do not suﬃce to determine the family.

174

Chapter 5. Intertwining Families

Proof. By deﬁnition, R3 (λ) is given by the sum 1 ∗ ∂3 i + 3! ∂ρ3

+ +

(− n−2 1 ∗ ∂2 2 )1 T0 (λ) + T1 (λ) i 1! 2! ∂ρ2

(− n−2 (− n−2 ∂ 2 )2 2 )1 T0 (λ) + T1 (λ) + T2 (λ) i∗ 2! 1! ∂ρ

(− n−2 (− n−2 (− n−2 2 )3 2 )2 2 )1 T0 (λ) + T1 (λ) + T2 (λ) + T3 (λ) i∗ , 3! 2! 1!

where T0 (λ) = 1, T1 (λ) = λ2 , T2 (λ) is given by (5.4.11), and T3 (λ) =

1 λ ΔS n−1 + ΔS n−1 12(n−4−2λ) 2 2(λ+2)(n−6−2λ) (ΔS n−1 + λ(λ+1)(n−4−2λ)) . + 8(n−3−2λ)

The latter formula for T3 (λ) can be simpliﬁed to λ+2 1 T3 (λ) = ΔS n−1 + λ(λ+1)(n−6−2λ) . 16(n−3−2λ) 3 Hence (by some calculation) 1 ∗ ∂3 n−2−λ ∗ ∂ 2 i i − 6 ∂ρ3 4 ∂ρ2

∂ 1 ΔS n−1 + (n−4−2λ)(λ−n+3)(λ−n+2) i∗ + 8(n−3−2λ) ∂ρ

(− n +1)(− n +2)(− n +3) (− n +1)(− n +2) λ 2 2 2 2 2 + + 6 2 2 n−2 1 − ΔS n−1 + λ(λ+1)(n−4−2λ) 2 8(n−3−2λ) λ+2 1 λ(λ+1)(λ+2)(n−6−2λ) ∗ ΔS n−1 + i . + 16(n−3−2λ) 48 (n−3−2λ)

R3 (λ) =

Another calculation yields the formula

∂ 1 ∗ ∂3 2−λ ∗ ∂ 2 1 i i ΔS n−1 − (λ−1)(λ−2)(n−4+2λ) i∗ − − 3 2 6 ∂ρ 4 ∂ρ 8(n−5+2λ) ∂ρ λ 1 1 Δ n−1 i∗ + (n−2+2λ)λ(λ−1)(λ−2)i∗ − 8(n−5+2λ) 2 S 48(n−5+2λ)

5.4. Families for (Rn , S n−1 )

175

for R3 (λ+n−4). Multiplying the latter formula with 16(n−5+2λ), gives 8 ∂3 ∂2 ∂ (2λ+n−5)i∗ 3 − 4(2−λ)(2λ+n−5)i∗ 2 + 2(λ−1)(λ−2)(2λ+n−4)i∗ 3 ∂ρ ∂ρ ∂ρ 1 ∗ ∂ ∗ − 2ΔS n−1 i − λΔS n−1 i + (2λ+n−2)λ(λ−1)(λ−2)i∗. ∂ρ 3 Finally, we use the identities ∂ψ ∂ϕ (1, b) = − (0, b), ∂ρ ∂r 2 2 ∂ ϕ ∂ϕ ∂ ψ (0, b), (0, b) − 4 2 (1, b) = ∂ρ ∂r2 ∂r ∂3ψ ∂3ϕ ∂ 2ϕ ∂ϕ 8 3 (1, b) = − 3 (0, b) + 3 2 (0, b) − 3 (0, b) ∂ρ ∂r ∂r ∂r 2

to return to the coordinates (r, b).

Lemma 5.4.4 re-proves the factorization identities D3b and

n−5 − 2

= P2 (S

n−1

)D1b

n−5 − 2

n n

D3b − +1 = D1b − −1 P2 (Rn ). 2 2

The following equivalent formula for D3b (λ) in terms of ΔRn , ΔS n−1 and ∂/∂r will be useful in Section 5.4.3. Lemma 5.4.5. # " ∂ 1 −D3b (λ) = − (2λ+n−5) i∗ ΔRn − (3λ+n−4)i∗ΔRn 3 ∂r " 1 ∂ + (2λ+n−2) ΔS n−1 i∗ − (3λ+n−5)ΔS n−1 i∗ 3 ∂r # ∂ + λ(λ−1)(λ−2)i∗ . − (3λ+n−4)λ+(n−3)(n−4) i∗ ∂r In turn, the formula for D3b (λ) becomes somewhat more beautiful if the family is expressed in terms of Yamabe operators on Rn and S n−1 , and D1b (λ) = −∂/∂r + λ. One should compare the following result with (5.3.9).

176

Chapter 5. Intertwining Families

Lemma 5.4.6. 1 1 (2λ+n−5)D1b (λ−2)P2 (Rn ) − (2λ+n−2)P2(S n−1 )D1b (λ) 3 3 n−5 n−2 + λ+ λ+ (2λ+n−3)D1b (λ) 2 2 n−1 1 n−3 n n−1 )− λ+ + (2λ+n−5)(2λ+n−2) P2 (R )−P2 (S λ+ . 3 2 2

− D3b (λ) =

5.4.3 D3b (0) for n = 4 and (P3 , T ) for (B4 , S 3 ) The constant term of the family D3b (λ) and the polynomial Q3 (λ) = D3b (λ)(1) are quantities which are of special interest. Lemma 5.4.5 implies D3b (0) =

(n−4)(n−5) ∂ n−5 ∂ n−2 ΔRn − ΔS n−1 − ΔRn 3 ∂r 3 ∂r 3 (n−2)(n−5) (n−2)(n−3)(n−4) ∂ ΔS n−1 + + 3 3 ∂r

(5.4.13)

(here we have omitted i∗ in order to simplify the statement of the formula). Moreover, we have 1 def Qb3 (λ) = D3b (λ)(1) = − (2λ+n−2)λ(λ−1)(λ−2), 3 and therefore Q˙ b3 (0) = − 23 (n−2). In particular, for n = 4, we ﬁnd 1 ∂ 2 ∂ + ΔS 3 i∗ D3b (0) = − i∗ ΔR4 − ΔS 3 i∗ 3 ∂r 3 ∂r

(5.4.14)

and 2 def Qb3 (λ) = D3b (λ)(1) = − (λ+1)λ(λ−1)(λ−2), 3

4 Q˙ b3 (0) = − . 3

(5.4.15)

For a compact four-manifold M with boundary, Chang and Qing ([68]) discovered a conformally covariant operator P3 (M, ∂M ) : C ∞ (M ) → C ∞ (∂M ) such that P3 (M, ∂M ; e2ϕg) = e−3ϕ P3 (M, ∂M ; g), and introduced an associated curvature invariant T ∈ C ∞ (∂M ). The formulas for P3 and T are rather complicated for general metrics. We refer to Section 6.26 for more details on (P3 , T ). In the special case of a vanishing second fundamental form L, these are given by 1 1 P3 (M, ∂M ; g) = i∗ Ng Δ(M,g) + Δ(Σ,g) i∗ Ng + tr(G) − τM (g) i∗ Ng (5.4.16) 2 3

5.4. Families for (Rn , S n−1 )

177

and

1 Ng τM (g), (5.4.17) 12 where normal derivatives are taken with respect to the inner normal Ng and T (M, ∂M ; g) = −

j Gji = RiN N

(5.4.18)

(see (2.5.18)). We emphasize that in [68] the sign convention for the Laplacian is opposite to ours. The pair (P3 , T ) satisﬁes the fundamental identity e3ϕ T (e2ϕ g) = T (g) + P3 (g)(ϕ). In the cases (S 4 , S 3 ) and (B4 , S 3 ), the operator P3 and the curvature T are given in the following lemma. Lemma 5.4.7. In the models (S 4 , S 3 ) and (B4 , S 3 ), we have the explicit formulas P3 (S 4 , S 3 ; gc ) = and

1 ∗ i grad(H0 )ΔS 4 + ΔS 3 i∗ grad(H0 ) − i∗ grad(H0 ), 2

1 ∂ ∂ − ΔS 3 i∗ , P3 (B4 , S 3 ; gc ) = − i∗ ΔR4 − ΔS 3 i∗ 2 ∂r ∂r

T =0

T = 2.

Proof. For the proof of the ﬁrst part we use (5.4.16) and (5.4.17). The vector ﬁeld grad(H0 ) has length 1 (on S 3 ) and points into the interior of the upper hemisphere. The second fundamental form L vanishes since S 3 is a totally geodesic submanifold. Hence H = 0. Moreover, tr(G) = 3 and τ (S 4 ) = 12. The vanishing of T follows from the constance of curvature of S 4 and the vanishing of L. This proves the assertions in the ﬁrst case. In the second case, we use the formulas for P3 and T as given in Theorem 6.26.1. Now L is the identity, i.e., H = 3. Moreover, tr(G) = 0 and τ (B4 ) = 0. On S 3 the vector ﬁeld ∂/∂r has length 1 and points into the exterior of the ball. This proves the formula for P3 in the second case. Finally, in order to ﬁnd T , we calculate T = 3H 3 −

1 tr(L3 ) = 3 − 1 = 2. 3

The proof is complete. The formula for the ball is also stated in [68]. Corollary 5.4.1. D3c (S 4 , S 3 ; 0) =

2 P3 (S 4 , S 3 ), 3

D3b (B4 , S 3 ; 0) =

2 P3 (B4 , S 3 ). 3

Proof. By (5.3.9), we have D3c (0) =

1 c 2 2 D1 ΔS 4 + ΔS 3 D1c − D1c 3 3 3

178

Chapter 5. Intertwining Families

on S 4 . The formula in the ﬁrst part of Lemma 5.4.7 and D1c = i∗ grad(H0 ) prove the ﬁrst identity. In the case of the ball B4 , we have, by Lemma 5.4.5, 1 ∂ 2 ∂ D3b (0) = − i∗ ΔR4 − (ΔS 3 i∗ + ΔS 3 i∗ ). 3 ∂r 3 ∂r Comparing the latter formula with the second formula of Lemma 5.4.7 completes the proof.

5.5 Automorphic distributions In the present section, we return to the theory of Selberg zeta functions for Kleinian groups as described in Section 1.2. We formulate a result from [150] on the structure of certain spaces of automorphic distributions on S n . It rests on the theory c of the families DN (λ) : C ∞ (S n ) → C ∞ (S n−1 ). The general situation is as follows. We consider a discrete subgroup Γ of Gn+1 = SO(1, n+1)◦ acting on the hyperbolic space Hn+1 = x ∈ Rn+1 | |x| < 1 and its geodesic boundary S n . We assume that Γ is torsion-free and convexcocompact, i.e., Γ acts without ﬁxed points on Hn+1 and with a compact quotient on the convex hull of the limit set Λ(Γ) ⊂ S n . Let Ω(Γ) ⊂ S n be the complement of Λ(Γ). Then the quotient Γ\Ω(Γ) is a compact smooth manifold which can be viewed as the boundary of a compactiﬁcation of the quotient space X n+1 = Γ\Hn+1 . Associated to this situation is the Selberg zeta function ZΓ (s) = det(id −S N ((Pcn−1 )− )e−s|c| ), (s) > δ(Γ) p.p.o. c N ≥0

of the geodesic ﬂow of X n+1 . Here the ﬁrst product runs over its prime periodic orbits. These can be identiﬁed with the closed oriented geodesics in X n+1 . But, in turn, closed oriented geodesics in X n+1 are in bijection with Γ-conjugacy classes of loxodromic elements, i.e., with Γ-orbits of attracting ﬁxed points of elements of Γ. The set of ﬁxed points is dense in Λ(Γ). Therefore, one can regard ZΓ as being naturally associated to the ﬁxed points of the action of Γ on S n . Then the natural question is to describe the divisor of its meromorphic continuation to C in terms of analysis on Λ(Γ). An ingredient of such a theory are theorems which relate the multiplicities of zeros and poles of ZΓ to the dimensions of spaces of Γ-automorphic distributions which are supported on Λ(Γ). This motivates us to describe the spaces C −∞ (Λ(Γ))Γλ , λ ∈ C of Γ-automorphic distributions supported on the limit set. The general problem will not be addressed here since we will be interested in that problem only for

5.5. Automorphic distributions

179

a very special choice of Γ. In fact, we assume that Γ ⊂ Gn+1 is a cocompact subgroup of the subgroup Gn ⊂ Gn+1 which leaves the equatorially embedded sphere S n−1 ⊂ S n invariant. Then the limit set of Γ acting on Hn+1 is Λ(Γ) = S n−1 ⊂ S n . Let X n = Γ\Hn . Theorem 5.5.1. Let Γ ⊂ Gn ⊂ Gn+1 be as above. Then if λ ∈ R the space C −∞ (Λ(Γ))Γλ of automorphic distributions is non-trivial iﬀ λ ∈ −μ−1−N0, where μ(n−1−μ) is an eigenvalue of the Laplacian on the compact hyperbolic manifold X n . Here any eigenvalue gives rise to two values for μ. More precisely, the space C −∞ (Λ(Γ))Γ−μ−1−N , N ∈ N0 has ﬁnite dimension, and is spanned by the distributions c (μ+N −(n−1))∗(ω), δN

where for each eigenfunction u of the Laplacian on X for the eigenvalue μ(n−1−μ), there is a canonically associated distribution ω ∈ C −∞ (S n−1 )Γ−μ . The distributions ω in Theorem 5.5.1 are the boundary distributions of (automorphic) eigenfunctions in the sense of the theory of Poisson transformations. Note that λ ∈ R implies μ ∈ R. Hence μ ∈ n−1 2 + iR. For λ ∈ R (in particular, for integers λ), the situation is more complicated due to possible topological contributions. We brieﬂy describe the main idea of the proof. The ﬁrst step is to prove that the relevant distributions arise in the range of the residues of an extension operator extλ : C ∞ (Ω(Γ))Γλ → C −∞ (S n )Γλ which extends Γ-automorphic functions on Ω(Γ) through the limit set to automorphic distributions on the sphere S n . The second step is to prove that this construction is exhaustive. The construction of the family extλ is very close to the residue method in the present work. It is also related to scattering theory of the hyperbolic manifold X n+1 . For details see [50] and [151]. n (see Section 2.2) to In order to deﬁne extλ , we ﬁrst use H0 ∈ C ∞ (S n )G 1 identify C ∞ (Γ\Ω(Γ)) C ∞ (Ω(Γ))Γ0 u → |H0 |λ u ∈ C ∞ (Ω(Γ))Γλ . Now for (λ) > −1 and u ∈ C ∞ (Γ\Ω(Γ)) deﬁne Γ extλ (|H0 |λ u), ϕ = |H0 |λ uπ∗,−λ−n (ϕ)db, ϕ ∈ C ∞ (S n ) Γ\Ω(Γ)

180

Chapter 5. Intertwining Families

with the average Γ π∗,λ (ϕ) =

πλc (γ)(ϕ).

γ∈Γ

The restriction on λ is imposed in order to guarantee the convergence of the series Γ (ϕ). No further condition is needed since Γ\Ω(Γ) is compact. But π∗,−λ−n Γ π∗,λ : C ∞ (S n ) → C ∞ (Ω(Γ))Γλ

is well deﬁned if (λ) < −δ(Γ) = −(n−1). Now we observe that for (λ) > −1, we have extλ (|H0 |λ u) = Mu (λ). In fact, using a fundamental domain F for the action of Γ on Ω(Γ), we calculate extλ (|H0 | u), ϕ =

λ

F

=

=

γ∈Γ

γ∈Γ

=

γ −1 (F )

γ∈Γ

γ −1 (F )

γ∈Γ

γ −1 (F )

γ∗ (db) n +1 λ

(|H0 | u)

λ

db

γ ∗ (|H0 |λ u) γ ∗ (|H0 |)λ u

γ∗ (ϕ)db

db γ ∗ (db) γ ∗ (db) db

ϕ

γ ∗ (db) db db

− nλ ϕdb

|H0 |λ uϕdb

|H0 |λ uϕdb = Mu (λ), ϕ.

=

nλ +1

(5.5.1)

Ω(Γ)

Since u is bounded (recall that Γ acts cocompactly on Ω(Γ)), the latter formula again proves that extλ (|H0 |λ u) is well deﬁned for (λ) > −1. By construction, extλ (|H0 |λ u) ∈ C −∞ (S n )Γλ . We are interested in a meromorphic continuation of the extension operator. The importance of such a continuation is due to the following simple but crucial observation. Since the composition resΩ(Γ) ◦ extλ of extλ with the restriction of distributions to the open set Ω(Γ) is the identity on C ∞ (Ω(Γ))Γλ , it follows that at a simple pole of extλ we have resΩ(Γ) ◦ rg Resλ (ext) = 0, i.e.,

rg Resλ (ext) ⊂ C −∞ (Λ(Γ))Γλ .

The theory for general convex-cocompact Γ provides a meromorphic continuation of extλ via its relation to the scattering operator of X n+1 ([50], [151]). In

5.5. Automorphic distributions

181

the present special case, the situation, however, is much easier. In fact, we use the relation (5.5.1) and a direct formal analysis of Mu (λ) to make the existence of the meromorphic continuation plausible and to derive formulas for the residues. The problem for general u ∈ C ∞ (Γ\Ω(Γ)) can be decomposed into the easier problems for eigenfunctions of the Laplacian on Ω(Γ) for the hyperbolic metric gN = |H0 |−2 gc . Since Γ operates with a compact quotient on Ω(Γ), the spectral decomposition of the Laplacian implies the desired result for general u. Now let u ∈ C ∞ (Γ\Ω(Γ)) C ∞ (Γ\Ω+ (Γ)) ⊕ C ∞ (Γ\Ω− (Γ)) C ∞ (Γ\Hn ) ⊕ ∞ C (Γ\Hn ) be given by an eigenfunction for the Laplacian of the constant curvature metric gN for the eigenvalue μ(n−1−μ) on Ω+ (Γ). We set u = 0 on Ω− (Γ). As before, we use a formal asymptotics of u near ker(H0 ) = Λ(Γ) = S n−1 to argue that Mu (λ), ϕ has a meromorphic continuation to C with simple poles in the ladder −μ−1−N0 and residues of the form Resλ=−μ−1−N (Mu (λ), ϕ) = ω, δN (μ+N −(n−1))(ϕ)S n−1

(5.5.2)

n

with certain G -equivariant families c δN (λ) : C ∞ (S n )λ → C ∞ (S n−1 )λ−N .

Here we have to use generalized asymptotic expansions of eigenfunctions with distributional coeﬃcients in C −∞ (S n−1 ) ([233]). The distribution ω ∈ C −∞ (S n−1 )Γ−μ is the leading term a0 in the generalized asymptotics n−1−μ+j μ+j H0 aj (μ) + H0 bj (μ). u∼ j≥0

j≥0

It is straightforward to verify the asserted Γ-invariance of ω = a0 . Now (5.5.2) gives the result c Resλ=−μ−1−N (Mu (λ), ϕ) = δN (μ+N −(n−1))∗(ω), ϕS n ,

i.e.,

c Resλ=−μ−1−N (Mu (λ)) = δN (μ+N −(n−1))∗(ω)

c (see also Lemma 6.5.1). Here we deﬁne the adjoint family δN (λ)∗ by c c δN (λ)∗ (ω), ϕS n = ω, δN (λ)ϕS n−1 , ϕ ∈ C ∞ (S n ). c Then δN (λ)∗ deﬁnes a Gn -equivariant operator

C −∞ (S n−1 )−λ+N −(n−1) → C −∞ (S n )−λ−n . In particular, we see that c δN (μ+N −(n−1))∗(ω) ∈ C −∞ (S n )Γ−μ−1−N

with support in the limit set S n−1 . In [150], the above arguments are replaced by arguments using analysis on Hn+1 .

Chapter 6

Conformally Covariant Families 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 the Euclidean metric on Rn+1 , (S n+1 , S n ) with the round metric on S n+1 and (Bn+1 , S n ) with the Euclidean metric on Bn+1 (Chapter 5) are now viewed as special cases in a much wider framework. The three model cases already indicate some aspects of the inﬂuence of curvature on the structure of such families. The ﬁrst model is Riemannian ﬂat, and the resulting families are not inﬂuenced by curvature, i.e., they are algebraic objects. The families in the second model reﬂect the non-trivial curvature of M and Σ in terms of higher polynomial degree. In contrast to these two models, the third model deals with a hypersurface with a non-vanishing second fundamental form. This leads to a further increase of the polynomial degree of the associated families. In the present chapter, we substantially reﬁne these observations. We start with a review of the contents. We shall call a family critical if its order coincides with the dimension of the target manifold Σ. In Section 6.1, we associate to any critical conformally covariant family Dn (M, Σ; g, λ) the fundamental pair

˙ n (M, Σ; g, 0)(1) . (Pn (M, Σ; g), Qn (M, Σ; g)) = Dn (M, Σ; g, 0), D We show that the conformal covariance of the family implies that the associated fundamental pair satisﬁes an identity which resembles the fundamental identity for the critical Q-curvature Qn of (Σn , g). We consider some low order special cases, and detect the GJMS-operator Pn (S n ) and Qn (S n ; gc ) in the family Dnc (S n+1 , S n ; λ). Of course, this leaves open the existence of critical conformally covariant families. The remainder of the chapter will be devoted to the discussion of two major constructions of such families and their consequences on Q-curvature.

184

Chapter 6. Conformally Covariant Families

In Section 6.2, we discuss the ﬁrst-order family D1 (M, Σ; g; λ). Section 6.3 and Section 6.4 are devoted to a discussion of an elementary construction of second-order families for an oriented codimension one submanifold Σn and a general background metric g. Section 6.3 deals with the critical case n = 2 and Section 6.4 describes the generalization to n ≥ 3. The residue method generates equivariant families from the asymptotics of eigenfunctions in the model cases Rn → Rn+1 , S n → S n+1 and S n → Rn+1 . In Section 6.5, we introduce a version of that method if (Σ, h) is the boundary of [0, ε) × Σ with the metric dr2 + hr so that r−2 (dr2 + hr ) is a Poincar´e-Einstein extension of h. In particular, we introduce the critical residue family Dnres (h; λ). It satisﬁes Dnres (h; 0) = Pn (Σ; h)i∗ . An evaluation of the relation n

−(−1) 2 D˙ nres (h; 0)(1) = Qn (h)

(6.0.1)

leads to the formula n

2 −1

( n2 −j −1)! ∗ ∗ n n ˙ ˙ 2(−1) Qn = Pn (0) − Pn (0) (1) + 2 ! P2j (0)(vn−2j ) 2 j=0 22j j! n 2

for the Q-curvature (Theorem 6.6.2). It expresses Qn (h) in terms of the holographic coeﬃcients v2j , which describe the asymptotics of the volume form of the Poincar´eEinstein metric, and the structure of harmonic functions for the corresponding Laplacian, up to the divergence term

δn (1) = P˙ n (0) − P˙ n∗ (0) (1). The result that the divergence term δn (1) also can be written in terms of the ∗ operators P2j (0) and the coeﬃcients v2j leads to the more explicit version n 2

Qn = (−1) 2

n−1

n

2 −1

( n2 −j)! ∗ −1 ! P (0)(vn−2j ) 2 22j j! 2j j=0

n

(6.0.2)

of the holographic formula (Theorem 6.6.6). The latter formula has an independent proof. This fact can be seen as a proof of the relation (6.0.1) used to derive it. The details are discussed in Section 6.6. (6.0.2) expresses Q-curvature Qn (h) of (Σ, h) in dimension n in terms of the asymptotic geometry of an associated Poincar´e-Einstein metric r−2 (dr2 + hr ) in n + 1 dimensions. Thus from a physical perspective, (6.0.2) says that Qn (h) can be viewed as part of an n-dimensional hologram of geometry in dimension n+1. This motivates the terminology. For closed Σ, the holographic formula (6.0.2) reproduces the result of Graham and Zworski ([128]) that the total Q-curvature is proportional to the integrated holographic anomaly. Moreover, it is compatible with a formula of Branson den scribing the contribution of Δ 2 −1 (J) to Qn , and it implies Gover’s formula ([110])

Chapter 6. Conformally Covariant Families

185

for the Q-curvature of a (Riemannian) Einstein metric. These issues are discussed in Section 6.6 and Section 6.16. Along the way, we derive formulas for the Paneitz operator P4,n and the curvature function Q4,n (from Chapter 4) using formal asymptotic series for eigenfunctions for the Poincar´e metric (see Remark 6.6.5). Moreover, we introduce the Q-polynomial Qres n (λ). In Section 6.7, we prove that in the framework of Section 6.6 the families of Section 6.4 coincide with the second-order residue families. In Section 6.8, we analyze the critical order 3 residue families. In order to make the holographic formulas more explicit, formulas for the holographic coeﬃcients v2j are required. In Section 6.9, we derive such formulas for the ﬁrst few of them: v2 , v4 and v6 . These formulas have been stated in one form or another in the literature (usually without proofs). Here we give proofs which rest on explicit formulas for the ﬁrst three terms in the Feﬀerman-Graham expansion (Theorem 6.9.1). In the formula for v6 , a version of the Bach-tensor in dimension n > 4 contributes. It is caused by the contribution of the Bach tensor to the term h(4) in the Feﬀerman-Graham expansion for the Poincar´e metric. The latter result is naturally reﬂected also in the curvature tensor of the FeﬀermanGraham ambient metric ([99], [96]). The results are used to derive explicit formulas for the ﬁrst two terms in the asymptotics of eigenfunctions of the Laplacian for the Poincar´e-Einstein metric. Finally, we formulate an extension of the holographic formula to all subcritical Q-curvatures (Conjecture 6.9.1). In Section 6.10, we give an alternative proof of the holographic formulas for the critical Q-curvature Q6 . Here we derive an explicit formula Q6 from the asymptotics of eigenfunctions of the Laplacian for the Poincar´e-Einstein metric. The method works in all dimensions n ≥ 6 and yields formulas for the subcritical Q6,n (n > 6) as well. We prove that the resulting formula for Q6,n coincides with the one derived in [116]. Finally, these formulas are used to prove the holographic formula n n

−3 (v4 ) − 2P4∗ −3 (v2 ) Q6,n = −8 · 48v6 − 32P2∗ 2 2 for the subcritical Q6,n (Theorem 6.10.4). The latter result supports Conjecture 6.9.1. The results in Section 6.11 have central signiﬁcance. They are consequences res (h; λ) of the fact that, for a conformally ﬂat metric h, the residue family D2N (2N ≤ n) satisﬁes a system of 2N factorization identities (Theorem 6.11.1). These identities are curved analogs of those for the family D2N (λ) of homomorphisms of Verma modules (Theorem 5.2.2). Partial results support the conjecture that the factorizations hold true for general metrics. The factorization identities give rise to recursive relations for Q-curvature, Q-polynomials and GJMS-operators. We describe some typical results. res (λ) exceeds its The fact that the number of factorization identities for D2N degree as a polynomial implies recursive relations which allow us to express the critical GJMS-operator Pn in terms of lower order GJMS-operators. In particular,

186

Chapter 6. Conformally Covariant Families

we ﬁnd a formula for P6 (h) in terms of P4 (h), P2 (h) and P¯4 (h) = P4 (dr2 +hr ) and P¯2 (h) = P2 (dr2 +hr ) (Corollary 6.11.6). One part of the discussion of recursive formulas for Q-curvatures is concerned with the recursive representation Qn =

|I|≤ n 2 −1

n

aI P2I (Qn−2|I| ) + (−1) 2 −1 (n)

(n−2)!! ∗ ¯ n2 −1 ¯ i P2 (Q2 ) (n−3)!!

(6.0.3)

of the critical Q-curvature in terms of lower order Q-curvature and lower order GJMS-operators (Conjecture 6.11.1). For n = 4 and n = 6, it takes the respective explicit forms ¯2) Q4 = P2 (Q2 ) − 2i∗ P¯2 (Q (6.0.4) and Q6 =

2 5 2 8 ¯ 2) P2 (Q4 ) + − P22 + P4 (Q2 ) + i∗ P¯22 (Q 3 3 3 3

(6.0.5)

(Theorem 6.11.5). The latter formula is equivalent to the explicit formula for Q6 in Theorem 6.10.3 (for n = 6). It holds true for all metrics. For conformally ﬂat h, the existence of the presentation (6.0.3) for the critical Q-curvature is a consequence of (6.0.1) combined with the factorization identities for residue families and analogs of (6.0.3) for all lower order, i.e., subcritical, Q-curvatures. Here a crucial role is played by the distinguished feature of the presentation (6.0.3) to hold true also for subcritical Q-curvatures. We refer to this property as universality. The conjectural status of (6.0.3) is due to the conjectural universality. For an alternative description of the coeﬃcients aI we refer to [95]. Furthermore, the factorization identities imply a recursive formula for the critical Q-polynomial n res 2 Qres n (λ) = −(−1) Dn (λ)(1) in terms of Qn , lower order GJMS-operators and lower order Q-polynomials (Theorem 6.11.10). Then Qres n (0) = 0, and Qn is its linear coeﬃcient. In addition, Qn contributes to all coeﬃcients of the polynomial Qres n (λ). More precisely, the holon graphic formula Q˙ res n (0) = Qn is only one identity among 2 similar ones which relate Qn and vn . In particular, for the critical Q6 we ﬁnd 6 Q6 = [P2 (Qres 4 (2)) + 2P4 (Q2 )] − 6 [Q4 + P2 (Q2 )] Q2 − 2 3!v6 ,

(6.0.6)

where Qres 4 (λ) = −λQ4 − (λ+1)P2 (Q2 ). (6.0.6) is both holographic (in view of the term v6 ) and recursive. In Section 6.13, we derive an analog of (6.0.6) for Q8 .

Chapter 6. Conformally Covariant Families

187

In Section 6.12, we apply the relation (6.0.6) to derive the following recursive formula for the critical GJMS-operator P6 : P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 48δ(P2 #du) − 8δ(B#du)

(6.0.7)

(Corollary 6.12.1). A similar formula yields the subcritical P6 (Corollary 6.12.2). The astonishing simplicity of these formulas suggests that we ask for their analogs for higher order GJMS-operators. In Section 6.13, we speculate on such a universal recursive formula for the GJMS-operators P8 which generalizes the formula P8 = 6P2 P6 − 32P22 P4 + 9P42 + 18P24 + 3!4!P2 on all round spheres S n . For an alternative but not explicit construction of a conformally covariant cube of the Laplacian see [56]. Finally, in Section 6.11 we discuss the meaning of the quadratic coeﬃcient in e-Einstein metrics (Theorem 6.11.15 Qres n (λ) for the renormalized volume of Poincar´ and Theorem 6.11.16). A part of the picture remains conjectural (see Conjecture 6.11.2). For conformally ﬂat metrics, explicit formulas for the holographic coeﬃcients v2j (Lemma 6.14.1) lead to a more explicit version of the holographic formula (6.0.2) (Theorem 6.14.1). It shows that the Euler form naturally splits oﬀ as the non-divergence part. This result suggests splitting oﬀ, for general metrics, the contributions n n Δj tr(∧ 2 −j (P)), j = 0, . . . , − 1 2 in Qn . The study of variational problems deﬁned by the traces tr(∧j (P)) was initiated in [235], and has led to an extensive literature around the σk -Yamabe problem (see [135], [134], [217]). By the classiﬁcation of Deser and Schwimmer (conjectured in [81] and established by Alexakis in [5], [6] and [3]), it is natural to relate vn in dimension n to the Pfaﬃan. This can be seen as an analog of the fact that the leading coeﬃcient in Weyl’s tube formula for a submanifold of Rn is given by the integral of the Pfaﬃan. In Section 6.14, we emphasize this analogy between the holographic coeﬃcients v2j and the coeﬃcients in Weyl’s formula for the volume of tubes. In Section 6.15, we discuss the Henningson-Skenderis test of the AdS/CFT duality which relates the holographic conformal anomaly v4 to a linear combination of quantum conformal anomalies. The calculations rest on explicit formulas for heat-equation coeﬃcients (conformal index densities). In Section 6.16, we discuss the meaning of the holographic formula for Qcurvature for Einstein metrics h. In that case, it is possible to calculate the Qcurvature explicitly. In fact, Gover ([110]) proved that Qn (h) = (n−1)!

τ (h) n(n−1)

n2 .

188

Chapter 6. Conformally Covariant Families

This generalizes the formula Qn (S n , gc ) = (n−1)!. Here we prove that the latter formula is a direct consequence of the holographic formula. The argument rests on an explicit formula for the Poincar´e-Einstein metric which also enables us to derive a formula ([110]) for the critical GJMS-operator for the Einstein metric. In the present special case, the relation to Weyl tube invariants is made explicit. In the remaining sections, we shall use a version of Eastwood’s method of curved translation ([19], [88], [90], [20]) in order to construct curved analogs of DN (λ) in full generality. As a preparation, we brieﬂy recall in Section 6.17 the relation between conformally covariant diﬀerential operators and homomorphisms of generalized Verma modules. This leads to a lifting problem (to semi-holonomic Verma modules) which has a negative solution within the framework of critical GJMS-operators. We formulate an analogous problem for families. In Section 6.18, we solve the lifting problem for all families DN (λ) using a version of Zuckerman translation ([243]). Here the key observation is that DN (λ) can be obtained by iterated translation. In particular, all families DN (λ) can be constructed from the embedding D0 (λ) = i and D1 (λ) = Yn− i. Now translation of homomorphisms of Verma modules has an analog for homomorphisms of semi-holonomic Verma modules. This yields the desired lifts. Moreover, we work out explicit formulas for the lifts of families of orders 2, 3 and 4. The resulting formulas in the semiholonomic category quickly become quite complicated with increasing order. In addition, we prove the existence of other lifts. The identiﬁcation of DN (λ) as an iterated Zuckerman translation is the key to the construction of the tractor family. The idea is to replace the embeddings and projections by tractor D-operators. In Section 6.19, we prepare the translation from Lie theory to the curved framework. The necessary tools from the conformal tractor calculus are developed in Section 6.20. The central objects are the conformally invariant tractor connection and the conformally covariant tractor D-operator. The discussion in these sections is self-contained, and can be read as an introduction to tractor calculus. After these preparations, in Section 6.21, we deﬁne the tractor families T (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ), DN

and formulate their relations to GJMS-operators and Q-curvature. To some extent, the material here is conjectural. The outlined perspective will be explicated in the following sections by a list of particular results. We summarize the main points. First of all, the quality of the situation depends on the parity of the dimension n of the target manifold Σ. T (·, Σ; g; λ) is regular at λ = − n2 + N . Its value For odd n, the family D2N n at λ = − 2 + N is a conformally covariant diﬀerential operator on Σ which only depends on i∗ (g). The critical family DnT (·, Σ; g; λ) has a simple pole at λ = 0. Its residue is a conformally covariant diﬀerential operator. For a vanishing residue, the value of the family at λ = 0 deﬁnes a conformally covariant operator. The value on the constant function 1 of its derivative at λ = 0 deﬁnes a notion of odd

Chapter 6. Conformally Covariant Families

189

order Q-curvature which satisﬁes a fundamental identity. It involves the value of the family at λ = 0. That notion of odd order Q-curvature is an invariant of the embedding. The obstructing residue is expected to vanish if the trace-free part L0 of L vanishes. This is the case if g = dr2 +hr on (0, ε) × Σn so that r−2 (dr2 +hr ) is a Poincar´e metric with conformal inﬁnity h0 = h. For another notion of odd order Q-curvature we refer to [97]. For even n, the behaviour of tractor families depends on the order. The critical family DnT (·, Σn ; g; λ) has a simple pole at λ = 0. However, T (·, Σ; g; λ) has its residue is expected to vanish. More generally, the family D2N a simple pole at λ = − n2 + N if N > n2 (supercritical case). The non-vanishing residue can be viewed as an obstruction to the existence of a conformally invariant power ΔN of the Laplacian on the manifold Σn . T (·, Σ; g; λ) is regular at λ = In the subcritical case N < n2 , the family D2N n ∗ − 2 + N . Its value only depends on i (g) and is given by a conformally covariant power of the Laplacian. In the critical case 2N = n, its value at λ = 0 is more complicated and depends on the embedding (Conjecture 6.21.3). The critical family DnT (·, Σn ; g; λ) induces the function QTn (·, Σn ; g; λ) = DnT (·, Σn ; g; λ)(1). For even n, we expect that QTn (·, Σn ; g; 0) = 0 and n Q˙ Tn (·, Σn ; g; 0) = −(−1) 2 (Qn (i∗ (g)) + Qen (g)) ,

where Qn (i∗ (g)) is a version of Q-curvature of (Σ, i∗ (g)). The additional term Qen depends on the embedding, and we shall call it the extrinsic Q-curvature of the embedding. It satisﬁes a fundamental identity which involves a conformally covariant operator Pne . Analogous results are expected in the supercritical case (if the obstructions vanish). For (M, Σ) = (Rn+1 , Rn ) with the Euclidean metric gc , the residue families nc and the tractor families both specialize to D2N (λ). Such relations continue to hold true for certain curved cases. In particular, we prove the holographic duality res T D2N (h; λ) = D2N (dr2 +hr ; λ)

for conformally ﬂat h (Theorem 6.21.2). We expect that the duality holds true for more general metrics h (possibly in a modiﬁed form). In Section 6.22, we collect results along the lines of the perspectives developed in Section 6.21. In particular, we prove that for n = 4, the value D4T (·, Σn ; g; − n2 + 2) does not depend on the metric oﬀ Σ and, more precisely, is given by the Paneitz operator P4 (i∗ (g))i∗ (Theorem 6.22.2). In the critical case n = 4, we prove that D4T (·; Σ4 ; g; λ) is regular at λ = 0, determine its value at λ = 0, and study the quantity Q˙ T4 (·, Σ4 ; g; 0). We prove that the intrinsic Paneitz pair (P4 , Q4 ) for (Σ4 , g) naturally appears in these data together with an additional extrinsic pair (P4e , Qe4 )

190

Chapter 6. Conformally Covariant Families

for which explicit formula are derived. It turns out that the extrinsic pair is characterized completely by the conformally invariant symmetric bilinear form 1 def J = (i∗ (PM ) − PΣ ) + HL − H 2 g 2 on Σ. J is conformally invariant in all dimensions n ≥ 3 and can be generalized to submanifolds of higher codimension. The trace of the invariant J is proportional to the Chen-Willmore quantity H 2 − τe |L0 |2 . It is related to the conformally invariant fundamental forms of Fialkow ([100]). In Section 6.23, we describe that relation, and prove that, for a hypersurface Σn , it can be written in the form 1 1 |L0 |2 g + c23 (C) , J = L20 − n−2 2(n−1) where c23 (C)ij = CiN N j . It follows that, for a hypersurface Σn → S n+1 in the M¨ obius space (S n+1 , gc ), the invariant J vanishes iﬀ Σ is totally umbilic (L0 = 0). More generally, for a conformally ﬂat background metric on M 5 , the extrinsic pair (P4e , Qe4 ) vanishes iﬀ Σ is totally umbilic. J and the extrinsic pair vanish for a background metric which is given by a Poincar´e-Einstein metric which extends a given metric on Σ (in these cases all three individual terms in the deﬁnition of J vanish, or equivalently, Σ is totally umbilic and C vanishes). This emphasizes the special role of the latter metrics. We prove directly that for such a background metric g the tractor Q-polynomial QT4 (·, Σ4 ; g; λ) actually coincides with the Q-polynomial introduced in Section 6.6. In Section 6.24, we prove that the second-order family D2 (M, Σ; g; λ) constructed in Section 6.4 actually is a composition of conformally covariant operators in terms of tractors. Moreover, we ﬁnd a generalization of the second-order family which acts on standard tractors instead of functions. This family appears in the proofs in Section 6.21. In Section 6.25 and Section 6.26, we prove that the order 3 family naturally yields the Chang-Qing pair (P3 , T ) (i.e., an order 3 operator together with a curvature invariant) if L0 = 0. In this case, there is only an extrinsic fundamental pair.

6.1 Fundamental pairs and critical families Here we describe a pattern by which any critical conformally covariant family Dn (M, Σ; g; λ) of diﬀerential operators gives rise to a conformally covariant linear diﬀerential operator and a curvature quantity with a conformal transformation law which is governed by the corresponding linear operator. Such pairs are called fundamental pairs. Let M be a manifold of dimension n+1 and let i : Σ → M be a hypersurface. For any Riemannian metric g, we ﬁx a unit normal vector ﬁeld N (g) on Σ. Let Dn (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ)

6.1. Fundamental pairs and critical families

191

be a holomorphic family of natural diﬀerential operators which is conformally covariant in the sense that ∗

e−(λ−n)i

◦ Dn (M, Σ; e2ϕ g; λ) ◦ eλϕ = Dn (M, Σ; g; λ)

(ϕ)

(6.1.1)

for all ϕ ∈ C ∞ (M ) and all λ ∈ C. For the following it will be enough to assume that the family is holomorphic at λ = 0. We deﬁne the fundamental pair

(6.1.2) (Pn (M, Σ; g), Qn (M, Σ; g)) = Dn (M, Σ; g, 0), D˙ n (M, Σ; g, 0)(1) consisting of a diﬀerential operator C ∞ (M ) → C ∞ (Σ) and a scalar function in C ∞ (Σ). The following simple but important observation is a direct consequence of the conformal covariance of the family. Lemma 6.1.1. For all ϕ ∈ C ∞ (M ), ∗

eni and

∗

eni

(ϕ)

(ϕ)

Pn (M, Σ; e2ϕ g) = Pn (M, Σ; g)

Qn (M, Σ; e2ϕ g) = Qn (M, Σ; g) − [Pn (M, Σ; g), ϕ] (1).

Proof. The ﬁrst assertion is obvious. (6.1.1) is equivalent to Dn (M, Σ; e2ϕ g; λ) = e(λ−n)ϕ ◦ Dn (M, Σ; g; λ) ◦ e−λϕ . Diﬀerentiating the latter identity with respect to λ at λ = 0 we get D˙ n (e2ϕ g; 0)(u) = ϕe−nϕ Dn (g; 0)(u) + e−nϕ D˙ n (g; 0)(u) − e−nϕ Dn (g; 0)(ϕu). Hence, for u = 1, we obtain D˙ n (e2ϕ g; 0)(1) = e−nϕ D˙ n (g; 0)(1) − e−nϕ [Dn (g; 0), ϕ] (1), i.e., enϕ Qn (M, Σ; e2ϕ g) = Qn (M, Σ; g) − [Pn (M, Σ; g), ϕ] (1).

The proof is complete.

Moreover, if Pn (M, Σ; g) annihilates constants, i.e., if Pn (M, Σ; g)(1) = 0, then

∗

eni

(ϕ)

Qn (M, Σ; e2ϕ g) = Qn (M, Σ; g) − Pn (M, Σ; g)(ϕ)

(6.1.3)

∞

for all ϕ ∈ C (M ). For even n, (6.1.3) resembles the fundamental identity n

enϕ Qn (Σ, e2ϕ g) = Qn (Σ; g) + (−1) 2 Pn (Σ; g)(ϕ), ϕ ∈ C ∞ (Σ)

(6.1.4)

for Q-curvature on Σ (see (3.1.6)). Although Qn (M, Σ; g; λ) and Qn (Σ; g) are curvature quantities on Σ, it is important to emphasize that (6.1.3) and (6.1.4) deal with diﬀerent situations: in (6.1.4) the metric g and the functions ϕ are

192

Chapter 6. Conformally Covariant Families

objects on Σ, but in (6.1.3) g and ϕ live in a neighbourhood of Σ in M . However, if Pn (M, Σ; g) = Dn (M, Σ; g; 0) lives on Σ, i.e., if it is of the form Pn (g)i∗ for a natural diﬀerential operator Pn on Σ, then (6.1.3) and (6.1.4) can be expected to be related. This case was discussed in Section 1.8 (for even n). Now we discuss three examples of fundamental identities from the point of view of the above pattern. Example 6.1.1. For n = 1, let D1 (g; λ) = i∗ ∇N (g) − λκ(g)i∗ : C ∞ (M ) → C ∞ (Σ), where κ(g) is the geodesic curvature of Σ in the surface M which corresponds to the choice of N (g). Then D1 (g; λ) is conformally covariant, i.e., e−(λ−1)ϕ ◦ D1 (e2ϕ g; λ) ◦ eλϕ = D1 (g; λ), ϕ ∈ C ∞ (M ). We diﬀerentiate the identity with respect to λ at λ = 0 and ﬁnd the well-known relation eϕ κ(e2ϕ g) = κ(g) + i∗ ∇N (g) (ϕ) (6.1.5) on Σ. Note that D1 (g; 0) = i∗ does not live on Σ. For the details we refer to Section 6.2. Example 6.1.2. Let n = 2 and assume that D2 (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ) is a holomorphic family which satisﬁes the identity e−(λ−2)ϕ ◦ D2 (e2ϕ g; λ) ◦ eλϕ = D2 (g; λ) for all λ ∈ C (near λ = 0) and all ϕ ∈ C ∞ (M ). Diﬀerentiating the identity at λ = 0 yields the equation e2ϕ D˙ 2 (e2ϕ g; 0)(1) = D˙ 2 (g; 0)(1) − D2 (g; 0)(ϕ), ϕ ∈ C ∞ (M )

(6.1.6)

if D2 (g; 0)(1) = 0. The latter identity is to be compared with the relation e2ϕ K(e2ϕ g) = K(g) − Δg (ϕ), ϕ ∈ C ∞ (Σ)

(6.1.7)

for the Gauß curvature K = J = τ2 (see (2.5.7)). Theorem 6.3.1 yields a conformally covariant second-order family. For this family, D2 (g; 0) = P2 (i∗ (g))i∗ = Δi∗ (g) i∗ , i.e., D2 (g; 0) lives in Σ. Moreover, the quantity D˙ 2 (g; 0)(1) coincides with K(g). Thus the fundamental identity (6.1.6) coincides with (6.1.7). Special cases are D2c (S 3 , S 2 ; λ) and D2b (B3 , S 2 ; λ). On S 3 we have D2c (0) = 2 ∗ P2 (S )i = ΔS 2 i∗ and D˙ 2c (0)(1) = 1 (Lemma 5.3.1). The Gauß curvature of S 2 is 1. On B3 we have seen that D2b (0) = ΔS 2 i∗ and D˙ 2 (0)(1) = 1 ((5.4.12)).

6.1. Fundamental pairs and critical families

193

Example 6.1.3. Let n = 3 and assume that D3 (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ) is a family which satisﬁes the identity e−(λ−3)ϕ ◦ D3 (e2ϕ g; λ) ◦ eλϕ = D3 (g; λ) for all λ ∈ C (near λ = 0) and all ϕ ∈ C ∞ (M ). Diﬀerentiating the identity at λ = 0 yields the relation e3ϕ D˙ 3 (e2ϕ g; 0)(1) = D˙ 3 (g; 0)(1) − D3 (g; 0)(ϕ), ϕ ∈ C ∞ (M ) if D3 (g; 0)(1) = 0. We compare the latter identity with the relation e3ϕ T (e2ϕ g) = T (g) + P3 (g)(ϕ), ϕ ∈ C ∞ (M )

(6.1.8)

for the operator P3 and the curvature function T introduced in [62], [68], [73] (see Section 6.26). In Theorem 6.26.2 we will establish the proportionality of both pairs (D3 (g; 0), D˙ 3 (g; 0)(1))

and

(−P3 (g), T (g))

(6.1.9)

for D3 (g; λ) given by the tractor family D3T (g; λ), if the trace-free part L0 of L vanishes. P3 (g) does not live on Σ, i.e., the fundamental pair is associated to the embedding i : Σ → M . This covers the special cases D3c (S 4 , S 3 ; λ) and D3b (B4 , S 3 ; λ). In fact, on S 4 we have • D3c (S 4 , S 3 ; 0) = 23 P3 (S 4 , S 3 ) (Corollary 5.4.1), • D˙ 3c (S 4 , S 3 ; 0)(1) = 0 (by (5.3.9)) and • T = 0 (Lemma 5.4.7). On B4 we have • D3b (B4 , S 3 ; 0) = 23 P3b (B4 , S 3 ) (Corollary 5.4.1), • D˙ b (B4 , S 3 ; 0)(1) = − 4 (by (5.4.15)) and 3

3

• T = 2 (by Lemma 5.4.7). In particular, the proportionality coeﬃcient is −3/2. Finally, we show that the GJMS-operator Pn (S n ) and the Q-curvature Qn (S n ) of (S n , gc ) can be read oﬀ from the family Dnc (S n+1 , S n ; λ) by forming the corresponding fundamental pair. We consider the family c c (λ) = D2N (S n+1 , S n ; λ) : C ∞ (S n+1 ) → C ∞ (S n ). D2N

By Lemma 5.2.11, we have c D2N (λ)(1) = Q2N (λ) = (−1)N λ(λ−1) · · · (λ−2N +1)

(6.1.10)

194

Chapter 6. Conformally Covariant Families

for any n, and by (5.2.25) (for j = N ), we know that n

c D2N − +N = P2N (S n )i∗ . 2 These two results imply n

c P2N (S n )(1) = P2N (S n )(i∗ (1)) = D2N − +N (1)

2 " n

n

# n −N (−1)N −N +1 · · · +N −1 . = 2 2 2 On the other hand, by deﬁnition of Q2N (S n ; gc ),

n −N (−1)N Q2N (S n ; gc ), P2N (S n )(1) = 2 i.e.,

n

n −N +1 · · · +N −1 . Q2N (S n ; gc ) = 2 2 In particular, we obtain Lemma 6.1.2. For even n, Dnc (S n+1 , S n ; 0) = Pn (S n )i∗ and n n D˙ nc (S n+1 , S n ; 0)(1) = (−1) 2 −1 Qn (S n ; gc ) = (−1) 2 −1 (n−1)!.

Proof. By (6.1.10), n

Dnc (S n+1 , S n ; λ)(1) = Qn (λ) = (−1) 2 λ(λ−1) · · · (λ−n+1). Hence n n D˙ nc (S n+1 , S n ; 0)(1) = (−1) 2 (n−1)!(−1)n−1 = (−1) 2 −1 (n−1)!.

Using Qn (S n ; gc ) = (n−1)!, the proof is complete.

c For odd order families, we have D2N +1 (λ)(1) = 0.

6.2 The family D1(g; λ) For any hypersurface i : Σ → M , there is an obvious conformally covariant family of ﬁrst order (in the sense of Section 1.8). Theorem 6.2.1. For a hypersurface Σ → M n (n ≥ 3) and a Riemannian metric g on M , let D1 (M, Σ; g; λ) = i∗ ∇N (g) − λH(g)i∗ , where N (g) is a unit normal vector ﬁeld and H(g) is the corresponding meancurvature (see (1.8.6)). Then D1 (M, Σ; g; λ) is conformally covariant, i.e., ∗

e−(λ−1)i

(ϕ)

◦ D1 (M, Σ; e2ϕ g; λ) ◦ eλϕ = D1 (M, Σ; g; λ)

for all λ ∈ C and all ϕ ∈ C ∞ (M ).

(6.2.1)

6.3. D2 (g; λ) for a surface in a 3-manifold

195

Proof. (6.2.1) is equivalent to ∗

e−(λ−1)i

(ϕ)

i∗ ∇N (e2ϕ g) (u) − λH(e2ϕ g)i∗ u = e−λϕ −λi∗ ∇N (g) (ϕ)u + i∗ ∇N (g) (u) − λH(g)i∗ u

for all u ∈ C ∞ (M ). Using N (e2ϕ g) = e−ϕ N (g), the latter identity, in turn, is equivalent to (6.2.2) H(e2ϕ g) = e−ϕ H(g) + ∇N (g) (ϕ) . In order to prove (6.2.2), we recall that ˆ X Y − ∇X Y = dϕ, XY + dϕ, Y X − g(X, Y ) grad(ϕ) ∇ (see (2.5.1)). The convention L(X, Y ) = −g(∇X (Y ), N ) implies ˆ Y ) − L(X, Y ) = g(X, Y )g(grad(ϕ), N ). e−ϕ L(X, Hence

ˆ − trg (L) = (n−1)∇N (g) (ϕ), e−ϕ trg (L)

i.e., eϕ H(ˆ g) − H(g) = ∇N (g) (ϕ). This proves (6.2.2). The proof is complete.

Remark 6.2.1. For a curve Σ in a surface M , an analogous critical family is given by D1 (M, Σ; g; λ) = i∗ ∇N (g) − λκ(g)i∗ . Here

˙ N (g)) ∈ C ∞ (Σ) κ(g) = g(∇c˙ (c),

(c is a natural parametrization of Σ) is the corresponding geodesic curvature. In that case, the conformal covariance of the family is equivalent to eϕ κ(e2ϕ g) = κ(g) + ∇N (g) (ϕ).

(6.2.3)

In the following, the identity (6.2.2), i.e., eϕ H(e2ϕ g) = H(g) + ∇N (g) (ϕ)

(6.2.4)

will be called the fundamental identity for H.

6.3 D2 (g; λ) for a surface in a 3-manifold Let i : Σ2 → M 3 be an oriented surface in a 3-manifold M . In the present section, we construct a second-order polynomial family D2 (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ) of diﬀerential operators so that e−(λ−2)ϕ ◦ D2 (M, Σ; e2ϕ g; λ) ◦ eλϕ = D2 (M, Σ; g; λ)

(6.3.1)

196

Chapter 6. Conformally Covariant Families

for all λ ∈ C and all ϕ ∈ C ∞ (M ), and D2 (M, Σ; g; 0) = P2 (Σ; g)i∗ ,

D˙ 2 (M, Σ; g; 0)(1) = Q2 (Σ; g).

(6.3.2)

This is a critical case: the order of the family coincides with the dimension of the target surface Σ. In Section 6.24, we will use tractor calculus for an alternative construction of its non-critical version (Theorem 6.4.1). Theorem 6.3.1. Let Ng be a unit geodesic normal vector ﬁeld in a neighborhood of Σ. Then the natural family D2 (M, Σ; g; λ) = −2λ3 Hg2 i∗ + λ2 (4Hg i∗ Ng + cg i∗ ) + λ −2i∗ Ng2 − 2Hg i∗ Ng + Kg i∗ + Δ(Σ,g) i∗ ,

(6.3.3)

with cg = 2Kg − Hg2 − 12 τg , satisﬁes (6.3.1) and (6.3.2). In Theorem 6.3.1 it is important to choose the geodesic extension of the unit normal vector ﬁeld on Σ to an open neighborhood. The choice of the extension inﬂuences the second normal derivative in the coeﬃcient of λ. The following result yields an equivalent formula in terms of Yamabe operators of (M, g) and (Σ, g). Theorem 6.3.2. Let Ng be a unit normal vector ﬁeld on Σ. Then D2 (M, Σ; g; λ) = −2λi∗ P2 (M, g) + (2λ+1)P2 (Σ, g)i∗ + 2λ(2λ+1)Hg i∗ Ng + λ(2λ+1) Q2 (Σ; g) − Q2 (M ; g) − λHg2 i∗ .

(6.3.4)

An advantage of Theorem 6.3.2 is that it does not require a choice of the normal ﬁeld in a neighbourhood of Σ. The second-order normal derivatives are covered by the Yamabe operator on M . Although the sign of the mean curvature depends on the choice of a normal vector ﬁeld, the family D2 (M, Σ; g; λ) does not depend on such a choice. If the mean curvature H vanishes, the polynomial degree of the family is at most 2. The coeﬃcient of λ2 is a combination of curvature contributions of both metrics and mean curvature. The family is of degree 1 iﬀ H = 0 and Q2 (M ; g) restricts to Q2 (Σ; g). Example 6.3.1. For the standard sphere Σ = S 2 ⊂ R3 , we have Hc = 1, Kc = 1 and ∂/∂nc = ∂/∂r. Therefore, (6.3.3) specializes to D2b (R3 , S 2 ; λ) (Lemma 5.4.3). Moreover, we recognize (5.4.12) for D2b (R3 , S 2 ; λ) as a special case of (6.3.4) (for g = gc ). Example 6.3.2. (6.3.3) and (6.3.4) reproduce the family D2c (λ) : C ∞ (S 3 ) → C ∞ (S 2 ). In fact, in this case H = 0, K = 1 and τ (S 3 ) = 6. Hence (6.3.3) yields the operator ∂2 ∂2 −λ2 + λ −2 2 + 1 + ΔS 2 = −2λ 2 + (ΔS 2 − λ(λ−1)) ∂n ∂n

6.3. D2 (g; λ) for a surface in a 3-manifold

197

(see (5.3.3)), where the normal derivative is the Lie derivative with respect to a unit length geodesic normal vector ﬁeld. On the other hand, (6.3.4) gives 6 ∗ −2λi∗ P2 (S 3 ) + (2λ+1)P2 (S 2 )i∗ + λ(2λ+1) 1− i 4 (see Lemma 5.3.1). The polynomial Q2 (M, Σ; g; λ) = D2 (M, Σ; g; λ)(1) will be called the Qpolynomial. Analogous polynomials will be associated to higher order families in Section 6.6. Theorem 6.3.1 implies Corollary 6.3.1. Q2 (M, Σ; g; λ) = λ

−2Hg2 λ2 + −Hg2 + 2Q2 (Σ; g) − 2Q2 (M ; g) λ + Q2 (Σ; g) .

The Q-polynomial Q2 (M, Σ; g; λ) has no constant term and the coeﬃcient of λ is Q2 (Σ; g). Moreover, the coeﬃcient of λ2 admits the following interpretation. We write −cg in the form

τ 2 g Hg − K g + K g − K g + (6.3.5) − Kg , 2 where K g denotes the sectional curvature of the planes T Σ ⊂ T M with respect to the metric g on M . Now the volume form 2 Hg − Kg + K g vol(g) ∈ Ω2 (Σ) (6.3.6) is conformally invariant. In fact, by the Gauß equation, the diﬀerence Kg − K g coincides with the product of the principal curvatures and thus Hg2 − Kg + K g is 14 times the squared diﬀerence of the principal curvatures. That observation implies the conformal invariance of (6.3.6). Integration of (6.3.6) (for compact Σ) yields 2 Kg vol(g) = W (Σ, g) − 2πχ(Σ), Hg + K g vol(g) − Σ

Σ

where W denotes the Willmore functional. The integral of the second term in (6.3.5) is conformally invariant, too. The third term in (6.3.5) gives rise to the form

τ g − K g vol(g) ∈ Ω2 (Σ). 2 τg j In view of 2 = K g +tr(Gg ), where G denotes the tensor Gji = RiN N (see (2.5.18)), the latter volume form can also be written in the form tr(Gg ) vol(g) ∈ Ω2 (Σ), i.e., we have proved the conformal invariance of the integral 1¨ Q2 (g; 0) + tr(Gg ) vol(g). Σ 2 We continue with the proof of Theorem 6.3.1. As a preparation, we recall the following well-known result.

198

Chapter 6. Conformally Covariant Families

Lemma 6.3.1. For (M, Σ; g) and geodesic normals, i∗ Δ(M,g) = i∗

∂2 ∂ + 2Hg i∗ + Δ(Σ,g) i∗ . ∂n2g ∂ng

Now we derive a formula for the conformal change of second-order geodesic normal derivatives Ng2 = ∂ 2 /∂n2g . The following results are valid in all dimensions. Lemma 6.3.2. For gˆ = e2ϕ g, Ngˆ = e−ϕ Ng and Ngˆ2

=e

−2ϕ

∂ϕ Σ 2 Ng + gradg (ϕ) Ng − ∂ng

as identities of diﬀerential operators C ∞ (M ) → C ∞ (Σ). Proof. The ﬁrst formula is obvious. For the proof of the second assertion we use Δ(M,ˆg) = e−2ϕ Δ(M,g) + e−2ϕ gradM g (ϕ) (see Lemma 4.2.1) and apply Lemma 6.3.1 to both sides. Then

Ngˆ2 + 2Hgˆ Ngˆ + Δ(Σ,ˆg ) = e−2ϕ Ng2 + 2Hg Ng + Δ(Σ,g) + gradM g (ϕ) . The assertion follows from Hgˆ Ngˆ = e−2ϕ (Hg + Ng (ϕ)) Ng .

(6.3.7)

The proof is complete. Using (6.3.3), we obtain e(λ−2)ϕ D2 (M, Σ; g; λ)(e−λϕ u) = λ3 e−2ϕ −2Hg2 u ∂u ∂ϕ u+ + cg u + λ2 e−2ϕ 4Hg −λ ∂ng ∂ng 2 2 ∂ ϕ ∂ϕ ∂ϕ ∂u ∂ 2u −2ϕ 2 + λe 2λ 2 u + 4λ − 2 2 − 2λ u ∂ng ∂ng ∂ng ∂ng ∂ng ∂ϕ ∂u + 2λHg u − 2Hg + Kg u ∂ng ∂ng + e(λ−2)ϕ Δ(Σ,g) (e−λϕ u)

6.3. D2 (g; λ) for a surface in a 3-manifold

8 3 −2ϕ

= −2λ e

Hg2

∂ϕ + 2Hg + ∂ng

199

∂ϕ ∂ng

2 ; u

2 ∂ϕ ∂u ∂ ϕ ∂u ∂ϕ +λ e + 4Hg + 2Hg u + cg u 2 2u+4 ∂ng ∂ng ∂ng ∂ng ∂ng ∂u ∂ 2u + λe−2ϕ −2 2 − 2Hg + Kg u ∂ng ∂ng 2 −2ϕ

+ e(λ−2)ϕ Δ(Σ,g) (e−λϕ u). In order to evaluate the term which involves the Laplacian on Σ, we apply Lemma 6.3.3.

Σ Δ(Σ,g) (eλϕ u) = Δ(Σ,g) (eλϕ )u + 2λeλϕ gradΣ + eλϕ Δ(Σ,g) (u) g (ϕ), gradg (u) g

and

12 1 1 1 Δ(Σ,g) (eλϕ ) = λeλϕ Δ(Σ,g) (ϕ) + λ2 eλϕ 1gradΣ g (ϕ)1 . g

and ﬁnd 12 1 1 1 e(λ−2)ϕ Δ(Σ,g) (e−λϕ u) = −λe−2ϕ Δ(Σ,g) (ϕ)u + λ2 e−2ϕ 1gradΣ g (ϕ)1 u g

Σ Σ −2ϕ −2ϕ gradg (ϕ), gradg (u) + e − 2λe Δ(Σ,g) (u). g

Thus we obtain the identity e(λ−2)ϕ D2 (M, Σ; g; λ)(e−λϕ u) (6.3.8) 8 2 ; ∂ϕ ∂ϕ + u = −2λ3 e−2ϕ Hg2 + 2Hg ∂ng ∂ng 2 1 12 ∂u ∂ϕ ∂ϕ ∂u ∂ ϕ 1 1 Σ 2 −2ϕ +λ e + 4Hg + 2Hg u + 1gradg (ϕ)1 u + cg u 2 2u+4 ∂ng ∂ng ∂ng ∂ng ∂ng g

2 ∂u ∂ u Σ Σ −2ϕ + λe − Δ(Σ,g) (ϕ)u − 2 gradg (ϕ), gradg (u) + Kg u −2 2 − 2Hg ∂ng ∂ng g + e−2ϕ Δ(Σ,g) (u). We claim that the coeﬃcients of the powers of λ in (6.3.8) coincide with the coeﬃcients of the corresponding powers in D2 (M, Σ; gˆ; λ). First of all, the identity e−2ϕ Δ(Σ,g) = Δ(Σ,ˆg ) on the surface Σ establishes the assertion for the constant term. The linear term of D2 (ˆ g ; λ) is given by −2

∂ ∂2 − 2Hg + Kgˆ . 2 ∂ngˆ ∂ngˆ

200

Chapter 6. Conformally Covariant Families

Now (6.3.7) and Kgˆ = e−2ϕ (Kg − Δ(Σ,g) (ϕ)),

(6.3.9)

together with Lemma 6.3.2, show that the linear term equals −2e−2ϕ

∂2 ∂ −2ϕ − 2e−2ϕ Hg − 2e−2ϕ gradΣ (Kg − Δ(Σ,g) (ϕ)). g (ϕ) + e ∂n2g ∂ng

The last sum coincides with the coeﬃcient of λ in (6.3.8). (6.2.4) also shows that the coeﬃcients of λ3 coincide. Thus it only remains to identify the coeﬃcients of λ2 . In other words, it only remains to prove that 12 " ∂2ϕ 1 # ∂ϕ ∂ ∂ϕ ∂ 1 1 e−2ϕ 2 2 + 1gradΣ (ϕ) + 2H + 4 + 4H + c 1 g g g g ∂ng ∂ng ∂ng ∂ng ∂ng g ∂ = 4Hgˆ + c(ˆ g) ∂ngˆ on Σ. (6.3.7) shows that the latter identity is equivalent to 1 12 " ∂2ϕ ∂ϕ 1 # 1 1 2 + 1gradΣ e−2ϕ 2 2 + 2Hg g (ϕ)1 + 2Kg − Hg − τg ∂ng ∂ng 2 g

1 = 2Kgˆ − Hgˆ2 − τgˆ 2

(6.3.10)

on Σ. But (6.2.4), (6.3.7), (6.3.9) and (2.5.6) imply 2Kgˆ −

Hgˆ2

2 ; ∂ϕ 2Kg − 2Δ(Σ,g) (ϕ) − Hg + ∂ng 1 12 1 1 1 + e−2ϕ 2Δ(M,g) (ϕ) + 1gradM − e−2ϕ τg g (ϕ)1 2 g 1 12 " 1 ∂ϕ 1 1 = e−2ϕ 2Kg − Hg2 − τg − 2Hg + 1gradM g (ϕ)1 2 ∂ng g 2 # ∂ϕ − + 2Δ(M,g) (ϕ) − 2Δ(Σ,g) (ϕ) ∂ng 1 12 " ∂ϕ # 1 ∂ 2ϕ 1 1 (ϕ) = e−2ϕ 2Kg − Hg2 − τg + 1gradΣ 1 + 2 2 + 2Hg g 2 ∂ng ∂ng g

1 − τgˆ = e−2ϕ 2

8

using Lemma 6.3.1. This proves (6.3.10) and the proof of Theorem 6.3.1 is complete. Theorem 6.3.2 is a direct consequence of Theorem 6.3.1 and Lemma 6.3.1. We omit the details.

6.4. Second-order families. General case

201

6.4 Second-order families. General case In the present section, we generalize Theorem 6.3.2 to oriented codimension one submanifolds Σn of an arbitrary manifold M n+1 of dimension n + 1 (n > 2) with a background metric g. The arguments are extensions of those in Section 6.3. Theorem 6.4.1. The family D2 (M, Σ; g; λ) = −(2λ+n−2)i∗P2 (M, g) + (2λ+n−1)P2(Σ, g)i∗ + (2λ+n−1)(2λ+n−2)Hgi∗ Ng n−1 n−2 + 2 λ+ Q2 (Σ; g) − Q2 (M ; g) − λHg2 i∗ λ+ 2 2

(6.4.1)

is conformally covariant, i.e., e−(λ−2)ϕ ◦ D2 (M, Σ; e2ϕ g; λ) ◦ eλϕ = D2 (M, Σ; g; λ)

(6.4.2)

for all ϕ ∈ C ∞ (M ). The family λ → D2 (g; λ) is of degree 3 and satisﬁes the factorization relations

n−1 n ∗ D2 g; − +1 = P2 (Σ, g)i and D2 g; − = i∗ P2 (M, g). 2 2 Example 6.4.1. Let (M, Σ) = (S n+1 , S n ) with the round metric g = gc . Then (6.4.1) specializes to D2 (S n+1 , S n ; gc ; λ) = −(2λ+n−2)i∗P2 (S n+1 , gc )

n−1 n−2 ∗ + (2λ+n−1)P2(S , gc )i − λ+ λ+ i 2 2 n

using Hgc = 0 and Q2 (S m ; gc ) = (see (5.3.2)).

m 2,

∗

i.e., the family coincides with D2c (S n+1 , S n ; λ)

Example 6.4.2. Let (M, Σ) = (Rn+1 , S n ) with the Euclidean metric g = gc . Then (6.4.1) specializes to

n n −1 i∗ D2 (Rn+1 , S n ; gc ; λ) = −(2λ+n−2)i∗ΔRn+1 + (2λ+n−1) ΔS n − 2 2 + (2λ+n−1)(2λ+n−2)i∗∂/∂r

n−2 n n−1 λ+ − λ i∗ + 2 λ+ 2 2 2 ∗ = −(2λ+n−2)i ΔRn+1 + (2λ+n−1) (ΔS n + λ(1 − λ)) i∗ + (2λ+n−1)(2λ+n−2)i∗∂/∂r

using H = 1 for the exterior normal derivative ∂/∂r. Hence the family coincides with D2b (Rn+1 , S n ; gc ; λ) (see (5.4.12)).

202

Chapter 6. Conformally Covariant Families

In Section 6.24, the families D2 (M, Σ; g; λ) will be recognized as tractor families. These are curved versions of families which are induced by homomorphisms of Verma modules. In particular, the latter observation that D2 (M, Σ; g; λ) specializes to the induced families D2c (λ) and D2b (λ) ﬁnds its natural explanation. In order to prove Theorem 6.4.1, we have to verify four identities for the four Taylor coeﬃcients of both sides of (6.4.2) at λ = − n2 + 1. Here we restrict to a detailed proof for the coeﬃcients of the powers 0, 1 and 3. The remaining identity can be proved similarly. An alternative proof will be given in Section 6.24. It relates the family D2 (M, Σ; g; λ) to a construction in terms of the conformally invariant tractor calculus. More precisely, D2 (M, Σ; g; λ) can be written as the composition of certain universal conformally covariant families (tractor D-operators). The fact that the coeﬃcients of (λ+ n2 −1)0 on both sides of (6.4.2) coincide is equivalent to

n n n n e−( 2 +1)ϕ ◦ D2 M, Σ; gˆ; − +1 ◦ e(− 2 +1)ϕ = D2 M, Σ; g; − +1 . 2 2 But this follows from D2 (M, Σ; g; − n2 +1) = P2 (Σ, g)i∗ . Next, we ﬁnd that on the left-hand side of (6.4.2) the coeﬃcient of (λ+ n2 −1)3 is given by < 2 = 2 ∂ϕ ∂ϕ ∂ϕ 2 2ϕ −2Hgˆ + 4Hgˆ −2 e2ϕ . e = −2 Hgˆ − ∂ngˆ ∂ngˆ ∂ngˆ In view of

∂ϕ ∂ϕ ϕ = e Hgˆ − Hg = e Hgˆ − , ∂ng ∂ngˆ ϕ

this coincides with the corresponding coeﬃcient −2Hg2 of the right-hand side of (6.4.2). Next, we diﬀerentiate (6.4.2) at λ = − n2 + 1 and ﬁnd

n n n n [P2 (Σ, g), ϕ] + e( 2 +1)ϕ ◦ D˙ 2 gˆ; − +1 ◦ e(− 2 +1)ϕ = D˙ 2 g; − +1 . (6.4.3) 2 2 On the other hand, (6.4.1) yields

n D˙ 2 g; − +1 = −2Δ(M,g) + 2Δ(Σ,g) 2

n

∂ −1 Hg2 . + 2Hg + (n−2)Q2 (M ; g) − (n−3)Q2 (Σ; g) + ∂ng 2

(6.4.4)

We prove that (6.4.3) is satisﬁed by the latter operator. In the ﬁrst step we verify (6.4.3) for u = 1. Let

n q2 (g) = D˙ 2 g; − +1 (1) ∈ C ∞ (Σ). 2

6.4. Second-order families. General case

203

Then q2 (g) = (n−2)Q2(M ; g) − (n−3)Q2(Σ; g) +

n 2

−1 Hg2 ∈ C ∞ (Σ).

(6.4.5)

In these terms, the assertion is

n ∂ n q2 (g) = Δ(Σ,g) (ϕ) + e( 2 +1)ϕ −2Δ(M,ˆg) + 2Δ(Σ,ˆg ) + 2Hgˆ + q2 (ˆ g ) e(− 2 +1)ϕ , ∂ngˆ i.e., 2 n ∂ ∂ (− n2 +1)ϕ

e + q2 (ˆ + (n−1)H g)e2ϕ q2 (g) = Δ(Σ,g) (ϕ) − 2e( 2 +1)ϕ g ˆ ∂n2gˆ ∂ngˆ using the appropriate analog of Lemma 6.3.1 (with geodesic normals). We apply Lemma 6.3.2 in order to make the latter formula more explicit. It follows that the diﬀerential operator on its right-hand side equals Δ(Σ,g) (ϕ) − 2e

(n 2 −1)ϕ

∂2 ∂ ∂ϕ ∂ Σ (− n 2 +1)ϕ e , + (n−1)H + (n−2) + grad (ϕ) g g ∂n2g ∂ng ∂ng ∂ng

i.e., n

Δ(Σ,g) (ϕ) − 2e( 2 −1)ϕ

∂ 2 (− n +1)ϕ

e 2 ∂n2g

+ (n−1)(n−2)Hg

∂ϕ + (n−2)2 ∂ng 2

∂ϕ ∂ng

2

1 12 1 1 + (n−2) 1gradΣ (ϕ) 1 g 2

2

g

∂ϕ ∂ ϕ (n−2) + ∂n2g 2 ∂ng 1 12 ∂ϕ 1 1 + (n−1)(n−2)Hg + (n−2) 1gradΣ g (ϕ)1 . ∂ng g

= Δ(Σ,g) (ϕ) + (n−2)

Therefore, the claim is q2 (g) = q2 (ˆ g )e

2ϕ

2

∂ 2ϕ (n−2)2 ∂ϕ + Δ(Σ,g) (ϕ) + (n−2) 2 + ∂ng 2 ∂ng 1 12 ∂ϕ 1 1 + (n−1)(n−2)Hg + (n−2) 1gradΣ (ϕ) 1 . g ∂ng g

(6.4.6)

But the latter identity (6.4.6) is a consequence of Yamabe’s equations (see (3.1.9)) for the scalar curvatures of M and Σ, and the fundamental identity for the mean curvature. In fact, we recall that, by Yamabe’s equation, n+2 n−2 n−2 n−2 −2ϕ τX (ˆ e g) = τX (g) − e− 2 ϕ Δ(X,g) (e 2 ϕ ) 4(n−1) 4(n−1)

204

Chapter 6. Conformally Covariant Families

on an n-manifold X. We write this in the form (n−2)Q2 (X; gˆ) = (n−2)Q2 (X; g)e−2ϕ − 2e−

n+2 2 ϕ

Δ(X,g) (e

n−2 2 ϕ

).

In particular, we have (for n > 2) (n−2)Q2 (M ; gˆ) = (n−2)Q2 (M ; g)e−2ϕ − 2

n−1 n−2 − n+3 ϕ e 2 Δ(M,g) (e 2 ϕ ) n−1

(6.4.7)

and (n−3)Q2(Σ; gˆ) = (n−3)Q2 (Σ; g)e−2ϕ − 2

n−2 n−3 − n+2 ϕ e 2 Δ(Σ,g) (e 2 ϕ ). n−2

(6.4.8)

Now, by Lemma 6.3.3, (6.4.7) is equivalent to (n−2)Q2(M ; gˆ)e2ϕ = (n−2)Q2 (M ; g) − (n−2)Δ(M,g) (ϕ) −

12 (n−1)(n−2) 11 1 1gradM g (ϕ)1 . 2 g

Similarly, (6.4.8) is equivalent to (n−3)Q2(Σ; gˆ)e2ϕ = (n−3)Q2(Σ; g) − (n−3)Δ(Σ,g) (ϕ) −

12 (n−2)(n−3) 11 1 (ϕ) 1gradΣ 1 . g 2 g

Hence we get 12 (n−1)(n−2) 11 1 M 1gradg (ϕ)1 2 g 12 (n−2)(n−3) 11 1 Σ + (n−3)Δ(Σ,g) (ϕ) + 1gradg (ϕ)1 2 g 2

n ∂ϕ −1 + − Hg2 . Hg + 2 ∂ng

g )e2ϕ = q2 (g) − (n−2)Δ(M,g) (ϕ) − q2 (ˆ

Now using Δ(M,g) =

∂ϕ ∂2ϕ + nHg + Δ(Σ,g) , 2 ∂ng ∂ng

we rewrite the above formula for the diﬀerence q2 (ˆ g )e2ϕ − q2 (g) as ∂ϕ ∂ 2ϕ − (n−2) 2 ∂ng ∂ng 12 (n−2)(n−3) 1 12 (n−1)(n−2) 11 1 1 1 M Σ − 1gradg (ϕ)1 + 1gradg (ϕ)1 2 2 g g 2 ∂ϕ n−2 ∂ϕ + (n−2)Hg + ∂ng 2 ∂ng

− Δ(Σ,g) (ϕ) − n(n−2)Hg

(6.4.9)

6.4. Second-order families. General case

= −Δ(Σ,g) (ϕ) − (n−2)

using

205

∂ϕ ∂ 2ϕ − (n−1)(n−2)Hg ∂n2g ∂ng 2 1 12 (n−2)2 ∂ϕ 1 1 − − (n−2) 1gradΣ (ϕ) 1 g 2 ∂ng g

12 1 12 ∂ϕ 2 1 1 1 1 1 M Σ . 1gradg (ϕ)1 = 1gradg (ϕ)1 + ∂ng g g

The latter sum coincides with the sum in (6.4.6), i.e., (6.4.6) is a consequence of the Yamabe equations and the fundamental identity for Hg . Now having established (6.4.3) for u = 1, it is easy to prove (6.4.3) for all u ∈ C ∞ (M ). In view of (6.4.4) and (6.4.9), we have to prove that ∂ ∂2 − 2(n−1)Hg 2 ∂ng ∂ng < = 2 n ∂ ∂ (n +1)ϕ 2 =e ◦ −2 2 − 2(n−1)Hgˆ + q2 (ˆ g ) ◦ e(− 2 +1)ϕ + Δ(Σ,g) , ϕ . ∂ngˆ ∂ngˆ

q2 (g) − 2

(6.4.10) By the above calculations, the right-hand side of (6.4.10) coincides with g )e2ϕ q2 (ˆ 2 " ∂ 2 ϕ (n−2)2 ∂ϕ + Δ(Σ,g) (ϕ) + (n−2) 2 + ∂ng 2 ∂ng + (n−1)(n−2)Hg < +e

2ϕ

−2

1 12 # ∂ϕ 1 1 + (n−2) 1gradΣ g (ϕ)1 ∂ng g =

∂ ∂ϕ ∂ ∂2 + 2(n−2) − 2(n−1)Hgˆ + 2 gradΣ g (ϕ). ∂n2gˆ ∂ngˆ ∂ngˆ ∂ngˆ

By (6.4.6), the ﬁrst two terms sum up to q2 (g). Therefore, it only remains to prove that ∂ ∂2 + (n−1)Hg ∂n2g ∂ng = e2ϕ

<

= ∂ ∂ϕ ∂ ∂2 − (n−2) + (n−1)Hgˆ − gradΣ g (ϕ). ∂n2gˆ ∂ngˆ ∂ngˆ ∂ngˆ

Now it suﬃces to apply Lemma 6.3.2 and (6.3.7). This proves (6.4.10). We close the present section with some comments. We consider the Q-polynomial Q2 (g; λ) more closely. Theorem 6.4.1 implies the formula

n n Q2 g; − +1 = (n−2)Q2 (M ; g) − (n−3)Q2(Σ; g) + −1 Hg2 . 2 2

206

Chapter 6. Conformally Covariant Families

Hence for n > 2 the function Q2 (g; − n2 + 1) is composed of Q-curvatures and (squared) mean curvature. However, in the critical case n = 2, Q2 (g; 0) is completely determined only by Q2 (Σ; g). It is immediate from the deﬁnition that

1¨ n Q2 g; − +1 = (n−3)Hg2 + 2 (Q2 (Σ; g) − Q2 (M ; g)) . 2 2 Thus we can identify the Q-polynomial. Lemma 6.4.1. The Q-polynomial Q2 (M, Σ; g; λ) is given by

3 n − 2Hg2 λ+ −1 2

2 n 2 + (n−3)Hg + 2Q2 (Σ; g) − 2Q2 (M ; g) λ+ −1 2

n n 2 −1 Hg + (n−2)Q2(M ; g) − (n−3)Q2(Σ; g) λ+ −1 + 2 n2

− −1 Q2 (Σ). 2 This result extends Corollary 6.3.1. We notice that (6.4.1) immediately yields D2 (M, Σ; g; 0)(1) = 0, i.e., D2 (g; 0) annihilates constants. Hence it is also natural to write the Q-polynomial as a Taylor polynomial at λ = 0 (as in Corollary 6.3.1). We ﬁnd Q2 (M, Σ; g; λ) = −2λ3 Hg2 + λ2 2Q2 (Σ; g) − 2Q2 (M ; g) − (2n−3)Hg2 (n−1)(n−2) 2 + λ (n−1)Q2 (Σ; g) − (n−2)Q2 (M ; g) − Hg . (6.4.11) 2 Example 6.4.3. In the case of the trivial embedding M n+1 = R × Σn with the metric gM = dr2 + gΣ , Q2 (M, Σ; gM ; λ) =

2λ(λ+n−1) Q2 (Σ; gΣ ). n

In particular, for n = 2, we ﬁnd Q2 (M, Σ; gM ; λ) = λ(λ+1)Q2 (Σ; gΣ ). Thus Q2 (g; λ) is a quadratic polynomial which is determined by Q2 (g; Σ). Of course, if H = 0 and i∗ Q2 (M ; g) = Q2 (Σ; g) then the degree of the polynomial is only 1. In Section 6.11, we shall consider a class of embeddings with this property. τΣ M , Q2 (gΣ ) = 2(n−1) . Hence Proof. We have H = 0, τM = τΣ and Q2 (gM ) = τ2n (6.4.11) yields 1 τ 1 1 n−2 1 2 Q2 (M, Σ; gM ; λ) = λ − − . τ +λ τ = λ(λ+n−1) n−1 n 2 2n n n−1

This implies the assertion.

6.4. Second-order families. General case

207

Although the quantity (n−1)(n−2) 2 Hg + (n−2)Q2(M ; g) − (n−1)Q2(Σ; g) 2 is not conformally invariant, it is related to a conformal invariant. We recall a result of Chen. Let 2 def λi λj , τe (g) = n(n−1) i<j where λi are the principal curvatures. τe is called the extrinsic scalar curvature. Chen ([76]) observed that 2 Hg − τe (g) g (6.4.12) is a conformal invariant. In fact, H 2 − τe is proportional to the squared norm |L0 |2 of the trace-free part L0 of L (see Lemma 6.22.4). The conformal invariance of the symmetric bilinear form (6.4.12) has been observed long before. In particular, it has been the basis of Fialkow’s profound study of conformal invariants of submanifolds ([100]). For more details see Section 6.23. For n = 2, (6.4.12) reduces to (Hg2 − Kg + K g )g used in Section 6.3. Lemma 6.4.2. (n−1)(n−2) 2 Hg + (n−2)Q2 (M ; g) − (n−1)Q2 (Σ; g) 2 1 (n−1)(n−2) 2 = Hg − τe (g) − (τΣ (g) + (n−2) tr(Gg )) , 2 n i j j where Gi = RiN N . The trace tr(G) = i Gi = i RicN N is the Ricci curvature in the direction of the unit normal. Proof. We have Hg2

2Q2 (M ; g) 2Q2 (Σ; g) − = + n−1 n−2

τM (g) τΣ (g) 2 Hg + . − n(n − 1) (n−1)(n−2)

But the Gauß equation RM (X, Y, Z, W ) = RΣ (X, Y, Z, W ) − L(X, W )L(Y, Z) + L(X, Z)L(Y, W ) for X, Y, Z, W ∈ X (Σ) implies 2 τM = τΣ + 2 tr(G) − λi + λ2i = τΣ + 2 tr(G) − n(n−1)τe . i

(6.4.13)

i

Hence the above sum can be written as 2 2 2 tr(Gg ) − τΣ (g). Hg − τe (g) + n(n−1) n(n−1)(n−2) This proves the assertion.

208

Chapter 6. Conformally Covariant Families

It follows that the diﬀerence 1 Q˙ 2 (M, Σ; g; 0)g − [τΣ (g)−(n−2) tr(Gg )] g n is conformally invariant and coincides with a multiple of (Hg2 − τe (g))g.

6.5 Families and the asymptotics of eigenfunctions We start with some motivating observations in the conformally ﬂat case (S n , S n−1 ) with the canonical metric gc . Let u ∈ C ∞ (H + ) be an eigenfunction −ΔN u = μ(n−1−μ)u,

(μ) = (n−1)/2

H0−2 gc

of the Laplacian ΔN of the metric with boundary value f ∈ C ∞ (S n−1 ) so that n−1−μ+N μ+N H0 aN (f ) + H0 bN (f ) u∼ N ≥0

N ≥0

with a0 (f ) = f . The eigenfunction u gives rise to the holomorphic family Mu (λ), ϕ = H0λ uϕ vol(S n , gc ), ϕ ∈ C ∞ (S n ) H+

of distributions on the half-plane (λ) > − n−1 2 . Lemma 6.5.1. Let the operator family c δN (μ) : C ∞ (S n ) → C ∞ (S n−1 )

be deﬁned by the residue formula Resλ=−μ−1−N (Mu (λ), ϕ) =

S n−1

c f δN (μ+N −(n−1))(ϕ) vol(S n−1 , gc ). (6.5.1)

Then c c c (μ) ◦ πμc (g) = πμ−N (g) ◦ δN (μ), g ∈ Gn . δN

(6.5.2) n

c Proof. The Gn -equivariance of δN (μ) is a consequence of H0 ∈ C ∞ (S n )G 1 . In fact, − n1 g∗ (vol(S n , gc )) c = H0 (6.5.3) π1 (g)(H0 ) = g∗ (H0 ) vol(S n , gc ) implies g∗ (vol(S n , gc )) λ Mu (λ), ϕ = g∗ (H0 ) g∗ (u)g∗ (ϕ) vol(S n , gc ) vol(S n , gc ) H+ λ +1 g∗ (vol(S n , gc )) n λ = H0 g∗ (u) g∗ (ϕ) vol(S n , gc ) vol(S n , gc ) H+ c = H0λ g∗ (u)π−λ−n (g)(ϕ) vol(S n , gc ). (6.5.4) H+

6.5. Families and the asymptotics of eigenfunctions

209

c Now (6.5.3) and the asymptotics u ∼ H0μ f + · · · yield g∗ (u) ∼ H0μ π−μ (g)f + · · · . We apply (6.5.1) to (6.5.4) and obtain the identity c f δN (μ+N −(n−1))(ϕ) vol(S n−1 , gc ) S n−1

c c c n−1 = π−μ (g)(f )δN (μ+N −(n−1)) πμ+N , gc ) −(n−1) (g)(ϕ) vol(S S n−1 c c c n−1 = f πμ−(n−1) (g −1 )δN (μ+N −(n−1))πμ+N , gc ). −(n−1)(g)(ϕ) vol(S S n−1

The proof is complete.

We refer also to (5.2.63) for the analogous results in the non-compact model. (6.5.2) can be rewritten as c c n g ∗ ◦ δN (g) ◦ g ∗ = Φμ−N ◦ δN (μ) ◦ Φ−μ g g , g ∈ G

(6.5.5)

c (μ) as being associated to for Φg so that g∗ (gc ) = Φ2g gc . Now we consider δN c c (gc ; μ). More generally, the metric gc and reﬂect this in the notation δN (μ) = δN c ∗ (6.5.5) suggests to regard g∗ ◦ δN (g) ◦ g as being associated to the metric g∗ (gc ) = c Φ2g gc . Therefore, we write also δN (g∗ (gc ); μ) for this composition. Now the residual c interpretation of δN (μ) (Lemma 6.5.1) suggests the following analogous residual c interpretation of δN (g∗ (gc ); μ).

Lemma 6.5.2. For u as in Lemma 6.5.1 and g ∈ Gn , let Mug (λ), ϕ = Hgλ uϕ vol(S n , g∗ (gc )), H+

where Hg = g∗ (H0 ). Then Resλ=−μ−1−N (Mug (λ), ϕ)

= ×

g∗ (vol(S n−1 , gc )) vol(S n−1 , gc )

μ − n−1

f

S n−1 c δN (g∗ (gc ); μ+N −(n−1))(ϕ) vol(S n−1 , g∗ (gc )).

Proof. In view of the relation 3 4 g Mu (λ), ϕ = H0λ g ∗ (u)g ∗ (ϕ) vol(S n , gc ) = Mg∗ (u) (λ), g ∗ (ϕ) , H+

(6.5.1) yields for the residues Resλ=−μ−1−N (Mug (λ), ϕ) c c = π−μ (g −1 )f δN (μ+N −(n−1))g ∗(ϕ) vol(S n−1 , gc ) S n−1

210

Chapter 6. Conformally Covariant Families

= S n−1

=

S n−1

g ∗ (vol(S n−1 , gc )) vol(S n−1 , gc ) vol(S n−1 , gc ) g∗ (vol(S n−1 , gc ))

μ n−1

μ n−1

c g ∗ (f )δN (μ+N −(n−1))g ∗(ϕ) vol(S n−1 , gc )

c f δN (g∗ (gc ); μ+N −(n−1))ϕ vol(S n−1 , g∗ (gc )).

The proof is complete. Notice that

g∗ (vol(S n−1 , gc )) vol(S n−1 , gc )

μ − n−1

f

is the coeﬃcient of Hgμ in the asymptotics of u. Thus Lemma 6.5.2 states that the c family δN (g∗ (gc ); μ) naturally arises from the asymptotics of u in powers of the function Hg . Now let M be a compact manifold with boundary Σn = ∂M . Assume that on M an asymptotically hyperbolic metric gAH is given. Let ΔAH be the corresponding Laplacian. Then to an eigenfunction u, n −ΔAH u = μ(n−μ)u, (μ) = 2 def

and a conformal compactiﬁcation g¯AH (ρ) = ρ2 gAH , ρ ∈ C ∞ (M ) we associate the family n def ρλ uϕ vol(¯ gAH (ρ)), ϕ ∈ C ∞ (M ), (λ) > − −1 Mu (λ; ρ), ϕ = 2 M of distributions on M . The existence of a formal asymptotic expansion u∼ ρμ+j aj (μ) + ρn−μ+j bj (μ), ρ → 0 j≥0

j≥0

with aj , bj ∈ C ∞ (∂M ) implies the existence of a meromorphic continuation of Mu (λ; ρ) to C with simple poles in the ladders −μ−1−N0,

−(n−μ)−1−N0.

Its residue at λ = −μ−1−N has the form ϕ → a0 δN (μ+N −n; g¯AH (ρ))(ϕ) vol(Σ, ρ),

(6.5.6)

Σ

where vol(Σ, ρ) is the Riemannian volume of the pull-back of g¯AH (ρ) to Σ and δN (·; g¯AH (ρ)) : C ∞ (M ) → C ∞ (Σ) is a family of diﬀerential operators. These families are conformally covariant with respect to a change of the conformal compactiﬁcation. More precisely, let g¯AH (ˆ ρ) = ρˆ2 gAH

6.5. Families and the asymptotics of eigenfunctions

211

be another conformal compactiﬁcation of gAH . Then for (λ) >> 0, def Mu (λ; ρˆ), ϕ = ρˆλ uϕ vol(¯ gAH (ˆ ρ)) M = ρλ uϕeλΦ e(n+1)Φ vol(¯ gAH (ρ)) M

= Mu (λ; ρ), e(λ+n+1)Φ ϕ

(6.5.7)

if ρˆ = eΦ ρ. We use the asymptotic expansions of u in powers of ρ and ρˆ to construct the families δN (μ+N −n; g¯AH (ρ))

and δN (μ+N −n; g¯AH (ˆ ρ)).

(6.5.7) implies that these satisfy the relation a ˆ0 δN (μ+N −n; g¯AH (ˆ ρ))(ϕ) vol(Σ, ρˆ) ∂M = a0 δN (μ+N −n; g¯AH (ρ))(e(−μ+n−N )Φ ϕ) vol(Σ, ρ). ∂M

But a ˆ0 ρˆ = a0 ρ , i.e., a ˆ0 = e−μΦ a0 on ∂M . Hence μ

μ

e(−μ+n)Φ ◦ δN (μ+N −n; g¯AH (ˆ ρ)) ◦ e(μ−n+N )Φ = δN (μ+N −n; g¯AH (ρ)), i.e.,

e−(λ−N )Φ ◦ δN (λ; e2Φ g¯AH (ρ)) ◦ eλΦ = δN (λ; g¯AH (ρ)).

(6.5.8)

(6.5.8) is the claimed conformal covariance. Note that the resulting families only depend on a ﬁnite part of the formal asymptotics of u near ∂M . In the above discussion, we also suppressed a discussion of the continuation in λ. In fact, the families are meromorphic and polynomial families arise by renormalization. Now we regard the families δN (λ; g¯AH (ρ)) as being associated to the conformal class on Σ which is deﬁned by the conformal compactiﬁcations of gAH . That perspective is reasonable if the resulting families depend naturally on that class. In Section 6.6, we shall deal with such a case in detail. Details of the residue method in the cases of the unit ball Bn and the hemisphere H + were given in Section 5.4.1 and Section 5.3. In these two cases, we worked with conformal compactiﬁcations with a large conformal group and we emphasized the equivariance property of the families with respect to the associated group representations. Finally, we illustrate the residue method for a conformal compactiﬁcation of the ball Bn which is natural from the point of view of the geometry of the Poincar´e-Einstein metric. Let 4 gE = (dx21 + · · · + dx2n ) (1−|x|2 )2

212

Chapter 6. Conformally Covariant Families

be the Poincar´e-Einstein metric on Bn . In contrast to the conformal compactiﬁcation g¯E (ρ) = dx21 + · · · + dx2n given by ρ(x) = deﬁned by

1−|x|2 2

(see Section 5.4.1), we consider now the compactiﬁcation ρˆ(x) =

Then g¯E (ˆ ρ) =

1−|x| . 1+|x|

4 (dx21 + · · · + dx2n ). (1 + |x|)4

We consider a formal asymptotic development 1 − |x| μ+j u∼ a ˆj 1 + |x| j≥0

of an eigenfunction −ΔgE u = μ(n−1−μ)u. Since 1 − |x| = exp −d(0, x), 1 + |x| the latter form of the asymptotics is the typical form of radial asymptotics used in harmonic analysis ([140], [233]). In order to determine the asymptotics, we write ΔgE in terms of polar coordinates (r, ·) ∈ (0, 1) × S n−1 and use the substitution n−1 s = 1−r , we have 1+r . In the coordinates (s, ·) ∈ (0, 1) × S 2 2 1−s2 1−s2 1 2 2 gE = 2 ds + gS n−1 , g¯E (ˆ ρ) = ds + gS n−1 , (6.5.9) s 2 2 and the corresponding Laplacian reads 2 4s2 2(n−1) ∂ 2 ∂ s + + n− Δ n−1 . s ∂s2 1−s2 ∂s (1−s2 )2 S ˆj yields The power series ansatz sμ j≥0 sj a j≥0

(6.5.10)

2(n−1) (μ+j)(μ+j −1)sj a ˆj + n− (μ+j)sj a ˆj 1−s2 j≥0

2

+ We write (1−s2 )−1 = a ˆj saying that

n≥0

4s (1−s2)2

sj ΔS n−1 a ˆj = −μ(n−1−μ)

j≥0

sj a ˆj .

j≥0

s2n and obtain a recursive relation for the coeﬃcients N (n−1−2μ−N )ˆ aN

6.5. Families and the asymptotics of eigenfunctions

213

is a linear combination of second-order diﬀerential operators with polynomial coˆN −4 , . . . . Thus the leading coeﬃcient a ˆ0 is free, eﬃcients in μ acting on a ˆN −2 , a a ˆ1 = 0, and all other coeﬃcients are determined by a ˆ0 . In particular, we have −2(n−3−2μ)ˆ a2 = 2(n−1)μˆ a0 − 4ΔS n−1 a ˆ0 , i.e., a ˆ2 =

1 (2ΔS n−1 a ˆ0 − μ(n−1)ˆ a0 ) . n−3−2μ

We use these results to determine δˆj (λ) = δj (λ; g¯E (ˆ ρ)) (j = 1, 2). We rewrite the integral ρˆλ uϕ vol(¯ gE (ˆ ρ)) Bn

ˆ : (0, 1)×S n−1 → Bn , in terms of geodesic normal coordinates based on S n−1 , i.e., α (t, x) → exp ˆ x (tN ), where exp ˆ denotes the exponential map for g¯E (ˆ ρ) and N is a unit normal vector ﬁeld. A simple calculation shows that α ˆ (t, x) =

1−t x. 1+t

Hence α ˆ ∗ (ˆ ρ)(t, x) = t, ∗

∗

gE (ˆ ρ)) = vol(ˆ α (¯ gE (ˆ ρ))) = α ˆ vol(¯

1−t2 2

n−1 dt vol(S n−1 , gc ).

Notice that geodesic normal coordinates coincide with the coordinates deﬁned by ρ). the gradient ﬂow of ρˆ with respect to the metric g¯E (ˆ Now the integral can be written as 1 2n−1

1

tλ uϕ(1−t2)n−1 dtdb.

0

S n−1

ˆ0 + tμ+2 a ˆ2 + · · · and The asymptotics u(t, ·) ∼ tμ a v(t) =

1−t2 2

n−1 = v0 + v2 t2 + · · ·

imply ∂f δˆ1 (μ+2−n)f = v0 ∂t and 1 ∂2f 1 n−1 (2Δ δˆ2 (μ+3−n)f = v0 + v + −(n−1)μ) 2 S 2! ∂t2 n−3−2μ

214

Chapter 6. Conformally Covariant Families

using a calculation similar to that in Section 5.4.1 (here restrictions to t = 0 are omitted in order to simplify the formulation). Thus 2n−1 δˆ1 (λ) =

∂ ∂t

and 2

∂ −2n (2λ+n−3)δˆ2(λ) = −(2λ+n−3) 2 + 4ΔS n−1 + 2λ(n−1). ∂t

(6.5.11)

We note that for n = 3, the family (6.5.11) reads ∂2 λ −2 2 + λ + 4ΔS 2 . ∂t This coincides with the natural family D2 (R3 , S 2 ; g¯E (ˆ ρ); λ) (Theorem 6.3.1). In fact, 1 g¯E (ˆ ρ)|S n−1 = gc , 4 K(S 2 , 14 gc ) = 4, and it is easy to prove that H = 0 and τ = 16 on S 2 .

6.6 Residue families and holographic formulas for Q-curvature We start with a summary of the content of the present section. Here we apply the method of Section 6.5 to the special case of the Poincar´e-Einstein metric gE = r−2 (dr2 +hr ), h0 = h on (0, ε) × M being associated to a given metric h on M (Section 3.2). In particular, the conformal compactiﬁcation g¯E = r2 gE of gE gives rise to the critical residue family Dnres (h; λ) : C ∞ ([0, ε) × M ) → C ∞ (M ).

(6.6.1)

The family Dnres (h; λ) is completely determined by the metric h and satisﬁes Dnres (h; 0) = Pn (h)i∗ . res (h; λ) of order N ≤ n for even n, More generally, we introduce residue families DN and any order for odd n, and show how they transform under conformal changes of the metric. In particular, that law suggests that residue families are specializations of conformally covariant families to g = g¯E . The main result of the present section is

Theorem 6.6.1. Let n be even. Then n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h).

(6.6.2)

6.6. Residue families and holographic formulas for Q-curvature

215

In order to prove Theorem 6.6.1, we evaluate the left-hand side of (6.6.2) (Theorem 6.6.2 and Corollary 6.6.1). Combined with an additional identity (Theorem 6.6.4), it follows that the assertion (6.6.2) is equivalent to a formula which will be called the holographic formula for Qn (Theorem 6.6.6). The latter result has an independent proof which in turn proves (6.6.2). In the remaining part of the section, we make the holographic formulas explicit for n ≤ 6. Let (M n , h) (n even) be a Riemannian manifold. Let gE = r−2 (dr2+hr ) with ¯ (n) + · · · hr = h(0) + r2 h(2) + · · · + rn h(n) + rn log rh be the Poincar´e-Einstein metric on (0, ε)×M n which is associated to h = h(0) = h0 . ¯ (n) has vanishing h-trace. The coeﬃcients h(2) , . . . , h(n−2) , h ¯ (n) and the trace Here h of h(n) are locally determined by h(0) = h0 = h. Now the ansatz rμ+2j a2j (h; μ) u∼ j≥0

for a formal approximate solution of −ΔgE u = μ(n−μ)u determines a sequence of diﬀerential operators Tj (h; μ) so that Tj (h; μ)a0 = aj (h; μ). Since hr is even in r (up to order n), we have Tj (μ) = 0 for odd j ≤ n. Moreover, Γ( n2 −μ−j) P2j (h; μ) 22j j! Γ( n2 −μ) 1 1 P2j (h; μ) = 2j n 2 j! ( 2 −μ−1) . . . ( n2 −μ−j)

T2j (h; μ) =

1

(6.6.3)

(see (3.2.16)), where P2j (h; μ) = Δjh + LOT and Pn (h; 0) = Pn (h). P2j (h; μ) is a polynomial family of diﬀerential operators of order 2j on M . Finally, let v(r, ·) =

vol(hr ) = v0 + r2 v2 + · · · + rn vn + · · · , v0 = 1. vol(h)

Note that there is no term rn log r in the formal Taylor series of v(r, b). In fact, that coeﬃcient coincides with the coeﬃcient of rn log r in det(id +(rn log r)h−1 ¯h(n) ) ¯ (n) ) = 0. which is given by tr(h−1 h Deﬁnition 6.6.1 (The critical residue family). For even n, let n n

! −λ−1 · · · (−λ) δn (h; λ) Dnres (h; λ) = 2n 2 2 with δn (h; λ) =

n j=0

∗ 1 n−j . Tj (h; λ) ◦ v0 + · · · + T0∗ (h; λ) ◦ vj ◦ i∗ ◦ (∂/∂r) (n−j)!

216

Chapter 6. Conformally Covariant Families

Here i∗ restricts functions to r = 0. The family Dnres (h; λ) : C ∞ ([0, ε) × M n ) → C ∞ (M n ) is called the critical residue family. Since Todd = 0 and vodd = 0, we can also write < j = n 2 1 n−2j ∗ δn (h; λ) = T2k (h; λ) ◦ vn−2k ◦ i∗ ◦ (∂/∂r) . (n−2j)! j=0

(6.6.4)

k=0

Theorem 6.6.2. (6.6.2) is equivalent to n

2 −1 n ( n −j −1)! ∗ ∗ ˙ − (−1) Qn (h) = Pn (h; 0)(1) − ! P2j (h; 0)(vn−2j ) 2n−2j 2 2 j=0 j! n n

∗ ! −1 !vn . (6.6.5) (h; 0)(v2 ) − · · · − 2n = P˙ n∗ (h; 0)(1) − 2nPn−2 2 2 n 2

Proof. The family δn (h; μ) : C ∞ ([0, ε) × M ) → C ∞ (M ) arises in the residue formula Resλ=−μ−1−n (Mu (λ; r), ϕ) = f δn (h; μ)(ϕ) vol(h), M

where a0 = f is the leading term in the asymptotics of u (see (6.5.6)) and the test function ϕ has compact support in [0, ε) × M . In fact, repeated partial integration in the integral ε rλ uϕ vol(¯ gE ) = rλ (uv)(r, b)ϕ(r, b)dr vol(h) Mu (λ; r), ϕ = (0,ε)×M

0

M

(see Section 5.4.1 for more details in a special case) yields the formula Resλ=−μ−1−n (Mu (λ; r), ϕ) n 1 ∂ n−j ϕ = [Tj (μ)(f )v0 + · · · + T0 (μ)(f )vj ] n−j (0, b) vol(h) (n−j)! M ∂r j=0 n n−j ∗ ∂ 1 ∗ = f Tj (μ) ◦ v0 + · · · + T0 (μ) ◦ vj (ϕ)(0, b) vol(h). (n−j)! M ∂r j=0 Hence δn (h; μ) n = j=0

∗ 1 n−j . Tj (h; μ) ◦ v0 + · · · + T0∗ (h; μ) ◦ vj ◦ i∗ ◦ (∂/∂r) (n−j)!

It follows that the renormalization n n

! −μ−1 . . . (−μ) δn (h; μ) Dnres (h; μ) = 2n 2 2

(6.6.6)

(6.6.7)

6.6. Residue families and holographic formulas for Q-curvature

217

is a family of the form n

2

Δjh ◦ i∗ ◦ (∂/∂r)

n−2j

j=0 n

=

2n ( n2 )! ( n2 −μ−1) . . . (−μ) 1 + LOT (n−2j)! 22j j! ( n2 −μ−1) . . . ( n2 −μ−j)

(n) n−2j + LOT, aj (μ) Δjh ◦ i∗ ◦ (∂/∂r)

2

j=0

where

n

(n) aj (μ)

−1 2

( n2 )! n 2n−2j −1−k−μ , = j!(n−2j)! 2

(n)

a n (μ) = 1

k=j

2

(see (5.1.4)). In particular, we have Dnres (h; 0) = Pn (h; 0)i∗ = Pn (h)i∗ .

(6.6.8)

Since the families Tn−2 (μ), . . . , T0 (μ) are regular at μ = 0, (6.6.6) implies n

D˙ nres (h; 0)(1)

=

P˙n∗ (h; 0)(1)

2 −1 n ( n −j −1)! ∗ ! P2j (h; 0)(vn−2j ). − 2n−2j 2 2 j=0 j!

Hence (6.6.2) is equivalent to (6.6.5). The proof is complete. Remark 6.6.1. (6.6.2) is equivalent to n

2 −1 n

∗ n n ˙ −(−1) Qn (h) = Pn (h; 0)(1) − 2 −1 ! ! T2j∗ (h; 0)(vn−2j ). 2 2 j=0 n 2

(6.6.9)

More generally, we introduce residue families of order ≤ n. Deﬁnition 6.6.2 (Residue families. General case). For even n and N ≤ n, let res (h; λ) : C ∞ ([0, ε) × M n ) → C ∞ (M n ) DN

be deﬁned as the product of 8 22M M ! − n2 −λ+2M −1 · · · − n2 −λ+M 22M M ! − n2 −λ+2M · · · − n2 −λ+M +1

if N = 2M , if N = 2M +1

(6.6.10)

with δN (h; λ+n−N ), where δN (h; λ) =

N j=0

∗ 1 N −j Tj (h; λ) ◦ v0 + · · · + T0∗ (h; λ) ◦ vj ◦ i∗ ◦ (∂/∂r) . (N −j)!

res The families DN (h; λ) with N < n will be called subcritical residue families.

218

Chapter 6. Conformally Covariant Families

Note that δN (h; λ) satisﬁes

Resλ=−μ−1−N (Mu (λ; r), ϕ) =

f δN (h; μ)(ϕ) vol(h)

(6.6.11)

M

and we have

n res D2N h; − +N = P2N (h)i∗ , 2

n ∂ res D2N +1 h; − +N +1 = P2N (h)i∗ . 2 ∂r

(6.6.12)

The latter two identities are examples of factorization identities (see Remark 6.11.1). res Remark 6.6.2. For even n, residue families D2N (h; λ) are well deﬁned for general res (h; λ) involves the Taylor coefh only if 2N ≤ n. In fact, the deﬁnition of D2N ﬁcients h(0) = h, h(2) , . . . , h(2N ) of hr , where h(2N ) enters only through its trace. ¯ (n) obstructs the conThese terms are completely determined by h. The coeﬃcient h struction of residue families of orders exceeding n. For odd n, residue families are deﬁned analogously for all orders since the construction of the Poincar´e-Einstein metric is not obstructed. We omit the details. res The residue families D2N (h; λ) give rise to Q-polynomials.

Deﬁnition 6.6.3 (Q-polynomials). For a Riemannian manifold (M, h) of even dimension n, the polynomials N res Qres 2N (h; λ) = −(−1) D2N (h; λ)(1), 1 ≤ N ≤

n 2

are called the Q-polynomials of (M, h). Sometimes we shall distinguish between the critical Q-polynomial Qres n (h; λ) res and the subcritical Q-polynomials D2N (h; λ) (N < n2 ). These polynomials are called Q-polynomials since, by Theorem 6.6.1, the linear term of the critical Q-polynomial yields Q-curvature. In Section 6.11, we will further analyze the relation between these polynomials and Q-curvature. In the critical case, (6.6.7) implies the relation n n

! −λ−1 . . . (−λ)δn (λ)(1) Dnres (λ)(1) = 2n 2 2 n

n n =2 ! −λ−1 . . . (−λ) [Tn∗ (λ)(v0 ) + · · · + T0∗ (λ)(vn )] 2 2 n 2 n

1 n ∗ = (λ)(vn−2j ). (6.6.13) !2n −λ−j −1 . . . (−λ)P2j 2j j! 2 2 2 j=0 Since P2j (λ) has polynomial degree j (in λ), it follows that Qres n (h; λ) has degree n 2 (in λ).

6.6. Residue families and holographic formulas for Q-curvature

219

More generally, (6.6.10) yields

n

n

res D2N (λ)(1) = 22N N ! − −λ+2N −1 . . . − −λ+N 2 2 ∗ (λ+n−2N ) (v0 ) + · · · + T0∗ (λ+n−2N ) (v2N )] × [T2N and it follows that Qres 2N (h; λ) has degree N (in λ). Example 6.6.1. The discussion in Section 5.2.4 shows that for M = Rn with the ﬂat res nc (hc ; λ) coincides with D2N (λ) : C ∞ (Rn+1 ) → metric hc , the residue family D2N C ∞ (Rn ). Example 6.6.2. For the sphere M = S n with the round metric 14 hc , the corresponding Poincar´e-Einstein metric is −2 2 2 21 dr +(1−r ) hc . gE = r 4 gE coincides with the usual hyperbolic metric 4 (dx21 + · · · + dx2n+1 ) (1−|x|2 )2 on the unit ball Bn+1 (see (6.5.9)). The conformal compactiﬁcation 1 dr2 +(1−r2 )2 hc , 4 i.e.,

1−|x| 1+|x|

2 gE ,

res 1 yields the residue families DN ( 4 hc ; λ). On the other hand in Section 5.4 an analogous residue constructions with respect to the conformal compactiﬁcation

(1−|x|2 )gE = 4(dx21 + · · · + dx2n+1 ) b (λ). The arguments in Section 6.5 show that the resulting led to the families DN families are conjugate, i.e., 1 b res hc ; λ , e−(λ−N )Φ ◦ DN (λ) ◦ eλΦ = DN (6.6.14) 4

where

1−|x| 1+|x|

eΦ = 1−|x|2 ,

i.e., eΦ = (1+|x|)2 . Thus (6.6.14) reads 2λ 1+|x| 1 b res 22N DN hc ; λ . (λ) ◦ = DN 2 4

220

Chapter 6. Conformally Covariant Families

Moreover, as for (M n , h) = (Rn , hc ), the renormalization (6.6.10) has the eﬀect that

n res h; − +N = P2N (h)i∗ (6.6.15) D2N 2 extending (6.6.8). In fact, it suﬃces to observe that (6.6.3) yields

n

n 22N N ! − −λ+2N −1 · · · − −λ+N T2N (λ+n−2N ) = P2N (λ+n−2N ) 2 2 and P2N ( n2 −N ) = P2N (by (3.2.17)). (6.6.15) is an extreme case of the factorization identities (see Section 6.11). Next, we discuss the behaviour of residue families under conformal changes ˆ = e2ϕ h are related by of the metric. The Poincar´e-Einstein metrics of h and h κ∗ r−2 (dr2 +hr ) = r−2 (dr2 + ˆhr ) using a diﬀeomorphism κ which ﬁxes the boundary r = 0 ([99], Theorem 2.3); for the present discussion we can ignore that all data are only determined to a certain order. Hence ∗ 2 κ (r) ∗ 2 κ (dr +hr ) = (dr2 + ˆhr ). r The latter identity implies

lim

r→0

κ∗ (r) r

= e−ϕ .

(6.6.16)

Now the arguments of Section 6.5 yield (for all N ≤ n if n is even) Theorem 6.6.3. res ˆ res (h; λ) = e(λ−N )ϕ ◦ DN (h; λ) ◦ κ∗ ◦ DN

κ∗ (r) r

λ .

(6.6.17)

Proof. Let g = r−2 (dr2 +hr ) be the Poincar´e-Einstein metric on (0, ε) × M associated to h and let u ∈ ker(Δg + μ(n − μ)) be an eigenfunction with smooth leading term f . Let ψ ∈ C0∞ (X), X = [0, ε) × M . We calculate the residue Resλ=−μ−1−N rλ κ∗ (u)ψ vol(r2 κ∗ (g)) X

in two ways. On the one hand, it is given by −μϕ ˆ ˆ ˆ μ)(ψ) vol(h). e f δN (h; μ)(ψ) vol(h) = e−(μ−n)ϕ f δN (h; M

M ∗

∗

In fact, by Δκ∗ (g) ◦ κ = κ ◦ Δg , the function κ∗ (u) is an eigenfunction of Δκ∗ (g) . Since κ restricts to the identity on r = 0, its leading term is ∗ μ κ (r) i∗ f = e−μϕ f r

6.6. Residue families and holographic formulas for Q-curvature

221

ˆ r = r2 κ∗ (r−2 (dr2 +hr )), we ﬁnd (using (6.6.16)). On the other hand, using dr2 + h r κ (u)ψκ (r) κ (vol(dr +hr )) Resλ=−μ−1−N X λ+n+1 −n−1 2 = Resλ=−μ−1−N κ∗ (r) uκ∗ (ψ)r vol(dr +hr ) X λ+n+1 κ∗ (r) λ 2 = Resλ=−μ−1−N r uκ∗ (ψ) vol(dr +hr ) r X −μ+n−N κ∗ (r) = f δN (h; μ) κ∗ (ψ) vol(h). r M

λ+n+1 ∗

∗

−n−1 ∗

2

Since f is arbitrary, we conclude

−μ+n−N κ∗ (r) ◦ κ∗ r ∗ μ−n+N κ (r) = δN (h; μ) ◦ κ∗ ◦ . r

ˆ μ) = δN (h; μ) ◦ e−(μ−n)ϕ ◦ δN (h;

For μ = λ+n−N , the latter formula reads ˆ λ+n−N ) = δN (h; λ+n−N ) ◦ κ ∗ ◦ e−(λ−N )ϕ ◦ δN (h;

κ∗ (r) r

λ .

(6.6.10) implies the assertion.

In particular, for the critical residue family, we ﬁnd the transformation formula

ˆ λ) Dnres (h;

=e

(λ−n)ϕ

◦

Dnres (h; λ)

◦ κ∗ ◦

κ∗ (r) r

λ .

(6.6.18)

Remark 6.6.3. Theorem 6.6.3 suggests that we ask for the construction of natural families DN (X, M ; g; λ) : C ∞ (X) → C ∞ (M ) with the properties ∗

e−(λ−N )i

(ϕ)

◦ DN (X, M ; e2ϕ g; λ) ◦ eλϕ = DN (X, M ; g; λ)

for all ϕ ∈ C ∞ (X) and all λ ∈ C (conformal covariance), and res (h; λ). DN (dr2 +hr ; λ) = DN

The transformation formula (6.6.17) would be a direct consequence of such a spe-

222

Chapter 6. Conformally Covariant Families

cialization result. In fact, we ﬁnd res ˆ (h; λ) = DN (dr2 + ˆ hr ; λ) DN −2 ∗ κ (r) ∗ 2 κ (dr +hr ); λ = DN r ∗ −(λ−N ) ∗ λ κ (r) κ (r) = i∗ ◦ DN κ∗ (dr2 +hr ); λ ◦ r r ∗ λ κ (r) res = e(λ−N )ϕ ◦ DN (dr2 +hr ; λ) ◦ κ∗ ◦ . r

Remark 6.6.4. (6.6.18) can be used for a proof of the fundamental identity. In fact, diﬀerentiating (6.6.18) at λ = 0 yields for u = 1 ˆ 0)(1) enϕ D˙ nres (h; = ϕDnres (h; 0)(1) + D˙ nres (h; 0)(1) + Dnres (h; 0) (κ∗ log(κ∗ (r)/r)) . Now Dnres (h; 0) = Pn (h)i∗ , i.e., Dnres (h; 0) is tangential. Hence we ﬁnd ˆ 0)(1) = D˙ res (h; 0)(1) − Pn (h)(ϕ), enϕ D˙ nres (h; n i.e.,

n

ˆ = Qn (h) + (−1) 2 Pn (h)(ϕ) enϕ Qn (h) using (6.6.2). It is actually this observation which suggests that we expect the validity of (6.6.2). The transformation formulas (6.6.17) can be regarded as an eﬀective and compact formulation of the behaviour of the Feﬀerman-Graham coeﬃcients h(2i) and the holographic coeﬃcients v2i with respect to conformal changes of the boundary metric. The (inﬁnitesimal) conformal variations of h(2i) and v2i have been used in [147], [211] and [212] to study structural properties of anomalies. In these references, the corresponding diﬀeomorphisms of bulk space appear under the name of PBH diﬀeomorphisms (see also [80] and [218]). n Using the relation P˙n (0)(1) = (−1) 2 Qn , we can write the formulas (6.6.9) and (6.6.5) also in the following more symmetric form. Corollary 6.6.1. (6.6.2) is equivalent to the identity n

2(−1) 2 Qn (h) = (P˙n (h; 0) − P˙n∗ (h; 0))(1) n

2 −1 n n

! −1 ! T2j∗ (h; 0)(vn−2j ) (6.6.19) +2 2 2 j=0

n

6.6. Residue families and holographic formulas for Q-curvature

223

or, equivalently, to n 2(−1) 2 Qn (h) = (P˙n (h; 0) − P˙n∗ (h; 0))(1) n

2 −1 n ( n −j −1)! ∗ ! P2j (h; 0)(vn−2j ). (6.6.20) 2n−2j 2 + 2 j=0 j!

In contrast to Pn (0), the operator P˙n (0) is not self-adjoint, in general. In order to evaluate the latter formulas further, it remains to describe the diﬀerence δn (h; 0)(1) = (P˙n (h; 0) − P˙n∗ (h; 0))(1) ∈ C ∞ (M n ). def

(6.6.21)

Note that if h is Einstein then P˙n (h; 0) is self-adjoint (see Section 6.16). Moreover, direct calculations show that δ2 (h; 0) vanishes and δ4 (h; 0)(u) = 2(Δ(Ju)−JΔ(u)). In particular, we ﬁnd δ4 (h; 0)(1) = 2ΔJ (see Example 6.6.4). Similarly, for n = 6, we shall prove in Section 6.10 that ∗ δ6 (h; 0)(1) coincides with a linear combination of the quantities P2j (h; 0)(v6−2j ), j = 1, 2. In fact, this is a general result which leads to an explicit representation of Qn for all n (see the discussion on page 26). The following result provides the desired formula for δn (h; 0)(1). 1 Theorem 6.6.4. Let n ≥ 2 be even. Then n

2 −1 n n

−1 ! ! nδn (h; 0)(1) = −2 2jT2j∗ (h; 0)(vn−2j ). 2 2 j=0

n

Proof. We consider the coeﬃcient of ε2λ in the asymptotics of the integral (uΔv − vΔu) vol(gE ), (6.6.22) δ>r>ε

where

n

u=

2

j=0

n

r

λ+2j

T2j (h; λ)(f ) and v =

2

rλ+2j T2j (h; λ)(1)

j=0

are approximate eigenfunctions of ΔgE for λ near 0. Here we choose f ∈ Cc∞ . On the one hand, the coeﬃcient of ε2λ vanishes in the asymptotic expansion of 1 Theorem

6.6.4 has been found by R. Graham during the Winter School “Geometry and Physics” at Srni in January 2007. At this conference I gave a lecture discussing formula (1.6.14). Partial results on δn (h)(1) for n ≤ 6 suggested that we expect a formula for Qn for general n without ∗ (0) and v . the contribution δn (1), i.e., a formula only in terms of P2j 2j

224

Chapter 6. Conformally Covariant Families

(6.6.22). On the other hand, by Green’s formula, the coeﬃcient coincides with the coeﬃcient of εn+2λ in the asymptotic expansion of ∂u ∂v − vr ur vol(hr ). ∂r ∂r r=ε The evaluation of that coeﬃcient yields the identity for λ → 0. For full details we refer to [125]. We combine Theorem 6.6.4 with Corollary 6.6.1 and ﬁnd Theorem 6.6.5. (6.6.2) is equivalent to n

2 −1 n n

(−1) 2nQn (h) = 2 ! −1 ! (n−2j)T2j∗ (h; 0)(vn−2j ), 2 2 j=0 n 2

n

i.e., Qn (h) = (−1)

n 2

n

2 −1

( n −j)! ∗ −1 ! P2j (h; 0)(vn−2j ). 2n−1−2j 2 2 j! j=0

n

Proof. Corollary 6.6.1 and Theorem 6.6.4 prove the equivalence of (6.6.2) and n

−1

2

1 n n −1 ! ! 2(−1) Qn (h) = 2n (−2j)T2j∗ (h; 0)(vn−2j ) n 2 2 j=0 n 2

n

2 −1 n

n n +2 −1 ! ! T2j∗ (h; 0)(vn−2j ) 2 2 j=0 n

2 −1 n n

1 ! −1 ! (n−2j)T2j∗ (h; 0)(vn−2j ). =2 2 2 n j=0

n

This proves the ﬁrst formula. The latter identity is equivalent to n

2(−1) 2 Qn (h) n

2 −1 n n

1 1 1 ! −1 ! P ∗ (h; 0)(vn−2j ). (n−2j) 2j =2 n 2 2 n j=0 2 j! ( 2 −1) · · · ( n2 −j) 2j

n

This implies the second assertion. Now the identities in Theorem 6.6.5 admit an independent proof ([125]).

6.6. Residue families and holographic formulas for Q-curvature

225

Theorem 6.6.6 (The holographic formula). For even n ≥ 2, n

2 −1

n n −1 ! (−1) 2nQn (h) = 2 (n−2j)T2j∗ (h; 0)(vn−2j ), ! 2 2 j=0 n 2

n

(6.6.23)

i.e., Qn (h) = (−1)

n 2

n

2 −1

( n −j)! ∗ −1 ! P2j (h; 0)(vn−2j ). 2n−1−2j 2 2 j! j=0

n

(6.6.24)

Theorem 6.6.6, in particular, completes the proof of Theorem 6.6.1. The proof of Theorem 6.6.6 ([125]) will be outlined below. As a consequence of (6.6.24), we give a simple proof of the following result. Corollary 6.6.2 ([128]). n n

n 2 Qn vol = (−1) 2 2n vn vol . ! −1 ! 2 2 Mn Mn Proof. (6.6.24) implies

n 2

Mn

Qn vol = (−1) 2

n−1

+ (−1)

But

Mn

n 2

n −1 ! ! vn vol 2 2 M

n

2 −1

( n −j)! ∗ −1 ! 2n−1−2j 2 P2j (0)(vn−2j ) vol . 2 j! n M j=1

n

n

∗ P2j (0)(vn−2j ) vol =

Mn

vn−2j P2j (0)(1) vol = 0

since P2j (0) annihilates constants. The proof is complete. The quantity

L= Mn

vn vol

appears as the coeﬃcient of the log-term in the asymptotics of the gE -volume of the set {r > ε} for ε → 0, ε

δ

r−n−1

v(r, b) vol dr Mn

∼ c0 ε−n + c2 ε−(n−2) + · · · + cn−2 ε−2 − L log ε + · · · (6.6.25)

226

Chapter 6. Conformally Covariant Families

with c2j

1 = n−2j

v2j vol

Mn

(6.6.26)

for j = 0, . . . , n2 −1. Therefore, Corollary 6.6.2 can be restated as

n n n ! −1 !L. Qn vol = (−1) 2 2n 2 2 2 Mn

(6.6.27)

In this form the result appears in [128]. n

Theorem 6.6.6 can be used to determine the contribution of Δ 2 −1 (J) to Qn . We recall that Q4 = 2(J2 − |P|2 ) − ΔJ. In all dimensions n ≥ 4, Qn contains an n analogous contribution Δ 2 −1 (J). By Theorem 6.6.6, all contributions of the form n −1 ∗ Δ 2 (τ ) come from Pn−2 (0)(v2 ), i.e., are given by n

n

n

n

(−1) 2 2Δ 2 −1 (v2 ) = (−1) 2 −1 Δ 2 −1 (J) using v2 = − 21 J. The fact that this is the correct contribution is a result of Branson (see [31], Corollary 1.4 or [32], Corollary 1.5). Corollary 6.6.3. For closed M n (n even) and u ∈ ker Pn , the sum n 2 −1

In (u) =

(n−2j)

j=0

Mn

vn−2j T2j (0)(u) vol

is conformally invariant. Proof. For u ∈ ker(Pn ), we ﬁnd that Theorem 6.6.6 implies that n 2 −1

j=0

(n−2j) Mn

uT2j∗ (0)(vn−2j ) vol

M

uQn vol is a conformal invariant. But n 2 −1

=

(n−2j)

j=0

Mn

vn−2j T2j (0)(u) vol

is conformally invariant.

It should be noted that Corollary 6.6.3 actually does not refer to Q-curvature. For u = 1, it reduces to the conformal invariance of L. The following comments on the nature of the various terms in the holographic formula (6.6.24) might be useful. The two main ingredients are the holographic coeﬃcients 1 = v0 , v2 , . . . , vn which describe the asymptotics of the volume of gE and the solution of the (approximate) Dirichlet problem for the Laplacian of gE . We explain the details. The following discussion is formal. h is ﬁxed and suppressed. We consider a genuine eigenfunction −ΔgE u = μ(n−μ)u

(6.6.28)

6.6. Residue families and holographic formulas for Q-curvature

of the form u=

rn−μ+2j a2j (n−μ) +

j≥0

227

rμ+2j b2j (n−μ).

(6.6.29)

j≥0

Then (by deﬁnition of S) b0 (n−μ) = S(μ)a0 (n−μ) and (see (6.6.3)) a2j (μ) =

22j j!

1 n P2j (μ)a0 (μ). −μ−1 . . . 2 −μ−j 2

n

In the special case μ = n, the function u is harmonic and (6.6.29) has the form u = a0 (0) + r2 a2 (0) + · · · + rn−2 an−2 (0) + HOT with a2j (0) =

1 22j j!( n2 −1) . . . ( n2 −j)

P2j (0)a0 (0), j = 0, . . . ,

n − 1. 2

Thus the operators P2j (0) determine the ﬁrst n2 terms in the asymptotics of harmonic functions. The operator an (μ) has a simple pole at μ = 0. For the boundary function a0 (0) = 1, its residue vanishes and we have an (0) = −

1 2n ( n2 )!( n2 −1)!

n P˙n (0)(1) = −c n2 (−1) 2 Qn .

(6.6.30)

As a preparation for the proof of Theorem 6.6.6, we recall some results of [97] and [128]. For the family u(μ) = P(μ)(1) (here P denotes the Poisson transformation of [128]) of eigenfunctions, we diﬀerentiate (6.6.29) and obtain u(n) ˙ = (− log r − Arn log r + B) + (Crn log r + Drn ) with coeﬃcients A, B, C and D which are even in r and satisfy A|r=0 = an (0), B|r=0 = 0, C|r=0 = b0 (0). Here we have used that a0 (0) = 1 and P2j (0)(1) = 0 for j = 1, . . . , n2 − 1. Hence u(n) ˙ = − log r + αrn log r + β with even α, β so that α|r=0 = −an (0) + b0 (0), But

β|r=0 = 0. n

b0 (0) = S(n)(1) = (−1) 2 c n2 Qn and (6.6.30) yield

n

α|r=0 = 2(−1) 2 Qn .

228

Chapter 6. Conformally Covariant Families

Diﬀerentiation of the eigenequation (6.6.28) at μ = n shows that ˙ = −nu(n) = −nP(n)(1) = −n. −ΔgE u(n) In other words, v = u(n) ˙ satisﬁes ΔgE v = n and has the asymptotics v = − log r + αrn log r + β with

n

α|r=0 = 2(−1) 2 c n2 Qn ,

(6.6.31)

β|r=0 = 0.

(6.6.32)

n

Feﬀerman and Graham ([97]) proved that v (mod O(r )) actually is uniquely determined by the requirements ΔgE v = n and (6.6.31) with smooth α and β so that β|r=0 = 0. β (mod O(rn )) and α|r=0 are locally determined by h. Moreover, α|r=0 yields Qn . An ambient metric version of that result is proved in [98]. Next, we recall the ideas of the proof of the identity n Qn vol = vn vol (6.6.33) 2(−1) 2 c n2 Mn

Mn

(Corollary 6.6.2) in [128] and [97], and outline an independent proof of Theorem 6.6.6 ([125]). In particular, this proves the relation (6.6.2) which has been used here as the starting point to derive the explicit holographic formula for Qn . The proof in [128] rests on the regularization of the integral |du(λ)|2 − λ(n−λ)u2 (λ) vol (6.6.34) X

for u(λ) = P(λ)(1) (λ near n). We compare the limiting behaviour for λ → n of the ﬁnite parts in two diﬀerent calculations of the integral r>ε . On the one hand, a version of Green’s formula implies that the ﬁnite part is given by S(λ)(1) vol . −n M n

By Theorem 3.2.2, the latter integral tends to −n(−1) 2 c n2 M Qn vol for λ → n. On the other hand, an evaluation of (6.6.34) using the asymptotics of u(λ) for r → 0 yields −n 21 M vn vol. This proves (6.6.33). The proof in [97] rests on the consideration of the coeﬃcient of log ε in the asymptotics of the integral (Δv) vol(gE ), (6.6.35) δ>r>ε

where v is as in (6.6.31). On the one hand, Green’s formula yields −n M α|r=0 vol= n −2(−1) 2 c n2 M Qn vol (using (6.6.32)). On the other hand, using Δv = n, we ﬁnd

6.6. Residue families and holographic formulas for Q-curvature

229

n r>ε vol and, by the deﬁnition of v(r), the coeﬃcient of log ε is −n M vn vol. This proves (6.6.33). A generalization of the latter argument can be used to prove Theorem 6.6.6 (see [125] for the details). The idea is to determine the coeﬃcient of log ε in the asymptotics for ε → 0 of the integral (uΔv − vΔu) vol(gE ), (6.6.36) δ>r>ε

where v is as in (6.6.31) and n 2 −1

u=

(T2j (0)f ) r2j , f ∈ Cc∞ (M ).

j=0

On the one hand, using Δu = O(rn ) we ﬁnd

n 2 −1

−n

j=0

n 2 −1

vn−2j (T2j (0)f ) vol = −n

M

j=0

f T2j∗ (0)(vn−2j ) vol .

M

On the other hand, by Green’s formula, it suﬃces to determine the coeﬃcient of log ε in ∂u ∂v −v −ε1−n u vol(hr ). ∂r ∂r r=ε We ﬁnd

− M

⎡ ⎣f nα +

n 2 −1

⎤ 2j(T2j (0)f )vn−2j ⎦ vol .

j=0

Using (6.6.32), this proves Theorem 6.6.6. In order to illustrate the holographic formulas, we work out these formulas in dimensions n = 2, n = 4 and n = 6. Example 6.6.3. For n = 2, we have Q2 = K. In that case, (6.6.5) states that Q2 = P˙ 2∗ (0)(1) − 4v2 . But

τ K 1 v2 = − J = − = − 2 4 2 ([119]) and P2 (μ) = Δ − μK. It is clearly better to use the more symmetric form (6.6.20) which reads 2Q2 = −4v2 since P2 (μ) is self-adjoint. Finally, Theorem 6.6.6 states directly Q2 = −2v2 .

230

Chapter 6. Conformally Covariant Families

Example 6.6.4. For n = 4, (6.6.5) states that Q4 = −P˙4∗ (0)(1) + 8P2∗ (0)(v2 ) + 32v4 . We conﬁrm the latter formula by explicit calculations using Theorem 6.9.2. First, we take a closer look at the contributions of ΔJ on both sides. For the left-hand side we know that Q4 = 2 J2 − |P|2 − ΔJ (see (4.1.1)), where J = tr(P) and P = 12 (Ric −Jg) (see (2.5.10)). On the righthand side we use the self-adjoint operator P2 (λ) = Δ − λJ and v2 = − 12 tr(P) = − 21 J. Thus the middle term contributes −4ΔJ. v4 is given by v4 =

1 2 (J − |P|2 ). 8

(6.6.37)

Finally, by (6.6.39), we have P˙ 4∗ (0)(1) = −3ΔJ + 2(J2 − |P|2 ). We also recall that P˙ 4 (0)(1) = Q4 (see (6.6.38)). Again, it is clearly more eﬃcient to use the more symmetric form (6.6.20)

2Q4 = P˙4 (0) − P˙4∗ (0) (1) + 8P2∗ (0)(v2 ) + 32v4 which reads

1 2Q4 = 2ΔJ + 8Δ − J + 4(J2 − |P|2 ) 2 = −2ΔJ + 4(J2 − |P|2 )

using (P˙4 (0) − P˙ 4∗ (0))(1) = 2ΔJ (see (6.6.40)). The latter formula is a special case of Theorem 6.6.4 which states 4(P˙ 4 (0) − P˙ 4∗ (0))(1) = −26 T2∗ (0)(v2 ). 1 Since T2 (λ) = 4(1−λ) (Δ − λJ) (see the proof of Lemma 6.6.1), we recover the above formula. Finally, Theorem 6.6.6 directly yields 1 Q4 = 8 2v4 + Δ(v2 ) , 4

i.e., Q4 = 2(J2 − |P|2 ) − ΔJ. The following result establishes the missing facts. Lemma 6.6.1. Let n = 4. Then P2 (λ) = Δ − λJ and P4 (λ) = (Δ−(λ+2)J) (Δ−λJ) + 2λ(λ−1)|P|2 + 4(λ−1)δ(P#d) + 2(λ−1)(dJ, d).

6.6. Residue families and holographic formulas for Q-curvature

Proof. Theorem 6.9.4 implies that the ansatz u ∼ −ΔgE u = λ(n − λ)u yields (see (6.6.3))

j≥0

231

rλ+2j a2j (λ) for u so that

1 P2 (λ)a0 , 4(1−λ) 1 1 a4 (λ) = T4 (λ)a0 = − P4 (λ)a0 4(1−λ) 8λ

a2 (λ) = T2 (λ)a0 =

with P2 (λ) = Δ − λJ and 1 1 2 (Δ−(λ+2)J)a2 − λ|P| a0 − δ(P#da0 ) − (dJ, da0 ) 2 2 1 (Δ−(λ+2)J)(Δ−λJ)a0 + 2λ(λ−1)|P|2 a0 =− 8λ4(1−λ) + 4(λ−1)δ(P#da0 ) + 2(λ−1)(dJ, da0 )] .

1 a4 (λ) = − 8λ

The proof is complete. In particular, we ﬁnd P4 (0)u = Δ2 u − 2JΔu − 4δ(P#du) − 2(dJ, du) = P4 u

by (4.1.3); note that in the present context the Laplacian Δ is deﬁned as −δd. In other words, the Paneitz operator is constructed by the asymptotics of eigenfunctions of the associated Poincar´e-Einstein metric. This is a special case of the results of Graham and Zworski (Section 3.2, (3.2.18)). Moreover, Lemma 6.6.1 implies P˙4 (0)u = −Δ(Ju) − JΔu + 2J2 u − 2|P|2 u + 4δ(P#du) + 2(dJ, du) and P˙4∗ (0)u = −JΔu − Δ(Ju) + 2J2 u − 2|P|2 u + 4δ(P #du) + 2δ(udJ). For u = 1, we get (by (4.1.1)) and

P˙4 (0)(1) = −ΔJ + 2J2 − 2|P|2 = Q4

(6.6.38)

P˙4∗ (0)(1) = −3ΔJ + 2J2 − 2|P|2 .

(6.6.39)

Note that

δ4 u = P˙4 (0) − P˙ 4∗ (0) u = 2(dJ, du) − 2δ(udJ) = 4(dJ, du) + 2uΔJ = 2(Δ(Ju) − JΔ(u)), i.e., the operator δ4 = P˙4 (0) − P˙4∗ (0) is only of ﬁrst order.

(6.6.40)

232

Chapter 6. Conformally Covariant Families

Example 6.6.5. We consider Q6 on M 6 . The holographic formula (6.6.5) says that 2Q6 = −δ6 (1) − (48 · 16)v6 − (16 · 6)P2∗ (0)(v4 ) − 12P4∗ (0)(v2 ) (for the deﬁnition of δ6 we refer to (6.6.21)). In Section 6.9 and Section 6.10, we will discuss explicit formulas for v6 and δ6 (1). Here we focus on the remaining two terms. As in Example 6.6.3 we have P2 (λ) = Δ − λJ (in all dimensions). In fact, P2 ( n2 −1) = P2 (see (3.2.17)). For the formula v4 =

1 2 J − |P|2 8

we refer to Theorem 6.9.2. Thus the third term in the above formula for Q6 yields 12Δ(|P|2 − J2 ). Next, we determine the family P4 (λ) which is given by the relations 1 P2 (λ)a0 , 4(2−λ) 1 P4 (λ)a0 a4 (λ) = T4 (λ)a0 = 32(2−λ)(1−λ)

a2 (λ) = T2 (λ)a0 =

(compare with the proof of Lemma 6.6.1 dealing with the case n = 4). Now Theorem 6.9.4 yields 1 1 1 a4 (λ) = (Δ−(λ+2)J)a2 − λ|P|2 a0 − δ(P#da0 ) − (dJ, da0 ) 8(1−λ) 2 2

1 (Δ−(λ+2)J)(Δ−λJ)a0 + 2λ(λ−2)|P|2 a0 = 32(1−λ)(2−λ) + 4(λ−2)δ(P#da0 ) + 2(λ−2)(dJ, da0 ) , i.e., P4 (λ)u = (Δ−(λ+2)J) (Δ−λJ) u + 2λ(λ−2)|P|2 u + 4(λ−2)δ(P#du) + 2(λ−2)(dJ, du) (6.6.41) on M 6 . In order to check the latter formula, we use the criterion P4 (1) = P4,6 (by (3.2.17) for j = 2 and n = 6). We ﬁnd P4 (1)u = Δ2 u − Δ(Ju) − 3JΔu + 3J2 u − 2|P|2 u − 4δ(P#du) − 2(dJ, du). It is easy to prove that this operator actually coincides with the Paneitz operator P4,6 u = Δ2 u + δ (4J − 4P#) du + 3J2 − 2|P|2 − ΔJ u

6.6. Residue families and holographic formulas for Q-curvature

233

(see (4.1.6) and (4.1.7)). Now (6.6.41) yields P4 (0) = (Δ−2J)Δ − 8δ(P#d) − 4(dJ, d) and hence P4∗ (0) = Δ(Δ−2J) − 8δ(P#d) − 4δ(·dJ). It follows that P4∗ (0)(J) = Δ2 J − 2Δ(J2 ) − 8δ(P#dJ) − 4δ(JdJ) = Δ2 J − 8δ(P#dJ).

(6.6.42)

Thus using v2 = − 12 J, we ﬁnd the formula 1 Q6 = − δ6 (1) − (8 · 48)v6 + 6Δ |P|2 − J2 + 3 Δ2 J − 8δ(P#dJ) . 2

(6.6.43)

In order to obtain a fully explicit formula for Q6 , it remains to work out the contributions δ6 (1) and v6 . As mentioned above, this will be done in Section 6.9 and Section 6.10. Moreover, we will conﬁrm the resulting formula for Q6 by an independent calculation of P˙ 6 (0)(1). Here we only note that the operator δ6 contains the contribution −4Δδ(udJ) which ﬁnally yields the contribution Δ2 J to Q6 (recall that the convention here is Δ = −δd). Remark 6.6.5. The analogous formula for a4 (λ) for general dimension n reproduces the Paneitz operator P4,n (see (4.1.6)). In fact, we ﬁnd a4 (λ) =

1

n (Δ−(λ+2)J)(Δ−λJ)a0 + 2λ λ− +1 |P|2 a0 2

n n +4 λ− +1 δ(P#da0 ) + 2 λ− +1 (dJ, da0 ) , 2 2

32( n2 −λ−1)( n2 −λ−2)

i.e., n

n

n −2 u = Δ− J Δ− −2 J u − (n−4)|P|2 u − 4δ(P#du) − 2(dJ, du). P4 2 2 2 It is easy to verify that the latter operator coincides with P4,n = Δ2 + δ ((n−2)J − 4P#) d +

n−4 n 2 J − 2|P|2 − ΔJ . 2 2

Remark 6.6.6. The details of the relation between the Paneitz operator P4,n and the square of the Laplacian of the ambient metric can be found in [149]. n Finally, we determine the critical Q-polynomial Qres n (M ; h; λ) in two special cases.

234

Chapter 6. Conformally Covariant Families

Example 6.6.6. Let n = 2. Then (by (6.6.13)) 2 ∗ ∗ Qres 2 (M ; h; λ) = −4λP0 (λ)(v2 ) + P2 (λ)(v0 ) = −4λv2 + (Δ − λJ)(1)

= 2λJ − λJ = λQ2 (h). Example 6.6.7. Let n = 4. We apply the holographic formula (6.6.13) in order to ﬁnd ∗ ∗ ∗ −Qres 4 (λ) = 32(1−λ)(−λ)P0 (v4 ) + 8(−λ)P2 (λ)(v2 ) + P4 (λ)(v0 ). Now Lemma 6.6.1 implies P4∗ (λ)(1) = −(λ+2)(Δ − λJ)J + 2λ(λ−1)|P|2 − 2(λ−1)ΔJ = −3λΔJ + λ(λ+2)J2 + 2λ(λ−1)|P|2 . Hence

2 2 −Qres 4 (λ) = λ 32(λ−1)v4 + 4(Δ−λJ)J − 3ΔJ + (λ+2)J + 2(λ−1)|P| = λ 4(λ−1)(J2 − |P|2 ) + ΔJ − 3λJ2 + 2J2 + 2(λ−1)|P|2 = λ λ(J2 − 2|P|2 ) + (−2J2 + 2|P|2 + ΔJ) ,

i.e., 2 2 2 Qres 4 (λ) = λ (2|P| − J ) + λQ4 .

(6.6.44)

The latter result can be written also in the form 2 Qres 4 (λ) = −λ(λ−1)Q4 − λ P2 Q2

since Q4 + P2 Q2 = J2 − 2|P|2 . For analogous results on critical Q-polynomials in higher dimensions we refer to Theorem 6.11.8. Remark 6.6.7. For odd n, we ﬁnd res Qres n (h; λ) = Dn (h; λ)(1) = 0

since

∗ (λ)(v1 ) + · · · + T0∗ (λ)(vn ) δn (h; λ)(1) = Tn∗ (λ)(v0 ) + Tn−1

and Todd = 0 and vodd = 0 (including vn in view of the vanishing trace of h(n) ). In particular, Q˙ res n (h; 0) = 0. The latter observation rests on the fact that hr is even in r (which implies that the boundary r = 0 has vanishing second fundamental form (by (6.16.10))). The situation is diﬀerent if the second fundamental form L is non-trivial. As an example, we note that the calculations in Section 5.4 yield families DN (Bn+1 , S n ; dr2 + r2 gS n ; λ) (the sphere corresponds to r = 1). In that case, L and D˙ n (dr2 + h2r ; 0)(1) do not vanish and the calculations in Section 5.4 yield a holographic formula for this quantity.

6.7. D2 (g; λ) as a residue family

235

6.7 D2 (g; λ) as a residue family For M n+1 = [0, ε) × Σn and a Poincar´e metric r−2 (dr2 +hr ) of h, the family D2 (dr2 +hr ; λ) : C ∞ (M ) → C ∞ (Σ) deﬁned in Section 6.4 (see (6.4.1)) is related to the residue family D2res (h; λ) deﬁned in Section 6.6. Here we prove that D2res (h; λ) arises by specialization of D2 (M, Σ; g; λ) to the background metric g = dr2 +hr . In Section 6.24, the family D2 (M, Σ; g; λ) will be recognized as a special tractor family. From that perspective, Theorem 6.7.1 appears as a version of the holographic duality (Section 6.21). We recall that the family D2res (h; λ) arises as follows. We consider the residue at λ = −μ−3 of the family of distributions ε rλ uϕ vol(dr2 +hr ) C0∞ (M ) ϕ → 0

Σ

which is deﬁned by a (formal) eigenfunction u(r, b) ∼ j≥0 rμ+2j aj (μ; b) of the Laplacian of r−2 (dr2 +hr ) for the eigenvalue μ(n − μ). Partial integration yields the formula a0 δ2 (h; μ)(ϕ) vol(h), h = h0 , Σ

where

1 ∂2 v0 ◦ i∗ 2 + [T2∗ (μ) ◦ v0 + T0∗ (μ) ◦ v2 ] i∗ 2 ∂r (Deﬁnition 6.6.2). Now using v0 = 1, h(2) = −P(h), v2 = 12 tr(h(2) ) = − 21 J(h) (Theorem 6.9.1, Theorem 6.9.2) and δ2 (h; μ) =

a2 (μ) = T2 (μ)a0 ,

T2 (μ) =

1 (Δh −μJ(h)), 2(n−2−2μ)

we ﬁnd

1 1 ∗ ∂2 1 (Δh −(μ+n−2)J(h)) + J(h) i∗ − δ2 (h; μ+n−2) = i 2 ∂r2 2(n−2+ 2μ) 2 2 1 ∗ ∂ ∗ = − (Δh +μJ(h))i . (n−2+2μ)i 2(n−2+ 2μ) ∂r2

Hence D2res (h; λ) = −2(n−2+2λ)δ2(h; λ+n−2) = −(n−2+2λ)i∗

∂2 + (Δh + λJ(h))i∗ . ∂r2

On the other hand, the relation J(h) = i∗ J(dr2 +hr ) (Lemma 6.11.1) implies D2 (M, Σ; dr2 +hr ; λ) = −(n−2+2λ)i∗(Δdr2 +hr +λJ(dr2 +hr )) + (n−1+2λ)(Δh +λJ(h))i∗ ,

236

Chapter 6. Conformally Covariant Families

i.e., D2 (M, Σ; dr2 +hr ; λ) = −(n−2+2λ)i∗Δdr2 +hr + (n−1+2λ)Δhi∗ +λJ(h)i∗ = −(n−2+2λ)i∗

∂2 + (Δh +λJ(h)) i∗ ∂r2

(6.7.1)

in view of H = 0 and (6.4.9). In fact, in the present case the general formula τ (ˆ g ) = e−2ϕ τ (g) − 2nΔgˆ (ϕ) + n(n−1)|dϕ|2gˆ yields τ (dr2 +hr ) = 2n tr(P(h)) + o(r) = 2nJ(h) + o(r). Thus we have proved Theorem 6.7.1. D2res (h; λ) = D2 (dr2 +hr ; λ). Note that the above discussion also proves the factorization identities n−2 n−1 D2res h; − = P2 (h)i∗ and D2res h; − = i∗ P2 (dr2 +hr ). (6.7.2) 2 2 In Section 6.11, we shall prove analogous factorizations for higher order residue families (for conformally ﬂat metrics).

6.8 D3res (h; λ) For a 3-dimensional Riemannian manifold (M, h), we analyze the critical order 3 residue family D3res (h; λ) = 2(1−2λ)δ3 (h; λ) : C ∞ ([0, ε) × M ) → C ∞ (M ). def

The following result identiﬁes the value of that family for λ = 0 with the ChangQing operator P3 (dr2 +hr ) : C ∞ ([0, ε) × M ) → C ∞ (M ) (see Section 6.26). Lemma 6.8.1. P3 (dr2 +hr ) = 3δ3 (h; 0) =

3 res D (h; 0). 2 3

Proof. We determine an explicit formula for the family δ3 (h; λ). Let h = h0 . (6.6.6) gives ∂ 1 ∂3 δ3 (h; λ) = i∗ 3 + (T2∗ (h; λ) + v2 )i∗ . 6 ∂r ∂r Now v2 = − 21 J(h) (by Theorem 6.9.2) and T2 (λ) =

1 (Δ−λJ) 2(1−2λ)

6.8. D3res (h; λ)

237

by (6.6.3). Hence for the renormalization D3res (h; λ) = 2(1−2λ)δ3 (h; λ) we ﬁnd D3res (h; λ) =

∂3 ∂ 1 (1−2λ)i∗ 3 + (Δh +(λ−1)J(h))) i∗ . 3 ∂r ∂r

(6.8.1)

In particular, D3res (h; 0) =

∂ 1 ∗ ∂3 i + (Δh −J(h)) i∗ . 3 ∂r3 ∂r

On the other hand, P3 (dr2 +hr ) =

∂ ∂ 1 ∂ Δdr2 +hr + Δh + tr(G) − 2J(dr2 +hr ) 2 ∂r ∂r ∂r

by (6.26.2) using H = 0; here we suppress the restriction i∗ to r = 0. Now Δdr2 +hr =

∂2 1 ˙ ∂ + Δh . + tr(h−1 r r hr ) ∂r2 2 ∂r

Thus using h˙ 0 = 0 and h0 = h, we ﬁnd P3 (dr2 +hr ) ∂ 1 ∂3 ∂ ∂ 1 ∂ −1 ¨ tr(h + Δ + tr(G) − 2J(dr2 +hr ) . = + ) h + Δh 0 h 3 2 ∂r 2 ∂r ∂r ∂r ∂r In view of Theorem 6.9.1, the latter formula simpliﬁes to ∂ 1 ∂3 ∂ 3 1 2 Δ + tr(G) − 2J(dr J(h) . + +h ) − h r 3 2 ∂r 2 ∂r 2 ∂r Now (6.8.2) implies τ (dr2 +hr ) = τ (h) + 2 tr(G) = 32 τ (h), i.e., J(dr2 +hr ) = J(h) for r = 0. Hence P3 (dr2 +hr ) =

3 ∂ 1 ∂3 3 ∂ − J(h) . + Δh 3 2 ∂r 2 ∂r 2 ∂r

This proves the assertion. It remains to verify the identity i tr(G) = RiN N = J(h). i

We use the formulas i RiN N =

s

ΓsN N Γiis −

s

ΓsiN ΓiN s + (ΓiN N )i − (ΓiiN )N

(6.8.2)

238

Chapter 6. Conformally Covariant Families

and Γsij =

1 ((gjk )i + (gki )j − (gij )k ) g ks . 2

(6.8.3)

k

In the present setting, we ﬁnd ΓiN N = 0 and ΓjiN =

1 ∂ ((hr )ik )hjk r . 2 ∂r k

Hence using h˙ 0 = 0, we ﬁnd i RiN N

=

−(ΓiiN )N

1 ∂ =− 2 ∂r

k

i.e., tr(G) =

1 ∂ ik 1 ((hr )ik )hr = Pik hik , r=0 ∂r k

i RiN N = tr(P(h)) = J(h).

i

This proves (6.8.2) and completes the proof. Remark 6.8.1. For the analogous family D3res (h; λ) = −2(n−4+2λ)δ3(h; λ+n−3) : C ∞ ([0, ε) × M n ) → C ∞ (M n ) in higher dimension, we ﬁnd δ3 (h; λ) = i∗ with T2 (λ) =

1 ∂3 ∂ + (T2∗ (h; λ) + v2 ) i∗ 6 ∂r3 ∂r

1 (Δh −λJ(h)), 2(n−2−2λ)

1 v2 = − J(h). 2

Hence ∂3 ∂ 1 D3res (h; λ) = − (n−4+2λ)i∗ 3 + (Δh +(λ−1)J(h)) i∗ . 3 ∂r ∂r

(6.8.4)

Remark 6.8.2. Using formula (6.8.4), we prove the factorization identities

n n−1 ∂ ∂ res D3 h; − = i∗ P2 (dr2 +hr ), D3res h; − +2 = P2 (h)i∗ . 2 ∂r 2 ∂r In fact, by (6.8.4) we have ∂3 ∂ n+1 n−1 D3res h; − J(h) i∗ . = i∗ 3 + Δh − 2 ∂r 2 ∂r

6.9. The holographic coeﬃcients v2 , v4 and v6

239

On the other hand, the product on the right-hand side of the ﬁrst identity is 2 ∂ 1 ∂ n−1 ∗ ∂ −1 ˙ 2 + Δhr − J(dr +hr ) + tr(hr hr ) i ∂r ∂r2 2 ∂r 2 n−1 ∂3 ∂ ∂ ∂ 1 + Δh i∗ − J(h)i∗ . = i∗ 3 − tr(P(h))i∗ ∂r 2 ∂r ∂r 2 ∂r This proves the ﬁrst identity. The second factorization is obvious. In Section 6.11, we shall prove analogous factorizations for higher order residue families (for conformally ﬂat metrics). Remark 6.8.3. In Section 6.25, we shall use the curved translation principle in order to construct an order 3 family D3T (g; λ) for a general background metric. By Lemma 6.25.3, this family satisﬁes analogs of both factorization identities which hold true for D3 (λ). The identities in Remark 6.8.2 are special cases. The holographic duality D3res (h; λ) = D3T (dr2 +hr ; λ) for order 3 families will be proved in Theorem 6.25.4.

6.9 The holographic coeﬃcients v2, v4 and v6 In the present section, we give detailed proofs of a number of results which can be found (usually without proofs) at various places and in varying form in the literature. More precisely, we determine the ﬁrst two terms in the FeﬀermanGraham expansion, derive the holographic coeﬃcients v2j , j = 1, 2, 3, and calculate the ﬁrst two terms in the asymptotics of eigenfunctions of the Laplacian for the Poincar´e-Einstein metric. Finally, we formulate an extension of the holographic formula to subcritical Q-curvatures (Conjecture 6.9.1). Theorem 6.9.1. The terms h(2) and h(4) in the formal Taylor series hr = h(0) + r2 h(2) + r4 h(4) + · · · of the Poincar´e-Einstein metric g = r−2 (dr2 +hr ) are given by h(2) = −P

(6.9.1)

(if n ≥ 4), where P denotes the Schouten tensor for h(0) , and 1 −Bij + (n−4)Pki Pkj 4(n−4)

(6.9.2)

Bij = Δ(P)ij − ∇k ∇j (P)ik + Pkl Ckijl .

(6.9.3)

h(4) ij = (if n ≥ 6), where def

B will be called the Bach tensor. Δ = ∇k ∇k is the Bochner-Laplacian on symmetric tensors.

240

Chapter 6. Conformally Covariant Families

Remark 6.9.1. The fact that the Bach tensor B appears in Theorem 6.9.1 is directly ˜ of the ambient metric g˜ which linked with the fact that the curvature tensor R corresponds to the Poincar´e-metric g satisﬁes 2 ˜ mijm = t Bij , n = 4 R n−4

on the ray bundle G = {(x, thx ) | t > 0} of the metric h (Proposition 3.2/(iv ) in [99]). In fact, the latter result follows by combining a calculation of the curvature ˜ with Theorem 6.9.1. The tensor R ˜ mijm describes the normal curvature tensor R of the embedding of G in the ambient space. In dimension n = 4 the Bach tensor yields the Feﬀerman-Graham obstruction. Note that Lemma 4.2.7 implies that div4 div1 (C)ij = (n−3) Δ(P)ij − ∇k ∇j (P)ik . Hence

1 1 ∇l ∇k (C)kijl + Rickl Ckijl . n−3 n−2 For n = 4, the latter formula yields the familiar Bach tensor Bij =

(6.9.4)

1 Rickl Ckijl 2 from general relativity which arises as the ﬁrst variation of the conformally invariant functional (see (2.5.15)) Bij = ∇l ∇k (C)kijl +

M4

|C|2 vol .

A direct calculation shows that e2ϕ Bˆij = Bij − (n−4)(Cikj + Cjki )ϕk + (n−4)Ckijl ϕk ϕl ,

(6.9.5)

where C(X, Y, Z) = ∇X (P)(Y, Z) − ∇Y (P)(X, Z)

(6.9.6)

is the Cotton tensor (see (4.2.14)) ([168], Section 5). Hence, for n = 4, the Bach tensor is conformally invariant: e2ϕ Bˆ = B.

(6.9.7)

It plays a central role in the characterization of conformally Einstein spaces ([158], [159], [168], [115]). By (6.9.4), the higher-dimensional Bach tensor B vanishes if C = 0, i.e., in the conformally ﬂat case. Thus Theorem 6.9.1 implies that for conformally ﬂat h = h(0) the FeﬀermanGraham expansion is of the form 2 1 r2 (6.9.8) hr = 1 − r2 P + r4 P2 + · · · = 1 − P + HOT, P = P(h), 4 2 where we identify P with P . In fact, there are no higher order terms (see (6.14.2)).

6.9. The holographic coeﬃcients v2 , v4 and v6

241

Theorem 6.9.2. The holographic coeﬃcients v2 , v4 and v6 are given by the formulas 1 v2 = − J if n ≥ 2, 2 1 1 2 v4 = J − |P|2 = tr(∧2 P) if n ≥ 4, 8 4 1 3 1 (B, P) if n ≥ 6. v6 = − tr ∧ P − 8 24(n−4) Now we combine the holographic formula (6.6.43) for Q6 with Theorem 6.9.2 and ﬁnd Corollary 6.9.1. On M 6 we have 1 Q6 = − δ6 (1) + 48 tr(∧3 P) + 8(B, P) + 6Δ |P|2 − J2 + 3 Δ2 J − 8δ(P#dJ) . 2 In order to compare the latter formula for Q6 with the formula for Q6 given in [116], it remains to work out the contribution δ6 (1). This will be done in Section 6.10 and can be considered as a test of the holographic formula. In terms of the coordinates r2 = ρ, the metric g reads g=

1 dρ2 + hρ , 4ρ2 ρ

hρ = h(0) + ρh(2) + ρ2 h(4) + · · · .

Then the condition Ric(g) + ng = 0 implies the system

˙ h˙ + (n−2)h˙ + tr(h−1 h)h ˙ + Ric(h) = 0, ¨ − 2hh ˙ −1 h˙ + tr(h−1 h) −ρ 2h ¨ − tr(h−1 h) and the conditions

1 ˙ −1 h) ˙ =0 tr(h−1 hh 2

˙ =0 ∇k h˙ ik − ∇i tr(h−1 h)

(6.9.9)

(6.9.10) (6.9.11)

(6.9.12)

for all i. (6.9.10) and (6.9.11) express the respective vanishing of the tangential and the normal components of Ric(g) + ng. Here dots denote derivatives with respect to ρ. Proof of Theorem 6.9.1. For ρ = 0, condition (6.9.10) yields (n−2)h(2) + tr(h−1 (0) h(2) )h(0) + Ric(h(0) ) = 0. Hence

−1 (n−2) tr(h−1 (0) h(2) ) + n tr(h(0) h(2) ) + τ (h(0) ) = 0,

i.e., tr(h(2) ) = −

1 τ (h(0) ) = −J(h(0) ), 2(n−1)

242

Chapter 6. Conformally Covariant Families

where the trace is with respect to h(0) . Therefore, h(2) = −

1 Ric(h(0) ) − J(h(0) )h(0) = −P(h(0) ). n−2

This proves the ﬁrst assertion of Theorem 6.9.1. Now (6.9.11) yields at ρ = 0, 2 tr(h−1 (0) h(4) ) −

1 2 tr((h−1 (0) h(2) ) ) = 0, 2

i.e., tr(h(4) ) =

1 tr(P2 ) 4

(6.9.13)

using h(2) = −P and (2.5.17). We diﬀerentiate (6.9.10) once with respect to ρ and obtain ˙ + tr(h−1 ˙ h)h ¨ + 2hh ˙ −1 h˙ − tr(h−1 h) ˙ h˙ + Ric(h) ˙ + tr(h−1 h)h ¨ + tr(h−1 h) ˙ h˙ = 0. (n−4)h For ρ = 0, we get 2(n−4)h(4) + 2Ph−1 P − tr(P2 )h +

1 tr(P2 )h + (d/dρ)|0 (Ric(hρ )) = 0 2

˙ = −h−1 hh ˙ −1 and (6.9.13). Here we have simpliﬁed the notation by using h−1 setting h(0) = h. This convention will be in force in what follows. Hence 2(n−4)h(4) −

1 tr(P2 )h + 2Ph−1 P + (d/dρ)|0 (Ric(hρ )) = 0, 2

i.e., we obtain the intermediate formula 1 2 1 1 k h(4) ij = |P| hij − Pi Pkj − (d/dρ)|0 (Ric(hρ ))ij . n−4 4 2

(6.9.14)

In order to evaluate the latter formula, we use the following formula for the ﬁrst variation of the Ricci tensor. ˙ Ric(g)[h] ij = (d/dt)|t=0 Ricij (g +th) 1 = − {ΔL (h)ij − [∇i (δg (h))j + ∇j (δg (h))i ] + Hessg (trg (h))ij } , 2 where ΔL : Γ(S 2 T ∗ M ) → Γ(S 2 T ∗ M ) is the Lichnerowicz-Laplacian on symmetric bilinear forms. ΔL is given by the formula ΔL (h)ij = ∇k ∇k (h)ij − Ricik (g)hkj + Ricjk (g)hki + 2Rrijs hrs .

6.9. The holographic coeﬃcients v2 , v4 and v6

243

For details we refer to [24], Theorem 1.174 and [77] (Chapter 3 and Appendix A). In the present application, we consider the variation h − tP. Hence ˙ (d/dρ)|0 (Ric(hρ ))ij = Ric(h)[−P] ij 1 = {ΔL,h (P)ij + Hessh (J)ij − (∇i (dJ)j + ∇j (dJ)i )} 2 1 1 = ΔL,h (P)ij − Hessh (J)ij 2 2 using δh (P) = dJ (Lemma 4.2.7). Now for the contribution of the LichnerowiczLaplacian, we ﬁnd ΔL,h (P)ij = Δ(P)ij − Ricik Pkj + Ricjk Pki + 2Rrijs Prs , where all curvature tensors are with respect to h. Hence we obtain the formula h(4) ij =

1 1 2 1 |P| hij − Pki Pkj + Hessij (J) n−4 4 4 1 1 1 Ricik Pkj + Ricjk Pki . (6.9.15) − Δ(P)ij − Rrijs Prs + 4 2 4

In the remaining step, we rewrite the Hessian of J using the identity ∇k ∇j (P)ki = Hessij (J) − Kjk (P)ki ,

(6.9.16)

where the curvature contribution K ∈ C ∞ (∧2 T ∗ M, End(S 2 T ∗ M )) is characterized by ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] (b) = K(X, Y )(b) for all symmetric bilinear forms b. In order to prove (6.9.16), we use normal coordinates at m and ﬁnd ∇k ∇j (P)(ek , ei ) = ∇j ∇k (P)(ek , ei ) − K(ej , ek )(P)(ek , ei ) k

k

= ∇j

∇k (P)(ek ) (ei ) − K(ej , ek )(P)(ek , ei )

k

= ∇j (δ(P))(ei ) − K(ej , ek )(P)(ek , ei ) = ∇j (dJ)(ei ) − K(ej , ek )(P)(ek , ei ) = Hess(J)(ei , ej ) − K(ej , ek )(P)(ek , ei ) using Lemma 4.2.7. Next, we determine an explicit formula for the curvature term. Using the general rule K(X, Y )(b)(U, V ) = −b(R(X, Y )U, V ) − b(U, R(X, Y )V ), b ∈ Γ(S 2 T ∗ M ),

244

Chapter 6. Conformally Covariant Families

we ﬁnd

Kjk (P)ki = −P( Rjkkl el , ei ) − P(ek , Rjkil el ) =−

l

l

Rjkkl Pil −

l

Cjkil Plk +

l

(P h)jkil Plk .

l

Hence Hessij (J) = ∇k ∇j (P)ki − Cjkil Plk − Ricjl Pli + (P h)jkil Plk .

(6.9.17)

(6.9.15) implies 1 2 |P| hij − Pki Pkj 4 1 k ∇ ∇j (P)ki − ∇k ∇k (P)ij + 4 1 1 1 − Cjkil Pkl − Ricjl Pli + (P h)jkil Plk 4 4 4 1 1 1 s r − Crij Ps + (P h)rijs Prs + (Ricik Pkj + Ricjk Pki ) 2 2 4 1 1 k k = ∇ ∇j (P)ki − ∇ ∇k (P)ij − Ckijl Pkl 4 4 1 2 1 1 k + |P| hij − Pi Pkj + (P h)rijs Prs + Ricik Pkj . 4 4 4

(n−4)h(4)ij =

Now using the identity (P h)rijs Prs = −Pij J + 2Pil Plj − |P|2 hij , we ﬁnd 1 2 1 1 |P| hij − Pki Pkj + (P h)rijs Prs + Ricik Pkj 4 4 4 1 1 1 1 1 1 = |P|2 hij − Pki Pkj − Pij J + Pil Plj − |P|2 hij + (n−2)Pik Pkj + JPij 4 4 2 4 4 4 1 = (n−4)Pki Pkj . 4 Hence (n−4)h(4)ij =

1 1 k n−4 k ∇ ∇j (P)ki − ∇k ∇k (P)ij − Ckijl Pkl + P Pkj . 4 4 4 i

In order to determine h(2) , we have used (6.9.10) at ρ = 0. (6.9.11) and (6.9.12) do not impose further conditions on h(2) . In fact, (6.9.11) at ρ = 0 yields a condition on h(4) and (6.9.12) at ρ = 0 states that ∇k (P)ik = ∇i (J)

6.9. The holographic coeﬃcients v2 , v4 and v6

245

and thus follows from (4.2.15). In order to determine h(4) , we have used the ﬁrstorder derivative of (6.9.10) at ρ = 0 and (6.9.11) at ρ = 0. The ﬁrst-order derivative of (6.9.12) at ρ = 0 is satisﬁed as well. Here we omit the detailed discussion of that point and refer to [122]. This completes the proof of Theorem 6.9.1. Theorem 6.9.1 implies that h(4) ij =

1 k P Pkj 4 i

(6.9.18)

if h is Bach-ﬂat (in dimension n ≥ 6). By (6.9.4), this is true if h is conformally ﬂat (C = 0). Now we use Theorem 6.9.1 for the calculation of the coeﬃcient v6 . The main step is the following technical result. Lemma 6.9.1. If the matrices A, B and C satisfy the relation det(I + tA + t2 B + t3 C + · · ·) = 1 + at + bt2 + ct3 + · · · , then 1 det I +tA+t2 B +t3 C + · · · 2 1 a a2 2 1 a a2 =1+ t+ b− t + c− b− t3 + · · · . (6.9.19) 2 2 4 2 2 4 and a = tr(A), b = tr(B) +

1 (tr A)2 − tr(A2 ) , 2

c = tr(C) + [tr A tr B − tr(AB)] +

1 1 1 tr(A)3 − tr A tr(A2 ) + tr(A3 ). 6 2 3

Proof. (6.9.19) follows by taking squares. For the proof of the remaining identities we compare the coeﬃcients of the power series of log det = tr log for the matrix (I + tA + t2 B + t3 C + · · ·) with log(1 + at + bt2 + ct3 + · · ·). The relation det(I + tA) = p tr(∧p A)tp shows that the formulas for the coeﬃcients a, b and c in Lemma 6.9.1 can be written also in the form a = tr(A), b = tr(B) + tr(∧2 A), c = tr(C) + [tr A tr B − tr(AB)] + tr(∧3 A).

246

Chapter 6. Conformally Covariant Families

The relations between both formulas are known as Newton’s relations σ1 = s1 , 2σ2 = s21 − s2 , 6σ3 =

s31

− 3s1 s2 + 2s3

(6.9.20) (6.9.21)

between the elementary symmetric functions σk = tr(∧k T ) and the functions sk = tr(T k ) of a linear map T . Lemma 6.9.1 and the series hr = h + r2 h(2) + r4 h(4) + · · · , i.e., hr = 1 − r 2 P + r 4

1 −B + (n−4)P2 + · · · 4(n−4)

imply that the ﬁrst three coeﬃcients in 1 1 det(hr ) 2 vol(hr ) 2 4 6 2 = v(r) = = det(h−1 0 hr ) = 1 + r v2 + r v4 + r v6 + · · · vol(h0 ) det(h0 ) are given by 1 1 v2 = − tr(P) = − J, 2 2 1 1 1 1 2 2 2 2 v4 = tr(P ) + tr(P) − tr(P ) − tr(P) 2 4 2 4 1 1 1 = − tr(P2 ) + tr(P)2 2 4 4 1 2 = (J − |P|2 ) 8 1 = (s21 (P) − s2 (P)) 8 1 = σ2 (P) 4 1 = tr(∧2 P) 4 using (6.9.20), and (after some calculation) v6 =

1 1 1 1 (tr(h(2) )3 − tr(h(2) ) tr(h2(2) ) + tr(h3(2) ) − tr(h(2) h(4) ) 48 16 12 6

(6.9.22)

using 2 1 tr(h(2) h(4) ) − tr(h3(2) ). (6.9.23) 3 6 The latter formula follows from (6.9.11) by diﬀerentiation at ρ = 0. In fact, diﬀerentiation yields −1 ˙ · −1 ˙ ¨ + tr(h−1 ... tr((h−1 )· h) (h h)). h ) = tr((h h) tr(h(6) ) =

6.9. The holographic coeﬃcients v2 , v4 and v6

247

Hence −2tr h−1 h(2) h−1 h(4) + 6tr h−1 h(6) = −tr (h−1 h(2) )3 + 2tr h−1 h(4) h−1 h(2) . This proves the claim. Thus we ﬁnd v6 = −

1 1 1 1 1 tr(P)3 + tr(P) tr(P2 ) − tr(P3 ) − tr(BP) + tr(P3 ) 48 16 12 24(n−4) 24

i.e., 1 1 tr(BP) + − tr(P)3 + 3 tr(P) tr(P2 ) − 2 tr(P3 ) 24(n−4) 48 1 1 (B, P) − tr(∧3 P). =− 24(n−4) 8

v6 = −

Here we have used Newton’s relation 6σ3 = s31 − 3s1 s2 + 2s3 . This completes the proof of Theorem 6.9.2. Finally, it is sometimes useful to rewrite the formula for v6 in Theorem 6.9.2 in terms of Riemannian curvature. Theorem 6.9.3. Let n = 6. Then v6 =

1 1 1 3 1 τ | Ric |2 − τ − Rikjl Ricij Rickl 64 12 100 6

1 1 1 τ Δτ . (6.9.24) + (Ric, Hess(τ )) − (Δ Ric, Ric) + 30 12 120

Since the formula in Theorem 6.9.3 involves the Hessian of τ , the most direct proof of that formula rests on the formula (6.9.14) for the metric h(4) and a calculation as in the proof of Theorem 6.9.2. Here are the details. Proof. Using (6.9.14) and (6.9.22), we ﬁnd v6 = −

1 1 1 tr(P)3 + tr(P) tr(P2 ) − tr(P3 ) 48 16 12 1 1 1 2 1 |P| J − tr(P3 ) + (Hess(J), P) − (ΔP, P) + 6(n−4) 4 4 4 1 s r ij 1 − Rrij Ps P + Ricik Pkj Pij + Ricjk Pki Pij . 2 4

Now we write

1 n−2

1 Ric = P + n−2 Jh. Then the contributions of tr(P3 ) cancel since

−

1 1 n−2 + −1 + = 0. 12 6(n−4) 2

248

Chapter 6. Conformally Covariant Families

For n = 6, we are left with 1 1 1 1 tr(P)3 + J|P|2 + (Hess(J), P) − (ΔP, P) 48 8 48 48 1 1 s − Rrij (Ricrs −Jgsr ) Ricij −Jg ij 24 16 1 1 1 1 3 2 = − tr(P) + J|P| − Rrijs Ricrs Ricij −2J| Ric |2 + J2 τ 48 8 24 16 1 1 (6.9.25) + (Hess(J), P) − (ΔP, P). 48 48

v6 = −

Now

1 7 2 2 tr(P ) = |P| = | Ric | − τ , 16 50 1 1 (Hess(τ ), Ric) − τ Δτ ( Hess(J), P) = 40 400 2

2

and (ΔP, P) =

1 16

7 (Δ Ric, Ric) − τ Δτ . 50

These results suﬃce to determine the coeﬃcients in (6.9.25). We ﬁnd 1 1 1 1 =− , 48 16 64 12 1 1 1 1 = , 48 40 64 30 1 1 1 7 1 1 − + = , 48 400 48 800 64 120 1 1 2 1 1 1 1 1 + = , 8 10 16 24 16 10 64 12 1 1 1 7 1 1 1 1 1 1 1 − − =− , − 48 1000 8 10 16 50 24 16 100 64 100 1 1 . − 64 6

(Δ Ric, Ric) : − ( Hess(τ ), Ric) : τ Δτ

:

τ | Ric |2

:

τ3

:

Rrijs Ricrs Ricij

:

The proof is complete.

Example 6.9.1. In order to check signs, we test Theorem 6.9.3 for S 6 with the canonical metric gc . Then Ric = 5gc , P = 12 gc , P = 12 id, J = 3 and R = −P gc = − 21 (gc gc ). Hence the formula in Theorem 6.9.3 gives 1 303 1 20 1 v6 = · 30 · 25 · 6 − − · 25 · 30 = − 64 12 100 6 64 using 1 1 Rkijl Ricij Rickl = − 52 (gc gc )kijl gcij gckl = 25(62 − 6). 2 2

6.9. The holographic coeﬃcients v2 , v4 and v6

249

On the other hand, we have 1 11 v6 = − tr ∧3 (P ) = − 8 88

20 6 =− 3 64

by Theorem 6.16.1. It also will be useful to collect the information on traces of h(2) , h(4) and h(6) . Lemma 6.9.2. tr(h(2) ) = − tr(P), 1 1 tr(h(4) ) = tr(P2 ) = |P|2 , 4 4 1 1 (B, P). tr(h(6) ) = 3! n−4 Proof. Theorem 6.9.1 directly implies the ﬁrst and the second equation. Moreover, combined with (6.9.23) it also yields the third equation. The following comments should facilitate a comparison with related results in the literature. We do not intend to be exhaustive in any sense and we do not discuss priorities. The full details of the Graham-Feﬀerman construction in [99] became available through [96]. Explicit formulas for h(2) , h(4) and h(6) are displayed in [96], Chapter 3. In addition, the work [117] describes the full expansion in terms of (extended) obstruction tensors, and derives formulas for the holographic coeﬃcients v2j in these terms. Explicit formulas are given up to v8 . We continue with comments on some earlier works. The original work [141] states formulas for v4 (for n = 4) as given in Theorem 6.9.2 and v6 (for n = 6) as given in Theorem 6.9.3 (up to normalizing coeﬃcients). Note that in [141], the sign conventions for curvature tensors diﬀer from ours: their curvature R coincides with ours, but Ric and τ have opposite signs. [8] gives formulas for h(2) and h(4) for general dimension n. These results coincide with (6.9.1) and (6.9.2). [8] rewrites the Henningson-Skenderis formula for v6 (n = 6) in terms of the Bach tensor. The result corresponds to Theorem 6.9.2. Again, the sign conventions diﬀer: R is opposite to ours, but Ric and τ coincide. In [8], the authors observe that the metric variation of the total holographic anomaly vn vol, Mn

or equivalently, the metric variation of the total critical Q-curvature, yields the Feﬀerman-Graham obstruction tensor. That result is proved in full generality in [122]. [8] also contains an explicit, but structurally unclear formula, for v8 . For a review see also [9]. The seminal work [119] gives formulas for v4 (for n = 4) and v6 (for n = 6). These results correspond to the results in Theorem 6.9.2. Here

250

Chapter 6. Conformally Covariant Families

the sign conventions on R, Ric and τ are as ours, but C has opposite sign. [80] gives a formula for h(4) in terms of R, Ric and τ for general n (see (A.1) in [80]). That formula is equivalent to the corresponding formula in Theorem 6.9.1. The sign conventions in [80] are the same as in [141]. The same formula for h(4) is also presented in [147]. Lemma 6.9.2 is also in [96]. In [175], the authors use the ambient metric to derive formulas for the GJMSoperators and conjecture the existence of a relation between the constant term of P2N and n

−N v2N 2 in any dimension n ≥ 2N , up to terms which are O(( n2 − N )2 ). The following comments should shed light on the situation. For N = 2 and n ≥ 4, we have n 2 J − 2|P|2 − ΔJ 2

n = 16v4 + 2Δv2 + −2 J2 . 2

Q4,n =

Here the term Δv2 is an additional divergence term. The appropriate way to write Q4,n seems to be the holographic formula

n −2 J v2 , Q4,n = 16v4 − 2 Δ− 2 where the operator Δ−( n2 −2)J is interpreted as a multiple of T2∗ ( n2 −2). Similarly, for N = 3 and n ≥ 6, (6.10.13) shows that Q6,n = −8 · 48v6 + (Δ2 J − 8δ(P#dJ)) + 4Δ(|P|2 − J2 ) up to terms which are O( n2 − 3). Theorem 6.10.4 seems to be the appropriate way to express Q6,n in terms of holographic data. The following formula for the subcritical Q2N,n generalizes these results and the holographic formula (Theorem 6.6.6) for the critical Q-curvature. Conjecture 6.9.1. For even n and 2N ≤ n, 4N cN Q2N,n (h) =

n

(2N −2j)T2j∗ h; −N (v2N −2j ), 2 j=0

N −1

where cN = (−1)N (22N N !(N −1)!)−1 .

(6.9.26)

6.9. The holographic coeﬃcients v2 , v4 and v6

251

It is convenient to summarize the holographic formulas in the matrix form ⎛ ⎞ nc n2 Qn ⎜ ⎟ ⎜(n−2)c n2 −1 Qn−2 ⎟ ⎜ ⎟ ⎜ ⎟ .. 2⎜ ⎟ . ⎜ ⎟ ⎟ ⎜ 4c2 Q4 ⎠ ⎝ 2c1 Q2 ⎛

1 T2∗ (0) T4∗ (0) . . . ⎜ 1 T2∗ (1) . . . ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ 0 ⎝

⎞⎛ ⎞ ∗ ∗ Tn−4 (0) Tn−2 (0) nvn ⎟⎜ ⎟ ∗ ∗ Tn−6 (1) Tn−4 (1) ⎟ ⎜(n−2)vn−2⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ .. .. .. ⎟⎜ ⎟ . (6.9.27) . . . ⎟⎜ ⎟ ⎟⎜ ⎟ 1 T2∗ ( n2 −2)⎠ ⎝ 4v4 ⎠ 1 2v2

(6.9.27) describes how on any Riemannian manifold (M, h) of even dimension n, the two systems of curvature quantities {Q2 , Q4 , . . . , Qn }

and

{v2 , v4 , . . . , vn }

are related by an invertible upper triangle matrix. Its entries are relatives of the operators T2N ( n2 − N ) which are constant multiples of the GJMS-operators P2N . For n = 6, (6.9.27) will be proved in Section 6.10. Conjecture 6.9.1 implies that Q2N diﬀers from a multiple of v2N by the sum of the divergence term N −1

(2N −2j)T2j∗ (0) (v2N −2j )

j=1

and a multiple of ( n2 − N ). This reﬁnes the idea of [175]. Finally, we prove the following result which has been used in the previous section. Similar results are stated in [80] (Appendix D, without proofs). Theorem 6.9.4. Let u ∼ j≥0 rλ+2j a2j (λ) be a formal solution of the eigenequation −Δg u = λ(n − λ)u for the Laplacian of the Poincar´e-Einstein metric g = r−2 (dr2 +hr ). Then 1 (Δ−λJ)a0 a2 (λ) = 2(n−2−2λ) and a4 (λ) =

1 1 1 (Δ−(λ+2)J) a2 (λ)− λ|P|2 a0 −δ(P#da0 )− (dJ, da0 ) . 4(n−4−2λ) 2 2

Here P, J, Δ and the scalar product are to be understood with respect to h0 = h.

252

Chapter 6. Conformally Covariant Families

Proof. In terms of the coordinates r2 = ρ, the metric is given by (6.9.9), and the Laplacian reads Δg = ρΔhρ + 4ρ2

∂2 ∂ ∂ + 2ρ2 (log det hρ ) . − 2(n−2)ρ 2 ∂ρ ∂ρ ∂ρ

λ

Now for u(ρ) = ρ 2 v(ρ) with v(ρ) = v0 + ρv1 + ρ2 v2 + · · · , the equation −Δg u = λ(n−λ)u is equivalent to λ λ λ λ λ λ λ λ +1 +2 +1 2 2 2 2 2 −1 ρ v + λρ ρ v+ρ v +ρ v v − 2(n−2) 4 2 2 2 λ λ λ λ λ +1 ρ 2 v + ρ 2 +2 v + ρ 2 +1 Δhρ (v) + λ(n−λ)ρ 2 v = 0, + 2(log det hρ ) 2 i.e., 4ρv − 2(n−2−2λ)v + Δhρ (v) + (log det hρ ) (λv + 2ρv ) = 0.

(6.9.28)

We compare the coeﬃcients of ρ0 and ρ1 and ﬁnd 1 −2(n−2−2λ)v1 + Δh (v0 ) + λ(log det hρ ) 1ρ=0 v0 = 0

(6.9.29)

and − 4(n−4−2λ)v2 + [Δh (v1 ) + (d/dt)|0 (Δh−tP )(v0 )] + λ [(log det hρ ) |0 v1 + (log det hρ ) |0 v0 ] + 2(log det hρ ) |0 v1 = 0. (6.9.30) But using (log det hρ ) |ρ=0 = (tr log hρ ) |ρ=0 = − tr(P) = −J, (6.9.29) reads −2(n−2−2λ)v1 + Δh (v0 ) − λJv0 = 0. This proves the ﬁrst assertion. In order to prove the second assertion, we apply the identity 1 (log det hρ ) |ρ=0 = − |P|2 . 2 In fact, 2 (det hρ ) (det hρ ) (log det hρ ) = − det hρ det hρ and

det hρ 1 tr(P)2 − tr(P2 ) + · · · = 1 − ρ tr(P) + ρ2 tr(h(4) ) + det h0 2 1 1 = 1 − ρJ + ρ2 − |P|2 + J2 + · · · 4 2

6.9. The holographic coeﬃcients v2 , v4 and v6

253

(by Lemma 6.9.1 and (6.9.13)) gives 1 1 1 (log det hρ ) |ρ=0 = 2 − |P|2 + J2 − J2 = − |P|2 . 4 2 2 Then (6.9.30) reads 1 −4(n−4−2λ)v2 + [Δh (v1 ) + (d/dt)|0 (Δh−tP )(v0 )] − λJv1 − λ|P|2 v0 − 2Jv1 = 0. 2 In order to evaluate the variation of the Laplacian, we apply the general formula 1 (d/dt)|0 (Δg+th )(u) = −(Hessg (u), h)g + (du, δg (h))g + (du, d(trg (h)))g (6.9.31) 2 ([24], Proposition 1.184). For the convenience of the reader, we sketch a proof of def

(6.9.31). For the family G(t) = g + th, we ﬁnd g ˙g d/dt|t=0 (G(∇G X Y, Z)) = h(∇X Y, Z) + g(∇X Y, Z),

˙ g Y = d/dt|t=0 (∇G Y ). Hence where ∇ X X 1 X(h(Y, Z)) + Y (h(Z, X)) − Z(h(X, Y )) 2 − h(X, [Y, Z]) + h(Y, [Z, X]) + h(Z, [X, Y ]) − h(∇gX Y, Z) 1 = {∇gX (h)(Y, Z) + ∇gY (h)(X, Z) − ∇gZ (h)(X, Y )} . (6.9.32) 2

˙ g Y, Z) = g(∇ X

Now write ΔG = −δG d = trG (∇G d). Then ˙ g du). ˙ g (∇g du) + trg (∇ d/dt|t=0 (ΔG u) = tr ˙ g (ω) = −g(h, ω) for ω ∈ Ω1 ⊗ Ω1 and (6.9.32) implies, for α ∈ Ω1 , But tr ˙ g α) = − trg (∇

i

∇gei (h)(ei , α ) +

1 trg (∇gα (h)) 2

1 1 = (δg (h), α)g + ∇gα trg (h) = (δg (h), α)g + (α, d(trg (h)))g . 2 2 Hence 1 d/dt|t=0 (ΔG u) = −(h, ∇g du)g + (δg (h), du)g + (du, d(trg (h)))g . 2 This proves the variational formula (6.9.31).

254

Chapter 6. Conformally Covariant Families

In our case, (6.9.31) yields 1 (d/dt)|0 (Δh−tP )(u) = (Hessh (u), P)h + (du, dJ)h − (du, dJ)h 2 using −δP = dJ (Lemma 4.2.7). Now Lemma 4.2.6 implies 1 (d/dt)|0 (Δh−tP )(u) = −δ(P#du) − (du, dJ). 2

(6.9.33)

The proof is complete.

6.10 The holographic formula for Q6 In the present section, we derive explicit formulas for Q6 in dimension n ≥ 6 from the asymptotics of eigenfunctions. We use the result to conﬁrm the holographic formula (6.6.24) for the critical Q6 . This also completes the discussion in Example 6.6.5. In particular, we evaluate here the term δ6 (1) in the formula (6.6.43). The latter result illustrates Theorem 6.6.4. Next, we prove an extension of the holographic formula to all subcritical Q-curvatures Q6,n . Finally, we relate the explicit formulas for all Q-curvatures Q6 to those given in [116]. The following calculations rest on the family P6 (λ) : C ∞ (M n ) → C ∞ (M n ) which is deﬁned by the asymptotics u∼ rλ+2j a2j (λ), −Δg u = λ(n−λ)u j≥0

and a6 (λ) =

1

P6 (λ) 23 3!(n−2λ−2)(n−2λ−4)(n−2λ−6)

(see (6.6.3)). In terms of the notation used in the proof of Theorem 6.9.4, we diﬀerentiate (6.9.28) twice at ρ = 0 and obtain the condition

1 − 6(n−6−2λ)v3 + Δ (v0 ) + Δ (v1 ) + Δ(v2 ) 2 1 + λ(log det h) v0 + (λ+2)(log det h) v1 + (λ+4)(log det h) v2 = 0, 2 where we use the abbreviations Δ = Δh ,

Δ = (d/dρ)|ρ=0 (Δhρ ),

Δ = (d2 /dρ2 )|ρ=0 (Δhρ )

and (log det h)(k) = (dk /dρk )|ρ=0 (log det hρ ), k ≥ 1.

6.10. The holographic formula for Q6

255

Using v1 =

1 P2 (λ)v0 2(n−2−2λ)

and v2 =

1 P4 (λ)v0 , (6.10.1) 8(n−4−2λ)(n−2−2λ)

we ﬁnd 1 48(n−6−2λ)(n−4−2λ)(n−2−2λ) " × [4(n−4−2λ)(n−2−2λ)Δ + 4(n−4−2λ)ΔP2 (λ) + ΔP4 (λ)] (v0 )

v3 =

+ 4λ(n−4−2λ)(n−2−2λ)(log det h) v0

# + 4(λ+2)(n−4−2λ)(log det h) P2 (λ)v0 + (λ+4)(log det h) P4 (λ)v0 , i.e., P6 (λ)u = 4(n−4−2λ)(n−2−2λ) [Δ + λ(log det h) ] (u) + 4(n−4−2λ) [(λ+2)(log det h) P2 (λ) + Δ P2 (λ)] (u) + (Δ−(λ+4)J)P4 (λ)u.

(6.10.2)

This formula will be the main tool in the proof of the following result. Theorem 6.10.1 (The explicit formula for Q6 ). On Riemannian manifolds of dimension n = 6, Q6 = −8 · 48v6 + Δ2 J − 8δ(P#dJ) + 4Δ |P|2 − J2 . Proof. We apply the identity Q6 = −P˙ 6 (0)(1) (see (3.2.20)) using (6.10.2) for n = 6. First of all, we observe that Δ (1) = 0 and P2 (0)(1) = P4 (0)(1) = 0 (if n = 6). Hence −Q6 = 32(log det h) + 16(log det h) P˙2 (0)1 + 8Δ P˙ 2 (0)1 + (Δ − 4J)P˙4 (0)1 = 32(log det h) − 16(log det h) J − 8Δ J + (Δ − 4J)P˙4 (0)1. (6.10.3) Now we recall that, by Lemma 6.9.1 and the discussion following it, det(hρ ) = 1 + tr(A)ρ + tr(B) + tr(∧2 A) ρ2 det(h0 ) + tr(C) + [tr(A) tr(B) − tr(AB)] + tr(∧3 A) ρ3 + · · · with A = −P,

1 1 B = − B + P2 8 4

(Theorem 6.9.1) and tr(C) =

1 2 tr(AB) − tr(A3 ). 3 6

((6.9.23))

256

Chapter 6. Conformally Covariant Families

Hence for (log det hρ ) =

(det hρ ) − det hρ

(det hρ ) det hρ

2

and (log det hρ ) =

(det hρ ) (det hρ ) (det hρ ) −3 +2 det hρ (det hρ )2

(det hρ ) det hρ

3 ,

we ﬁnd 1 (log det hρ ) |ρ=0 = 2 tr(B) + tr(∧2 A) − (tr A)2 = − |P|2 2 (as in the proof of Theorem 6.9.4) and (log det h) = 6 tr(C) + [tr(A) tr(B) − tr(AB)] + tr(∧3 A) − 3 · 2 tr(B) + tr(∧2 A) tr(A) + 2(tr A)3 . Therefore, − Q6 = 32 6 tr(C) − 6 tr(AB) + 6 tr(∧3 A) − 6 tr(∧2 A) tr(A) + 2(tr A)3 1 2 + 8|P| J + 8 δ(P#dJ) + (dJ, dJ) 2 + (Δ − 4J)(−ΔJ + 2J2 − 4|P|2 ) using (6.9.33) and P˙4 (0)1 = −ΔJ + 2J2 − 4|P|2 (see (6.6.41) for n = 6). Separating terms without derivatives, we ﬁnd that −Q6 equals 32 4 tr(AB) − tr(A3 ) − 6 tr(AB) + 6 tr(∧3 A) − 6 tr(∧2 A) tr(A) + 2(tr A)3 − 24 tr(A) tr(A2 ) + 8(tr A)3 + 8δ(P#dJ) + 4(dJ, dJ) − Δ2 J + 2ΔJ2 − 4Δ|P|2 + 4JΔJ. Now Newton’s identities 1 (tr A)2 − tr(A2 ) , 2 1 3 (tr A)3 − 3 tr(A) tr(A2 ) + 2 tr(A3 ) tr(∧ A) = 6 tr(∧2 A) =

(see (6.9.20), (6.9.21)) show that 6 tr(∧3 A) − 6 tr(∧2 A) tr(A) + 2(tr A)3 = 2 tr(A3 ).

6.10. The holographic formula for Q6

257

Hence − Q6 = 32 −2 tr(AB) + tr(A3 ) − 24 tr(A) tr(A2 ) + 8(tr A)3 − (Δ2 J − 8δ(P#dJ)) − 4Δ(|P|2 − J2 ). Using the explicit formulas for the coeﬃcients A and B, it follows that the latter formula is equivalent to −Q6 = −6 · 8 tr(∧3 P) − 8(P, B) − (Δ2 J − 8δ(P#dJ)) − 4Δ(|P|2 − J2 ). Now

1 1 v6 = − tr(∧3 P) − (B, P) 8 48

(see Theorem 6.9.2) completes the proof. Corollary 6.10.1. Q6 can be written in the form Q6 = −8 · 48v6 − 32P2∗ (0)(v4 ) − 2P4∗ (0)(v2 )

(6.10.4)

since P2∗ (0) = Δ and P4∗ (0)J = Δ2 J − 8δ(P#dJ) (see (6.6.42)). This shows that Theorem 6.10.1 is a special case of Theorem 6.6.6 stating that 1 1 Q6 = −26 (6v6 + P2∗ (0)(v4 ) + P4∗ (0)(v2 )). 2 32 In order to relate Theorem 6.10.1 and Corollary 6.10.1 to (6.6.43), it remains to determine the quantity δ6 (1). Theorem 6.10.2.

δ6 (1) = P˙6 (0) − P˙ ∗ (0) (1) = 4 Δ(|P|2 − J2 ) + (Δ2 J − 8δ(P#dJ)) 6

= −32P2∗ (0)(v4 ) − 8P4∗ (0)(v2 ). Theorem 6.10.2 is a special case of Theorem 6.6.4. In fact, Theorem 6.6.4 states that 6δ6 (1) = −26 3!2! (2T2∗ (0)(v4 ) + 4T4∗ (0)(v2 )) . In view of T2 (0) =

1 P2 (0) 8

and T4 (0) =

1 P4 (0) 64

(see (6.10.1)), this formula is equivalent to 1 ∗ 1 P2 (0)(v4 ) + P4∗ (0)(v2 ) = −32P2∗(0)(v4 ) − 8P4∗ (0)(v2 ). δ6 (1) = −27 4 16

258

Chapter 6. Conformally Covariant Families

Proof. (6.10.2) yields (for n = 6) P˙ 6 (0)u = −48Δ u + 32(log det h) u − 8(log det h) P2 (0)u + 16(log det h) P˙2 (0)u − 8Δ P2 (0)u + 8Δ P˙2 (0)u − JP4 (0)u + (Δ − 4J)P˙4 (0)u. (6.10.5) The last two terms in (6.10.5) contribute to δ6 (u) by −JP4 (0)u + P4∗ (0)(Ju) + (Δ − 4J)P˙4 (0)u − P˙4∗ (0)(Δ − 4J)u, i.e., for u = 1 by P4∗ (0)J + (Δ − 4J)(P˙4 (0)(1)) − P˙4∗ (0)(Δ − 4J)(1).

(6.10.6)

In order to ﬁnd explicit formulas for these terms, we use the formula in Remark 6.6.5. We ﬁnd P4∗ (0)J = Δ(Δ − 2J)J − 8δ(P#dJ) − 4δ(JdJ) and P˙4 (0)u = −Δ(Ju) − JΔu + 2J2 u − 4|P|2 u + 4δ(P#du) + 2(dJ, du), i.e., P˙4∗ (0)u = −JΔu − Δ(Ju) + 2J2 u − 4|P|2 u + 4δ(P#du) + 2δ(udJ). Therefore, (Δ − 4J)(P˙4 (0)(1)) = (Δ − 4J)(−ΔJ + 2J2 − 4|P|2 ) −P˙ ∗ (0)(Δ − 4J)(1) = 4P˙ ∗ (0)J 4

4

= −4JΔJ − 4Δ(J2 ) + 8J3 − 16|P|2 J + 16δ(P#dJ) + 8δ(JdJ). Hence (6.10.6) yields

Δ2 (J) − 2Δ(J2 ) − 8δ(P#dJ) − 4δ(JdJ) + −Δ2 (J) + 2Δ(J2 ) − 4Δ(|P|2 ) + 4JΔJ − 8J3 + 16J|P|2 − 4JΔJ − 4Δ(J2 ) + 8J3 − 16|P|2 J + 16δ(P#dJ) + 8δ(JdJ) = −6ΔJ2 − 4Δ|P|2 + 8δ(P#dJ). Next, the ﬁrst two terms in the last line of (6.10.5) contribute to δ6 (u) by

− 8Δ P2 (0)u + 8P2∗ (0)Δ∗ u + 8Δ P˙ 2 (0)u − 8P˙2∗ (0)Δ∗ u = −8Δ Δu + 8ΔΔ∗ u − 8Δ Ju + 8JΔ∗ u.

6.10. The holographic formula for Q6

259

Now by (6.9.33), 1 Δ u = −δ(P#du) − (du, dJ) 2

1 and Δ∗ u = −δ(P#du) − δ(udJ). 2

It follows that for u = 1, the sum −Δ Δu + ΔΔ∗ u − Δ Ju + JΔ∗ u yields 1 1 1 1 1 − Δ(δdJ) + δ(P#dJ) + (dJ, dJ) − JδdJ = Δ2 J + δ(P#dJ) + Δ(J2 ), 2 2 2 2 4 i.e., we ﬁnd the contribution 4Δ2 J + 8δ(P#dJ) + 2ΔJ2 .

(6.10.7)

The second term in the second line and the second term in the ﬁrst line of (6.10.5) trivially contribute to δ6 (u). The ﬁrst term in the second line of (6.10.5) contributes to δ6 (u) by 4|P|2 Δu − 4Δ|P|2 u, i.e., for u = 1, by −4Δ|P|2 .

(6.10.8)

It remains to determine the contribution of the second variation Δ , i.e., Δ (1) − Δ∗ (1). However, we avoid determining the second variation Δ by using the following argument. The self-adjointness of P6 (0) (known by the general theory) implies an explicit formula for the diﬀerence Δ (u) − Δ∗ (u) using (6.10.2). Here are the details. We have (for n = 6)

P6 (0)u = 32Δ u − 8|P|2 Δu − 8δ(P#dΔu) − 4(dΔu, dJ) + (Δ − 4J)P4 (0)u. Hence P6∗ (0)u = 32Δ∗ u − 8Δ(|P|2 u) − 8Δδ(P#du) − 4Δδ(udJ) + P4∗ (0)(Δ − 4J)u. Thus P6 (0) = P6∗ (0) is equivalent to 32(Δ − Δ∗ )u = 8 |P|2 Δu − Δ(|P|2 u) + 8 (δ(P#dΔu) − Δδ(P#du)) + 4 ((dΔu, dJ) − Δδ(udJ)) − (Δ − 4J)P4 (0)u + P4∗ (0)(Δ − 4J)u. In particular, for u = 1, we ﬁnd 32(Δ − Δ∗ )(1) = −8Δ(|P|2 ) + 4Δ2 J − 4P4∗ (0)J = −8Δ(|P|2 ) + 32δ(P#dJ), i.e.,

4(Δ − Δ∗ )(1) = −Δ(|P|2 ) + 4δ(P#dJ).

(6.10.9)

260

Chapter 6. Conformally Covariant Families

Summarizing the results, we obtain δ6 (1) = P˙6 (0)1 − P˙6∗ (0)1 = 12Δ(|P|2 ) − 48δ(P#dJ)

(by (6.10.9))

− 4Δ(|P| ) 2

(by (6.10.8))

+ 4Δ J + 8δ(P#dJ) + 2ΔJ 2

2

(by (6.10.7))

− 6ΔJ − 4Δ|P| + 8δ(P#dJ) 2

2

= 4Δ2 J − 4ΔJ2 − 32δ(P#dJ) + 4Δ(|P|2 ) = 4 Δ((|P|2 − J2 ) + (Δ2 J − 8δ(P#dJ) .

This completes the proof. Now using (6.10.4) and Theorem 6.10.2, we ﬁnd (2Q6 + 16 · 48v6 ) + δ6 (1) = [−64P2∗(0)(v4 ) − 4P4∗ (0)(v2 )] + [−32P2∗ (0)(v4 ) − 8P4∗ (0)(v2 )] = −96P2∗ (0)(v4 ) − 12P4∗ (0)(v2 ).

The latter formula is equivalent to (6.6.43). This completes the discussion in Example 6.6.5. These results for Q6 (for n = 6) are completely analogous to results for Q4 (for n = 4). In fact, the relation δ4 (1) = 2ΔJ = −4P2∗ (0)(v2 ) (see (6.6.40)) implies the equivalence of both formulas 2Q4 = δ4 (1) + 32v4 + 8P2∗ (0)(v2 ) (see Example 6.6.4) and Q4 = 16v4 + 2P2∗ (0)(v2 )

(6.10.10)

for the critical Q4 . The latter formula is equivalent to the deﬁnition of Q4 as 2(J2 − |P|2 ) − ΔJ. We continue with a discussion of the subcritical Q6 . The starting point is the formula n

n−6 −3 1 = − Q6,n P6 2 2 (see (3.1.3)). For n ≥ 6, (6.10.2) yields P6 (λ)1 = 4(n−4−2λ)(n−2−2λ)λ(log det h) + 4(n−4−2λ) [−λ(λ+2)(log det h) J − λΔ J] + (Δ−(λ+4)J) −λ(Δ−(λ+2)J)J − λ|P|2 (n−2−2λ) ,

6.10. The holographic formula for Q6

261

i.e., P6

n−6 " −3 1 = 32(log det h) − 4(n−2)(log det h) J − 8Δ J 2 2 n

n

# n − Δ− +1 J Δ− −1 J J − 4 Δ−( +1)J |P|2 . 2 2 2

n

Hence − Q6,n = 32(log det h) − 4(n−2)(log det h) J − 8Δ J

n

n

n − Δ− +1 J Δ− −1 J J − 4 Δ− +1 J |P|2 . 2 2 2

(6.10.11)

Note that for n = 6, we recover −Q6,6 = 32(log det h) − 16(log det h) J − 8Δ J − (Δ−4J) Δ−2J2 + 4|P|2 which by P˙4 (0)1 = −ΔJ + 2J2 − 4|P|2 coincides with (6.10.3). Now let A = −P and B = −

1 1 B + P2 . 4(n−4) 4

Similarly as in the proof of Theorem 6.10.1, (6.10.11) implies that −Q6,n coincides with 32 6 tr(C) − 6 tr(AB) + 6 tr(∧3 A) − 6 tr(∧2 A) tr(A) + 2(tr A)3 + 2(n−2)|P|2J + 8δ(P#dJ) + 4(dJ, dJ)

n

n

n −1 ΔJ2 + +1 JΔJ− +1 −1 J3 −4Δ|P|2 +2(n+2)J|P|2 , −Δ2 J+ 2 2 2 2 n

i.e., with 32 4 tr(AB) − tr(A3 ) − 6 tr(AB) + 6 tr(∧3 A) − 6 tr(∧2 A) tr(A) + 2(tr A)3

n

n +1 −1 (tr A)3 − 4n tr(A) tr(A2 ) + 2 n

2

n 2 + 8δ(P#dJ) + 4(dJ, dJ) − Δ J + −1 ΔJ2 − 4Δ|P|2 + +1 JΔJ. 2 2 The latter result implies Theorem 6.10.3. For n ≥ 6, 16 n2 −4 3 n−6 J + (dJ, dJ) + (B, P) Q6,n = 16 tr(P3 ) − 4nJ|P|2 + 4 2 n−4 1 + (Δ2 J − 8δ(P#dJ)) + Δ(16|P|2 − (3n−2)J2). 4

262

Chapter 6. Conformally Covariant Families

Theorem 6.10.3, obviously, generalizes Theorem 6.10.1 (using (6.9.21)). For closed M n , Theorem 6.10.3 implies Q6,n vol Mn n2 −4 3 n−6 16 3 2 16 tr(P ) − 4nJ|P| + = J + (dJ, dJ) + (B, P) vol . 4 2 n−4 Mn (6.10.12) If C = 0 (and hence B = 0), (6.10.12) coincides with Theorem 6.4 in [31] or [32]. Branson’s proof rests on invariance arguments and explicit calculations in test cases. In Section 6.11, we will give an alternative proof of the formula for Q6 for n = 6. The following result extends the holographic formula (6.10.4) to all subcritical Q6,n , i.e., conﬁrms Conjecture 6.9.1 for N = 3. Theorem 6.10.4. For n ≥ 6, Q6,n = −8 · 48v6 − 32P2∗

n

n

−3 (v4 ) − 2P4∗ −3 (v2 ). 2 2

Proof. First of all, a calculation using Theorem 6.9.2 shows that Theorem 6.10.3 can be written in the form Q6,n = −8 · 48v6 + (Δ2 J − 8δ(P#dJ)) + 4Δ(|P|2 −J2 ) n−6 −3ΔJ2 + 2(dJ, dJ) − 16J|P|2 + (n+6)J3 . + 4

(6.10.13)

Thus the assertion is equivalent to the relation Δ2 J − 8δ(P#dJ)) + 4Δ(|P|2 −J2 ) n−6 + −6JΔJ − 4(dJ, dJ) − 16J|P|2 + (n+6)J3 4 n n

−3 (J2 −|P|2 ) + P4∗ −3 (J). = −4P2∗ 2 2

(6.10.14)

But using P2∗ (λ) = Δ − λJ and P4∗ (λ) = (Δ−λJ)(Δ−(λ+2)J) − λ(n−2−2λ)|P|2 − 2(n−2−2λ)δ(P#d) − (n−2−2λ)δ(Jd) (see Theorem 6.9.4), we ﬁnd P2∗ ( n2 −3) = Δ−( n2 −3)J and n

− 3 (J) P4∗ 2

n

n −3 J Δ− − 1 J − 2(n − 6)|P|2 − 8δ(P#d) − 4δ(JdJ). = Δ− 2 2

6.10. The holographic formula for Q6

263

Hence the right-hand side of (6.10.14) equals

n

n

n n −1 ΔJ2 − −3 JΔJ + −3 −1 J3 − 2(n−6)J|P|2 Δ2 J− 2 2 2 2

n −3 J (J2 −|P|2 ). − 8δ(P#dJ) + 4JΔJ + 4(dJ, dJ) − 4 Δ− 2

From here the assertion follows by a direct calculation. Remark 6.10.1. critical Q6 to a

In Theorem 6.11.7 we extend the relation D˙ 6res (0)(1) = Q6 relation between Q6,n and the quantity D˙ 6res (− n2 + 3)(1).

Remark 6.10.2. Theorem 6.10.4 is an analog of the extension n

Q4,n = 16v4 + 2P2∗ −2 (v2 ) 2

for the

(6.10.15)

of the holographic formula for the critical Q4 to n ≥ 4. For the proof we observe that the right-hand side of (6.10.15) equals

n n 2(J2 −|P|2 ) − Δ− −2 J J = J2 −2|P|2 . 2 2 These results imply that for n = 6 the curvature quantities {Q2 , Q4 , Q6 } and {v2 , v4 , v6 } are related by ⎛ 1 ⎞ ⎛ ⎞⎛ ⎞ − 64 Q6 1 T2∗ (0) T4∗ (0) 6v6 ⎜ 1 ⎟ ⎜ ⎟⎜ ⎟ ∗ 1 T2 (1)⎠ ⎝4v4 ⎠ . (6.10.16) ⎝ 4 Q4 ⎠ = ⎝0 −Q2 2v2 0 0 1 This is a special case of (6.9.27). We close the present section with a discussion of the relation to a formula of Gover and Peterson ([116]). Using tractor calculus (and computer aided calculations), Gover and Peterson derived formulas for the Q-curvature Q6,n and Q8,n , together with formulas for the corresponding GJMS-operators P6 and P8 , in terms of Riemannian invariants. Their formula for Q6 = Q6,6 is 16(P, ΔP) + 8(∇P, ∇P) − 8JΔJ + Δ2 J − 32 tr(P3 ) − 16|P|2 J + 8J3 + 16Pij Pkl Ckijl .

(6.10.17)

Note that in [116] the sign convention for C is opposite to ours. In Table 6.1 we compare the formula for Q6,n (equation (23) in [116]) with Theorem 6.10.3. Both results diﬀer by the sum 16 2 |P| J − n tr(P3 ) − (P, Hess(J)) + Pij Pkl Ckijl + ∇k ∇j (P)ik Pij . (6.10.18) n−4 The following result shows that both formulas actually coincide.

264

Chapter 6. Conformally Covariant Families

term (P, ΔP) Δ2 J (∇P, ∇P) JΔJ (dJ, dJ) (P, Hess(J)) J3 tr(P3 ) |P|2 J Pij Pkl Ckijl ∇k ∇j (P)ik Pij

[116]

Theorem 6.10.3

16 n−4

8+ 1 8

16 8 + n−4 1 8

2−3n 2

2−3n 2

6−n 16 8 − n−4

6−n 8

n2 −4 4 64 − n−4 16 n−4 − 4n 32 n−4

n2 −4 4

0

16 −4n 16 n−4 16 − n−4

Table 6.1: Q6,n Lemma 6.10.1. The sum in (6.10.18) vanishes. Proof. By (6.9.17), we have ∇k ∇j (P)ki = Hessij (J) + Cjkil Plk + Ricjl Pli − (P h)jkil Plk . Hence ∇k ∇j (P)ki Pij = (P, Hess(J)) − Ckijl Plk Pij + ((n−2)P + Jh)jl Pli Pij − (P h)jkil Plk Pij . But ((n−2)P + Jh)jl Pli Pij = (n−2) tr(P3 ) + J|P|2 and (P h)jkil Plk Pij = 2J|P|2 − 2 tr(P3 ). Thus ∇k ∇j (P)ki Pij = (P, Hess(J)) − Pij Pkl Ckijl + n tr(P3 ) − J|P|2 . The proof is complete.

6.11 Factorization identities for residue families. Recursive relations For M n = Rn with the Euclidean metric hc , the residue family res (hc ; λ) : C ∞ ([0, ε) × M n ) → C ∞ (M n ) D2N

6.11. Factorization identities for residue families. Recursive relations

265

nc (Deﬁnition 6.6.2) coincides with the diﬀerential intertwining operator D2N (λ) for (the non-compact models of) spherical principal series. It is induced by a homomorphism of Verma modules which is given by (a normalization of) the element D2N (λ). According to Theorem 5.2.2, the family D2N (λ) of order 2N satisﬁes a system of 2N factorization relations which induce corresponding factorizations for the diﬀerential operator families. In the present section, we prove that for a conformally ﬂat metric h the res (h; λ) satisfy analogous systems of factorization identities. residue families D2N These identities give rise to recursive relations for

Q-curvatures,

Q-polynomials and GJMS-operators.

The methods are the following. The critical family Dnres (h; λ) satisﬁes a system of n factorization identities. One half of the system, together with the formula n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h),

allows us to determine the critical Q-polynomial Qres n (h; λ) in terms of Qn (h), lower order Q-polynomials and lower order GJMS-operators. Applying analogous arguments to the lower order Q-polynomials, yields a formula for Qres n (λ) in terms of Qn , lower order Q-curvatures and lower order GJMS-operators. That formula n for Qres n (λ) is equivalent to a set of 2 − 1 relatives of the holographic formula for Qn which relate Qn to vn . Similarly, one half of the system, together with the factorization identity n−1 n+3 res Dnres h; − (6.11.1) h; − = Dn−2 ◦ P2 (dr2 +hr ), 2 2 yields a formula for D˙ nres (h; 0)(1), i.e., Qn (h), in terms of lower order residue families and lower order GJMS-operators. This formula involves the Yamabe operator P2 (dr2 +hr ) of the extension dr2 + hr of h oﬀ M . A repeated application of factorization identities for lower order residue families yields a formula for Qn (h) in terms of lower order GJMS-operators (acting on 1) and powers of P2 (dr2 + hr ) (acting on 1). If we express all these powers, except the highest one, in terms of lower order Q-curvatures, we arrive at the recursive formula (6.11.30) for Qn . Another source of relations is the fact that the number of identities for res (h; λ) exceeds the number of coeﬃcients of the polynomial (if N ≥ 2). It D2N yields recursive relations which allow us to determine the critical GJMS-operator Pn (h) using lower order GJMS-operators. We brieﬂy review the main results. The factorization identities are stated in Theorem 6.11.1. Here we assume that h is conformally ﬂat, but the result is expected to extend to general metrics. For those factorization identities which contain a GJMS-operator on the target manifold M as a factor, this follows from Theorem 6.11.18. However, the remaining identities have not yet been established res (h; λ) of order 2N ≤ 6, Theorem 6.11.2 in full generality. For residue families D2N

266

Chapter 6. Conformally Covariant Families

and Theorem 6.11.4 establish some of these identities. We use these results to derive a recursive formula for the critical Q6 (Theorem 6.11.5). Theorem 6.11.6 states that the same formula yields Q6 in all dimensions n ≥ 6 (universality). An extension of the method leads to Conjecture 6.11.1 (for conformally ﬂat h). Its extension to general metrics would require, in particular, to prove (6.11.1) in full generality. Theorem 6.11.10 states a recursive formula for the critical Qpolynomial Qres n (λ). Its formulation motivates a close consideration of subcritical Q-polynomials. Such formulas are discussed in Theorem 6.11.11. They generalize Theorem 6.11.7. The critical Q-polynomial Qn (λ) contains n2 relations between Qn and vn . Among these relations, Corollary 6.11.5 emphasizes the one which arises by consideration of the leading power of λ. Explicit versions for Q4 and Q6 are given in (6.11.44) and (6.11.48), respectively. The corresponding recursive formula for Q8 is discussed in Section 6.13. Finally, Theorem 6.11.15 relates the quadratic coeﬃcient of the critical Q-polynomial to the renormalized volume, and Theorem 6.11.17 illustrates for P6 how to derive recursive formulas for GJMS-operators. Theorem 6.11.17 should be compared with the alternative recursive formula in Theorem 6.12.1 which rests on (6.11.48). We start with the discussion of the factorization identities. First of all, we notice that for N = 1, the only factorization identities are formulated in (6.7.2), i.e., these are n−2 n−1 res ∗ res h; − h; − D2 = P2 (h)i and D2 = i∗ P2 (dr2 +hr ). 2 2 Both formulas are valid for all metrics h. For the convenience of the reader, we repeat the argument. Deﬁnition 6.6.2 implies ∂2 + (Δ+λJ)i∗ . ∂r2 The ﬁrst identity is obvious. The second identity asserts that D2res (λ) = −(n−2+2λ)i∗

i∗

∂2 n−1 J(h)i∗ = i∗ P2 (dr2 +hr ). + Δh i∗ − 2 ∂r 2

But this follows from H = 0 and i∗ J(dr2 +hr ) = J(h) (Lemma 6.11.1). Now we prove the factorization identities for residue families of arbitrary even order for conformally ﬂat metrics. Theorem 6.11.1 (Factorization). Let h be conformally ﬂat. Then the residue family res (h; λ) factorizes for D2N " # n+1 n+1 n n λ∈ − ,...,− +N ∪ − +N, . . . , − +2N 2 2 2 2 into products of (lower order ) residue families and GJMS-operators. More precisely, for j = 1, . . . , N the identities

n n res res D2N h; − +2N −j = P2j (h) ◦ D2N h; − +2N −j (6.11.2) −2j 2 2

6.11. Factorization identities for residue families. Recursive relations

and res D2N

n+1 n+1 res +j = D2N −2j h; − −j ◦ P2j (dr2 +hr ) h; − 2 2

267

(6.11.3)

hold true. Proof. The proof rests on the transformation formula (Theorem 6.6.3) ∗ −λ κ (r) res ˆ res (h; λ) = D2L (h; λ) ◦ ◦ κ∗ . e−(λ−2L)ϕ ◦ D2L r

(6.11.4)

Here κ is a diﬀeomorphism which restricts to the identity on M and satisﬁes

κ∗ r−2 (dr2 + ˆ hr ) = r−2 (dr2 +hr ) and i∗

κ∗ (r) r

= eϕ ,

ˆ = e2ϕ h. h

We use local coordinates on M n so that h = e2ϕ hc with hc = i dx2i . In order to prove the ﬁrst set of identities, we apply the transformation formula (6.11.4) and the factorization identities (6.11.2) for h = hc . These hold true in view of res nc (hc ; λ) = D2N (λ). We ﬁnd D2N res D2N

−(− n2 +2N −j) κ∗ (r) ◦ κ∗ r

κ∗ (r) −(− n2 +2N −j) n res h +2N −j ◦ = P2j (hc ) ◦ D2N ; − ◦ κ∗ , c −2j 2 r

n hc ; − +2N −j ◦ 2

i.e.,

n n res h; − +2N −j e( 2 +j)ϕ ◦ D2N 2

n n res h; − +2N −j . = P2j (hc ) ◦ e( 2 −j)ϕ ◦ D2N −2j 2 Now, by the covariance of P2j , we get

n n res res h; − +2N −j = P2j (h) ◦ D2N D2N −2j h; − +2N −j . 2 2 This proves (6.11.2) for h = e2ϕ hc . Similarly, we apply (6.11.4) and (6.11.3) for h = hc , and ﬁnd res D2N

∗ n+1 −j κ (r) 2 n+1 +j ◦ ◦ κ∗ hc ; − 2 r ∗ n+1 −j κ (r) 2 n+1 res 2 −j ◦ P2j (dr +hc ) ◦ = D2N −2j hc ; − ◦ κ∗ , 2 r

268

Chapter 6. Conformally Covariant Families

i.e., n+1 +j e ◦ h; − 2 ∗ n+1 2 +j κ (r) 2 κ∗ (r) n+1 res 2 = D2N −2j hc ; − −j ◦ ◦ P2j (dr +hc ) ◦ κ∗ 2 r r ∗ n+1 +j κ (r) 2 n+1 res −j ◦ = D2N −2j hc ; − ◦ κ∗ ◦ κ∗ ◦ P2j (κ∗ (dr2 +hr )) ◦ κ∗ 2 r n+1 n+1 res −j ◦ P2j (dr2 +hr ). = e( 2 +2N −j)ϕ ◦ D2N h; − −2j 2 ( n+1 2 +2N −j)ϕ

res D2N

This proves (6.11.3) for h = e2ϕ hc . The proof is complete.

Remark 6.11.1. An analogous proof shows factorization identities for odd order res families D2N +1 (h; λ) for conformally ﬂat metrics. Here we apply Theorem 5.2.4. The factorizations

n res h; − +N = P2N (h)i∗ , D2N 2

n n ∂ res D2N +1 h; − +N +1 = P2N (h)D1res h; − +N +1 = P2N (h)i∗ 2 2 ∂r (see (6.6.12)) directly follow from the deﬁnitions and are valid for all metrics h. In particular, for a conformally ﬂat metric h, we have the n factorizations n

n

n res Dnres h; −j = P2j (h) ◦ Dn−2j h; −j , j = 1, . . . , 2 2 2 and

n n+1 n+1 res +j = Dn−2j −j ◦ P2j (dr2 +hr ), j = 1, . . . , h; − Dnres h; − 2 2 2

of the critical residue family. res (h; λ) is a polynomial of degree N , the residue family is completely Since D2N determined by the N +1 operators

n res P2j (h) ◦ D2N −2j h; − +2N −j , j = 1, . . . , N 2 and res D2N −2

n+3 h; − ◦ P2 (dr2 +hr ). 2

(6.11.5)

res The lower order families D≤2N −2 (h; λ) in these compositions, in turn, can be written as linear combinations of compositions of P≤2N −2 (h) and residue families

6.11. Factorization identities for residue families. Recursive relations

269

res of order ≤ 2N −4. Thus D2N (λ) recursively can be written as a linear combination of compositions of GJMS-operators P2 (h), . . . , P2N (h) and P2 (dr2 +hr ). In particular, this has the consequence that the critical Q-curvature n Qn (h) = −(−1) 2 D˙ nres (h; 0)(1)

(see (6.6.2)) can be written as a linear combination of composition of the GJMSoperators P2 (h), . . . , Pn−2 (h) and the Yamabe operator P2 (dr2 + hr ) (acting on u = 1). Remark 6.11.2. The only factorization identity in the system (6.11.3) which will be used in connection with the study of the recursive structure of Q-curvature has the right-hand side (6.11.5). This is a curved analog of the identity n−1 n+3 0 0 − − = D2N Δ− D2N −2 n+1 , 2 2 i.e., N

n−1 − j )2N −2j (Δ− − (Yn+1 n) 2 ⎞ ⎛ N −1 n+3 (N −1) − − j⎠ ((Yn+1 =⎝ aj )2N −2j (Δ− )2 + Δ− − (Yn+1 n) n ) (6.11.6) 2 j=0

(N )

aj

j=0

in U(n− n+2 ), where the coeﬃcients are given by (M)

aj

(λ) =

M−1 M! (−2)M−j (2λ−4M +2k+n+2), j = 0, . . . , M −1 j!(2M −2j)! k=j

(M)

and a0 (λ) = 1 (see (5.1.4)). For the convenience of the reader we add a direct proof. It diﬀers from the general (but complicated ) argument in the proof of Theorem 5.2.2. (6.11.6) is equivalent to the relations n−1 n+3 n+3 (N ) (N −1) (N −1) − − − aj = aj−1 + aj , j = 1, . . . , N −1 2 2 2 and a corresponding relation for j = 0 with the undeﬁned term a−1 removed. But these identities are equivalent to the obvious relations 1 N (2N −1) = j(4N −2j −1) + (2N −2j −1)(2N −2j). 2 res Since D2N (h; λ) has polynomial degree N , Theorem 6.11.1 implies that there are 2N −(N +1) = N −1 linear relations among the right-hand sides.

270

Chapter 6. Conformally Covariant Families

In particular, for N = 2 (and n = 4) we ﬁnd a linear relation among the four operators P4 (h)i∗ ,

P2 (h)D2res (h; 1),

D2res (h; −7/2)P2 (dr2 +hr ),

i∗ P4 (dr2 +hr ).

Its content will be discussed in Lemma 6.11.6. Moreover, for even n ≥ 6, we ﬁnd a presentation " Pn (h)i∗ = a2 Pn−2 (h)D2res (h; 1)

# n res + a4 Pn−4 (h)D4res (h; 2) + · · · + an−2 P2 (h)Dn−2 h; −1 2 n+3 n+5 res res + b2 Dn−2 h; − h; − P2 (dr2 +hr ) + b4 Dn−4 P4 (dr2 +hr ) 2 2

of the critical GJMS-operator Pn (h) in terms of lower order residue families res (h; λ), lower order GJMS-operators Pn−2 (h), . . . , P2 (h) and P4 (dr2 + hr ), D≤n−2 2 P2 (dr + hr ). In turn, all residue families which appear here as factors can be written in terms of Pn−2 (h), . . . , P2 (h) and P2 (dr2 +hr ). Thus we ﬁnd a formula for Pn (h)i∗ (n ≥ 6) in terms of lower order GJMSoperators Pn−2 (h), . . . , P2 (h) and P2 (dr2 +hr ), P4 (dr2 +hr ). It would be interesting to work out the combinatorics of the recursion. The case n = 6 will be discussed below. Next, we consider residue families of order 4 for arbitrary metrics h. The following result deals with the critical case. Theorem 6.11.2. For any Riemannian four-manifold (M, h), D4res (h; 0) = P4 (h)i∗ , D4res (h; 1) = P2 (h) ◦ D2res (h; 1), 3 7 res res D4 h; − h; − = D2 ◦ P2 (dr2 +hr ), 2 2 1 res D4 h; − = i∗ P4 (dr2 +hr ). 2 In Theorem 6.11.2, the complexity of the identities increases starting with the ﬁrst identity. The four values λ ∈ {−3/2, −1/2, 0, 1} are characterized by the property that the conformal covariance of the compositions on the right-hand sides are consequences of the conformal covariance of the factors. Theorem 6.11.2 shows that the residue family D4res (h; λ) interpolate the Paneitz operators P4 (h)i∗ and i∗ P4 (dr2 +hr ). In that respect, it resembles the family D2res (h; λ) which interpolates the Yamabe operators P2 (h)i∗ and i∗ P2 (dr2 +hr ) (see the formulas at the end of Section 6.7). There is an analogous result for the tractor family D4T (dr2 +hr ; λ) (Theorem 6.22.4). For the proof of Theorem 6.11.2 we need the following technical result.

6.11. Factorization identities for residue families. Recursive relations

271

Lemma 6.11.1. For (M n , h) and the associated Poincar´e-Einstein metric r−2 (dr2 +hr ), the function r → J(dr2 +hr ) satisﬁes i∗ J(dr2 +hr ) = J(h),

J˙ = 0

and

¨J = |P|2 .

Here the dots denote the derivatives with respect to r at r = 0. Proof. The formula

τ (ˆ g ) = e−2ϕ τ (g) − 2nΔg (ϕ) − n(n−1)|dϕ|2g

implies

τ (g) = e2ϕ τ (ˆ g ) + 2nΔgˆ (ϕ) − n(n−1)|dϕ|2gˆ ,

i.e.,

τ (ˆ g ) = e−2ϕ τ (g) − 2nΔgˆ (ϕ) + n(n−1)|dϕ|2gˆ . 2

We apply (6.11.7) to gˆ = dr +hr , g = r −n(n+1), we ﬁnd for small r > 0

−2

(6.11.7)

2

(dr +hr ) and ϕ = log r. Since τ (g) =

n(n+1) − 2nΔdr2 +hr (log r) + n(n−1)|d log r|2dr2 +hr r2 2 n(n−1) n(n+1) 1 ∂ ∂ −1 ˙ =− − 2n (log r) + tr(hr hr ) (log r) + 2 2 r ∂r 2 ∂r r2 ˙ 1 = −n tr(h−1 r hr ) . r

τ (dr2 +hr ) = −

Hence

1 ˙ (6.11.8) tr(h−1 r hr ). 2r Now hr = h0 + r2 h(2) + r4 h(4) + · · · yields −1 −1 −1 −1 2 −1 4 (h0 h(2) )2 − h−1 h−1 r = h0 − r h0 h(2) h0 + r 0 h(4) h0 + · · · J(dr2 +hr ) = −

and

−1 3 −1 2 3 −1 ˙ tr(h−1 r hr ) = tr 2rh0 h(2) − 2r (h0 h(2) ) + 4r h0 h(4) + · · ·

i.e., −1 1 −1 2 2 ˙ 1 − tr(h−1 − 2r2 tr(h−1 r hr ) = − tr(h0 h(2) ) + r tr (h0 h(2) ) 0 h(4) ) + · · · . 2 r Now (6.11.8) and h(2) = −P imply i∗ J(dr2 + hr ) = J(h) and J˙ = 0. Moreover, using tr(h(4) ) = 14 |P|2 , it follows that ∂2 1 ∗ ∂2 2 i J(dr i +h ) = − r ∂r2 2 ∂r2 ∗

This proves ¨J = |P|2 .

1 −1 ˙ tr(hr hr ) = |P|2 . r

272

Chapter 6. Conformally Covariant Families

The results of Lemma 6.11.1 can be seen also as consequences of the following useful formula for P(dr2 +hr ). Lemma 6.11.2. In the situation of Lemma 6.11.1, 1 ˙ hr . 2r

(6.11.9)

i∗ P(dr2 +hr ) = P(h)

(6.11.10)

P(dr2 +hr ) = − In particular, and PN N (dr2 +hr ) = 0,

|P(dr2 +hr )|2 = |P(h)|2

on r = 0. Proof. Applying the conformal transformation law (2.5.9) for the metrics gˆ = dr2 +hr and g = r−2 (dr2 +hr ) yields 1 1 P(dr2 +hr ) = P(g) − |d log r|2g g − Hessg (log r) + 2 dr2 2 r 1 1 1 2 g = − g − g − Hess (log r) + 2 dr 2 2 r 1 g = − 2 hr − Hess (log r). r In the second line we used that the Schouten tensor of the Einstein metric g is − 21 g. Now the tangential components of the Hessian are given by Hessg (log r)ij =

5 6 dr 1 ∂ ∇gi . = − ΓN , r ∂xj r ij

The usual formulas for the Christoﬀel symbols (see (6.8.3)) imply 1 1 1 NN ΓN = − h˙ r + hr . ij = − (∂/∂r)(gij )g 2 2 r Hence Hessg (log r)ij =

1 ˙ 1 (hr )ij − 2 (hr )ij . 2r r

The remaining components of the Hessian vanish: 1 = 0 and Hessg (log r)iN = − ΓN r iN

Hessg (log r)N N = −

1 1 − ΓN = 0. 2 r r NN

These results prove (6.11.9). The remaining assertions are direct consequences.

6.11. Factorization identities for residue families. Recursive relations

273

We continue with the proof of Theorem 6.11.2. Deﬁnition 6.6.1 yields the formula D4res (λ) = 32(λ−1)λ 2 1 ∗ ∂4 1 ∗ ∗ ∂ ∗ ∗ ∗ i × + (T2 (λ) + v2 )i + (T4 (λ) + T2 (λ)v2 + v4 )i , 24 ∂r4 2 ∂r2 i.e., D4res (λ) =

4 ∂4 ∂2 (λ−1)λi∗ 4 − 4λ(Δ + (λ−2)J)i∗ 2 3 ∂r ∂r + P4∗ (λ) + 4λ(Δ−λJ)J + 4λ(λ−1)(J2 −|P|2 ) i∗ ,

(6.11.11)

where P4 (λ) = (Δ−(λ+2)J)(Δ−λJ) + 2λ(λ−1)|P|2 + 4(λ−1)δ(P#d) + 2(λ−1)(dJ, d) (see also Lemma 6.11.8). Here we have used Lemma 6.6.1 and Theorem 6.9.2. In particular, we ﬁnd D4res (0) = P4∗ (0)i∗ = P4 (0)i∗ . This proves the ﬁrst assertion. It is a special case of (6.6.8). Next, we ﬁnd ∂2 + [(Δ − J)(Δ − 3J) + 4(Δ − J)J] i∗ ∂r2 2 ∗ ∂ ∗ = (Δ − J) −4i + (Δ + J)i = P2 (h) ◦ D2res (h; 1). ∂r2

D4res (h; 1) = −4(Δ − J)i∗

This proves the second assertion of Theorem 6.11.2. We split the proof of the remaining two identities. Lemma 6.11.3. D4res (h; −3/2) = D2res (h; −7/2) ◦ P2 (dr2 +hr ). Proof. By (6.11.11), we ﬁnd D4res (h; −3/2) = 5i∗

∗ ∂2 ∂4 + 6 Δ − 7/2J i 2 ∂r4 ∂r ∗ + P4 (−3/2) − 6 Δ + 3/2J J + 15(J2 − |P|2 ) i∗

(6.11.12)

with P4∗ (−3/2)u = (Δ + 3/2J)(Δ − 1/2J)u + 15/2|P|2u − 10δ(P#du) − 5δ(udJ); the convention here is Δ = −δd. We have to compare this operator with the composition D2 (dr2 +hr ; −7/2)P2 (dr2 +hr ) ∂2 7 3 = 5i∗ 2 + Δi∗ − Ji∗ Δdr2 +hr − J(dr2 +hr ) . ∂r 2 2

(6.11.13)

274

Chapter 6. Conformally Covariant Families

We apply the formula Δdr2 +hr =

∂2 1 ˙ ∂ + Δh . + tr(h−1 r r hr ) ∂r2 2 ∂r

The coeﬃcients of ∂ 4 /∂r4 in (6.11.12) and (6.11.13) both are equal to 5. In view ¨ 0 = −2P (Theorem 6.9.1), the of i∗ J(dr2 + hr ) = J(h) (Lemma 6.11.1) and h 2 2 coeﬃcient of ∂ /∂r in (6.11.13) is given by −

∂ 1 7 15 ˙ J + 5 1r=0 tr(h−1 r hr ) − J + 6Δ = −21J + 6Δ. 2 ∂r 2

This result coincides with the coeﬃcient of ∂ 2 /∂r2 in (6.11.12). It remains to compare the tangential components of both operators. For u ∈ C ∞ (M 4 ), (6.11.12) yields the formula 3 3 1 15 Δ2 u − Δ(Ju) + JΔu − J2 u + |P|2 u − 10δ(P#du) + 5(du, dJ) + 5(ΔJ)u 2 2 4 2 − 6Δ(Ju) − 9J2 u + 15J2 u − 15|P|2 u 15 3 21 = Δ2 u − 5JΔu − 8(du, dJ) − 10δ(P#du) − (ΔJ)u − |P|2 u + J2 u. 2 2 4 Now recall that δ(P#du) = −(dJ, du) − (P, Hess(u)) (Lemma 4.2.5). Thus we obtain 15 3 21 Δ2 u − 5JΔu + 10(P, Hess(u)) + 2(du, dJ) − (ΔJ)u − |P|2 u + J2 u. 2 2 4 Next, we determine the tangential part of (6.11.13). We ﬁnd −

7 21 15 ¨ 3 ∂2 1 Ju + Δ2 u − Δ(J)u − JΔu + J2 u + 5 2 1r=0 (Δhr u), 2 2 2 4 ∂r

¨ = −2P, a calculation shows where ¨J = (∂ 2 /∂r2 )|r=0 (J(dr2 +hr )). But using h 2 1 ¨ = ∂ 1 (Δhr u) Δu ∂r2 r=0 = −JΔu − δ(Jdu) + 2(P, Hess(u)) = (dJ, du) + 2(P, Hess(u)).

Moreover, using ¨J = |P|2 (Lemma 6.11.1), it follows that the tangential part coincides with 21 3 15 Δ2 u − 5JΔu + 10(P, Hess(u)) + 2(du, dJ) − (ΔJ)u + J2 u − |P|2 u. 2 4 2

The proof is complete. Lemma 6.11.4.

D4res (h; −1/2) = i∗ P4 (dr2 +hr ).

6.11. Factorization identities for residue families. Recursive relations

275

Proof. (6.11.11) gives D4res (−1/2)u = i∗

2 ∂4u ∗∂ u + (2Δ − 5J) i ∂r4 ∂r2 ∗ + P4 (−1/2) − 2(Δ + 1/2J)J + 3(J2 − |P|2 ) i∗ u, (6.11.14)

where P4∗ (−1/2)u = (Δ + 1/2J)(Δ − 3/2J)u + 3/2|P|2 u − 6δ(P#du) − 3δ(udJ). On the other hand, the Paneitz operator for dr2 +hr is given by 1 P4,5 = Δ2 + 3δ(Jd) − 4P#d + Q4,5 2 with

5 2 J − 2|P|2 − ΔJ; 2 here all constructions refer to the metric dr2 +hr . In order to ﬁnd a formula for i∗ Q4,5 , we use i∗ J = J, J˙ = 0, ¨J = |P|2 and i∗ P = P (Lemma 6.11.2 and Lemma 6.11.1). Hence Q4,5 =

i∗ Q4,5 =

5 2 5 J − 2|P|2 − (|P|2 + ΔJ) = J2 − 3|P|2 − ΔJ. 2 2

We compare the coeﬃcients of ∂ 2 /∂r2 . On the left-hand side we have 2Δ − 5J (by (6.11.14)). On the other hand, i∗ P4 (dr2 +hr )u is given by the restriction of 2 2 ∂ ∂ 1 ∂ −1 ˙ tr(h + Δ + Δ + ) h h r hr u r ∂r2 ∂r2 2 ∂r 2 ∂ u 1 − 3J + Δu − 3(dJ, du) − 4P#du + Q4,5 u (6.11.15) ∂r2 2 to r = 0; here we used the notation Δ = Δh , P = P(h) etc. and we applied the fact that PN N = 0 (by Lemma 6.11.2). Now the latter formula contributes 2 2 ∂ 11 −1 ˙ ∗∂ u ∗∂ u tr(h ) − 3J i = (2Δ − 5J)i . h 2Δ + r r ∂r r=0 ∂r2 ∂r2 This proves that the coeﬃcients of the normal derivatives on both sides of the assertion coincide. Finally, we verify the coincidence of the tangential parts. The residue family contributes 3 1 1 Δ − J u − 2 Δ + J Ju − 6δ(P#du) − 3δ(udJ) Δ+ J 2 2 2 3 2 + |P| u + 3 J2 − |P|2 u 2 5 2 3 2 1 2 J − |P| u = Δ u − 3JΔu + 2(du, dJ) + 6(P, Hess(u)) − (ΔJ)u + 2 4 2

276

Chapter 6. Conformally Covariant Families

using δP = −dJ. It remains to analyze the tangential part of (6.11.15). Using the ¨ from the proof of Lemma 6.11.3, we ﬁnd formula for Δ Δ2 u + 2(P, Hess(u)) + (dJ, du) 1 − 3JΔu − 3(dJ, du) + 4(P, Hess(u)) + 4(dJ, du) + 2

5 2 2 J − 3|P| − ΔJ u. 2

The proof is complete.

We emphasize that the proofs of Lemma 6.11.3 and Lemma 6.11.4 rest on the relations 0 0 ∗ 2 i P(dr +hr ) = , J˙ = 0 and ¨J = |P|2 . 0 P(h) The family D4res (λ) is polynomial of degree 2 in λ, i.e., can be written in the form Aλ2 + Bλ + C with three operator-coeﬃcients A, B, C. On the other hand, Theorem 6.11.2 yields four relations, i.e., the right-hand sides of the four relations satisfy one linear relation i∗ P4 (dr2 +hr ) = αP4 (h)i∗ + βP2 (h)D2res (h; 1) + γD2res (h; −7/2)P2 (dr2 +hr ) with rational coeﬃcients. In order to determine (α, β, γ), it suﬃces to consider the ﬂat case. Then 1 (α, β, γ) = (5, −1, 1). 5 Thus D4res (h; λ) can be written as a linear combination of compositions of P4 (h), P2 (h) and P2 (dr2 +hr ). Moreover, the factorizations D4res (h; 0) = P4 (h)i∗ , D4res (h; 1) = P2 (h)D2res (h; 1), 3 7 D4res h; − = D2res h; − P2 (dr2 +hr ) 2 2 imply C(h) = P4 (h)i∗ and ( − 32 )2 − 32 A(h)(1) D2res (h; − 27 )P2 (dr2 +hr )(1) = , P2 (h)D2res (h; 1)(1) 1 1 B(h)(1) i.e.,

A(h)(1) B(h)(1)

=

4 15 4 − 15

2 5 3 5

D2res (h; − 27 )P2 (dr2 +hr )(1) . P2 (h)D2res (h; 1)(1)

In particular, we can write Q4 (h) = −B(h)(1) as the linear combination 4 3 7 Q4 (h) = − D2res h; − P2 (dr2 +hr )(1) + P2 (h)D2res (h; 1)(1). (6.11.16) 15 2 5

6.11. Factorization identities for residue families. Recursive relations

277

But using 7 h; − = 5i∗ P2 (dr2 +hr ) − 4P2 (h)i∗ , 2

D2res

D2res (h; 1) = −4i∗ P2 (dr2 +hr ) + 5P2 (h)i∗ , we ﬁnd the following formula for Q4 in terms of Yamabe operators P2 and Qcurvatures Q2 = J. Corollary 6.11.1. For any Riemannian four-manifold (M, h), 4 Q4 (h) = P2 (h)Q2 (h) + i∗ P22 (dr2 +hr )(1) 3 = P2 (h)Q2 (h) − 2i∗ P2 (dr2 +hr )Q2 (dr2 +hr ). Proof. (6.11.16) yields Q4 (h) = −3P22 (h)(1) +

4 P2 (h)i∗ + i∗ P2 (dr2 +hr ) P2 (dr2 +hr )(1). 3

Now using P2 (h)(1) = −J(h), P2 (dr2 +hr )(1) = − 23 J(dr2 +hr ) and i∗ J(dr2 + hr ) = J(h), we ﬁnd Q4 (h) = 3P2 (h)J(h) − 2 P2 (h)i∗ + i∗ P2 (dr2 +hr ) J(dr2 +hr ) = P2 (h)Q2 (h) − 2i∗ P2 (dr2 +hr )J(dr2 +hr ). The proof is complete.

It is instructive to observe how the various terms in Corollary 6.11.1 are organized. The proof of the following somewhat more general result rests on such explicit formulas. Lemma 6.11.5. The second identity in Corollary 6.11.1, i.e., Q4 (h) = P2 (h)Q2 (h) − 2i∗ P2 (dr2 +hr )Q2 (dr2 +hr ) holds true in all dimensions n ≥ 3. Proof. We recall that Q4,n = n2 J2 − 2|P|2 − ΔJ (see (4.1.7)). On the other hand, the right-hand side of the assertion is given by

n n−1 ∗ 2 Δ−( −1)J J − 2i Δdr2 +hr − J(dr +hr ) J(dr2 +hr ) 2 2 n−1 2 n n 2 ¨ = ΔJ − ( −1)J − 2 J + ΔJ − J = −ΔJ − 2|P|2 + J2 . 2 2 2

The proof is complete.

278

Chapter 6. Conformally Covariant Families

The following result extends Theorem 6.11.2. Theorem 6.11.3. For (M n , h) (n ≥ 3),

n D4res h; − +2 = P4 (h)i∗ , 2

n n res D4 h; − +3 = P2 (h) ◦ D2res h; − +3 , 2 2 n−1 n+3 D4res h; − = D2res h; − ◦ P2 (dr2 +hr ), 2 2 n−3 res D4 h; − = i∗ P4 (dr2 +hr ). 2 We omit the analogous proof. For the remainder of the present section it will be convenient to introduce the abbreviations ¯ 2j (h) = Q2j (dr2 +hr ). P¯2j (h) = P2j (dr2 +hr ) and Q

(6.11.17)

The four right-hand sides in Theorem 6.11.3 satisfy one linear relation. The following result is equivalent to that relation. Lemma 6.11.6. For any Riemannian manifold (M n , h) (n ≥ 3), i∗ P¯4 (h) − P¯22 (h) = P4 (h) − P22 (h) i∗ .

(6.11.18)

Proof. The formula D2res (λ) = −(n−2+2λ)i∗P¯2 + (n−1+2λ)P2i∗ yields n

D2res − +3 = −4i∗ P¯2 + 5P2 i∗ 2

n+3 and D2res − = 5i∗ P¯2 − 4P2 i∗ . 2

Now the right-hand sides of the identities in Theorem 6.11.3 satisfy the linear relation n

1 1 n+3 ¯ P2 + i∗ P¯4 . P4 i∗ = P2 D2res − +3 − D2res − 5 2 5 2 In fact, for the ﬂat metric h = hc on Rn it specializes to the obvious relation Δ2n i∗ =

1 1 Δn (−4i∗ Δn+1 + 5Δn i∗ ) − (5i∗ Δn+1 − 4Δn i∗ )Δn+1 + i∗ Δ2n+1 . 5 5

But for a general metric it implies 1 ∗ 1 P2 −4i∗P¯2 + 5P2 i∗ − 5i P¯2 − 4P2 i∗ P¯2 + i∗ P¯4 5 5 = P22 i∗ − i∗ P¯22 + i∗ P¯4 .

P4 i∗ =

The proof is complete.

6.11. Factorization identities for residue families. Recursive relations

279

A direct calculation shows that for all n, P4 − P22 = −4δ(P#)d − (n−4)|P|2 + J2 − ΔJ . The latter identity can be used for a more direct proof of Lemma 6.11.6. We continue with the discussion of the critical case of order 6 residue families, i.e., of D6res (h; λ) for (M 6 , h). Theorem 6.11.4. For any Riemannian 6-manifold (M, h), D6res (h; 0) = P6 (h)i∗ , D6res (h; 1) = P4 (h) ◦ D2res (h; 1), D6res (h; 2) = P2 (h) ◦ D4res (h; 2), 5 9 D6res h; − = D4res h; − ◦ P2 (dr2 +hr ). 2 2 Proof. The complexity of the identities increases starting with the ﬁrst one. The ﬁrst assertion directly follows from the deﬁnition of D6res (h; λ) using the selfadjointness of P6 = P6 (0). Deﬁnition 6.6.1 yields the formula D6res (λ) = 26 3!(2−λ)(1−λ)(−λ) " 1 ∂ 6 ∗ ∂ 4 1 ∗ ∗ × i + T (λ) + v2 i 6! ∂r 4! 2 ∂r 2 # ∂ 1 ∗ T4 (λ) + T2∗ (λ)v2 + v4 i∗ + + T6∗ (λ) + T4∗ (λ)v2 + T2∗ (λ)v4 + v6 i∗ 2! ∂r with T2 (λ) =

1 (Δ−λJ) 4(2−λ)

and 1 32(2−λ)(1−λ) × (Δ−(λ+2)J)(Δ−λJ) + 4(2−λ) −λ/2|P|2 − δ(P#d) − 1/2(dJ, d) .

T4 (λ) =

Now (6.10.2) implies ∂2 D6res (1) = −6P4∗ ◦ i∗ 2 + (P4∗ ◦ P2∗ (5) + 6P4∗ ◦ J) i∗ ∂r ∂2 = P4 ◦ −6i∗ 2 + (Δ + J)i∗ ∂r = P4 ◦ D2res (1). We omit the details of the proof of the third identity. It rests on the following result (for n = 6).

280

Chapter 6. Conformally Covariant Families

Lemma 6.11.7. For (M n , h) we have the factorizations n n

P4 h; −1 = P2 h; +1 ◦ P2 (h) 2 2 (if n ≥ 4) and

n n P6 h; −2 = P2 h; +2 ◦ P4 (h), 2 2 n n

P6 h; −1 = P4 h; +1 ◦ P2 (h) 2 2 (if n ≥ 6). Proof. The factorization of P4 (λ) is a direct consequence of Theorem 6.9.4. For λ = n2 −2, (6.10.2) yields P6

n

n

n

n

−2 = Δ− +2 J P4 −2 = P2 +2 P4 . 2 2 2 2

This proves the ﬁrst factorization for P6 (λ). For λ = P6

n 2 −1,

(6.10.2) yields

n

−1 2 n

n

n n +1 (log det h) + Δ P2 −1 + Δ − ( +3)J P4 −1 . = −8 2 2 2 2

But using (log det h) = − 12 |P|2 and 1 Δ = −δ(P#d·) − (d·, dJ) 2 (these formulas extend the corresponding results in Section 6.10 to general dimension), the formula simpliﬁes to

n

n n 4 +1 |P|2 + 8δ(P#d·) + 4(d·, dJ) P2 + Δ−( +3)J Δ − ( +1)J P2 . 2 2 2 Here we have used the factorization of P4 (λ). Now it suﬃces to apply Theorem 6.9.4. Lemma 6.11.7 is a special case of Theorem 6.11.18. The remaining part of the proof of Theorem 6.11.4 concerns the proof of the last identity. We verify that the normal components of both sides coincide. A calculation shows that for λ = −5/2, the respective coeﬃcients of i∗ ∂ 6 /∂r6 , i∗ ∂ 4 /∂r4 and i∗ ∂ 2 /∂r2 are given by the operators 21, 18 · 35 [1/18(Δ + 5/2J) − 1/2J] = 35(Δ − 13/2J)

(6.11.19)

6.11. Factorization identities for residue families. Recursive relations

281

and 15 (Δ + 5/2J)(Δ + 1/2J) + 18(5/4|P|2 − δ(P#d) − 1/2δ(·dJ)) − 14 · 15(Δ + 5/2J)J + 63 · 15(J2 − |P|2 ).

(6.11.20)

These operators are to be compared with the respective coeﬃcients of i∗ ∂ 6 /∂r6 , i∗ ∂ 4 /∂r4 and i∗ ∂ 2 /∂r2 in the composition D4res (h; −9/2) ◦ P2 (dr2 + hr ). For that purpose, we apply the following formula for D4res (h; λ). Lemma 6.11.8. For (M n , h) (n ≥ 3), D4res (λ) =

∂4 ∂2 1 (n+2λ−4)(n+2λ−6)i∗ 4 − 2(n+2λ−4)(Δ+(λ−2)J)i∗ 2 3 ∂r ∂r

+ P4∗ (n+λ−4) + 2(n+2λ−4) (Δ−(n+λ−4)J) J

+ (n+2λ−4)(n+2λ−6)(J2 −|P|2 ) i∗ .

We omit the proof. Now Remark 6.6.5 and Lemma 6.11.8 yield ∂4 ∂2 D4res (−9/2) = 21i∗ 4 + 14(Δ − 13/2J)i∗ 2 ∂r ∂r

+ (Δ + 5/2J)(Δ + 1/2J) + 18 5/4|P|2 − δ(P#d) − 1/2δ(·dJ)

− 14(Δ + 5/2J)J + 63(J2 −|P|2 ) i∗ .

Using P2 (dr2 +hr ) =

∂2 1 ˙ ∂ + Δh − 5 J(dr2 +hr ), + tr(h−1 r r hr ) 2 ∂r 2 ∂r 2

we ﬁnd that the respective coeﬃcients of i∗ (∂/∂r)6 and i∗ (∂/∂r)4 in the composition are 21 and 1 ˙ 42(∂/∂r)1r=0 tr(h−1 r hr ) + 21(Δ − 5/2J) + 14(Δ − 13/2J) = 35(Δ − 13/2J). The latter result coincides with (6.11.19). Finally, it remains to compare (6.11.20) with the coeﬃcient of i∗ (∂/∂r)2 in the composition D4res (−9/2)P2 , i.e., with

1 ˙ r ) + 6Δ ¨ − 15¨J + 14(Δ − 13/2J)(Δ − 9/2J) 21 2(∂ 3 /∂r)3 1r=0 tr(h−1 h r

+ (Δ + 5/2J)(Δ + 1/2J) + 18 5/4|P|2 − δ(P#d) − 1/2δ(·dJ) − 14(Δ + 5/2J)J + 63(J2 −|P|2 ) i∗ . (6.11.21)

282

Chapter 6. Conformally Covariant Families

Now (6.11.20) coincides with (6.11.21) iﬀ (Δ + 5/2J)(Δ + 1/2J) + 18 5/4|P|2 − δ(P#d) − 1/2δ(·dJ) − 14(Δ + 5/2J)J + 63(J2 −|P|2 ) = (Δ − 13/2J)(Δ − 9/2J)

1 ˙ ¨ ¨ + 3 (∂ 3 /∂r)3 1r=0 tr(h−1 r hr ) + 3Δ − 15/2J .

(6.11.22)

Using the formulas 1 2 ˙ (∂ 3 /∂r)3 1r=0 tr(h−1 r hr ) = −6|P| ,

¨J = |P|2

(by Lemma 6.11.1 and its proof) and ¨ = 2(P, Hess) + (dJ, d) Δ (see the proof of Lemma 6.11.3), we determine the coeﬃcients of Δ2 u, J(Δu), u(ΔJ), (P, Hess(J)), J2 u, |P|2 u and (dJ, du) on both sides of (6.11.22). The results are listed in Table 6.2. It shows that all coeﬃcients coincide. term

left-hand side

right-hand side

Δ2 u u(ΔJ) JΔu (P, Hess(u)) J2 |P|2 (dJ, du)

1 + 9 − 14 3 − 14 18 5 4 − 35 + 63 45 2 − 63 1 + 18 + 9 − 28

1 − 92 −11 18

1 2

117 4

3(−6 − 15 2 ) −9 + 9

Table 6.2: Evaluation of (6.11.22) It remains to prove the coincidence of the tangential components of both sides of the last factorization identity. That proof, in particular, involves the operator T6 (− 52 ). We omit the details of the direct calculation. An alternative proof of the fact that the constant terms of both operators coincide can be derived from the observation below that the factorization identities imply the correct for mula for Q6 . We do not discuss here the remaining two factorization identities D6res (h; −1/2) = i∗ P6 (dr2 +hr ), D6res (h; −3/2) = D2res (h; −11/2) ◦ P4 (dr2 +hr )

(6.11.23)

for metrics which are not conformally ﬂat. These hold true for conformally ﬂat metrics by Theorem 6.11.1.

6.11. Factorization identities for residue families. Recursive relations

283

(6.11.23) has the consequence that the ﬁve operators P6 (h)i∗ and P4 (h) ◦ D2res (h; 1),

P2 (h) ◦ D4res (h; 2),

D4res (h; −9/2) ◦ P2 (dr2 +hr ),

D2res (h; −11/2) ◦ P4 (dr2 +hr )

are linearly dependent. More precisely, P6 (h)i∗ can be presented as a linear combination of the last four, i.e., P6 (h)i∗ = αP4 (h)D2res (h; 1) + βP2 (h)D4res (h; 2) + γD4res (h; −9/2)P2 (dr2 +hr ) + δD2res (h; −11/2)P4(dr2 +hr )

(6.11.24)

for certain rational coeﬃcients α, β, γ, δ. In order to determine the numerical coeﬃcients, it suﬃces to choose them in the ﬂat case. The result is (α, β, γ, δ) =

1 (18, −5, −4, 12). 21

In turn, the families D4res (h; λ) and D2res (h; λ) can be written as linear combinations of compositions of the Yamabe operators P2 (h), P2 (dr2 + hr ) and the Paneitz operator P4 (h). Hence P6 (h)i∗ is a linear combination of compositions of the lower order GJMS-operators P2 (h), P4 (h), P2 (dr2 +hr ) and P4 (dr2 +hr ). The explicit result is given in Theorem 6.11.17 and Corollary 6.11.6. On the other hand, the four identities in Theorem 6.11.4 imply that the family D6res (h; λ) can be written as a linear combination of the right-hand sides, i.e., as a linear combination of compositions involving P2 (dr2 +hr ), P2 (h), P4 (h) and P6 (h). Finally, we show how Theorem 6.11.4 can be applied to determine Q6 . The family D6res (h; λ) is a polynomial of degree 3 in λ, i.e., can be written in the form Aλ3 + Bλ2 + Cλ + D with four operator-coeﬃcients A, B, C, D. Now, by Theorem 6.11.4, we have D = P6 (h)i∗ . We determine Q6 by combining the formula Q6 (h) = D˙ 6res (h; 0)(1) = C(h)(1) with the formula for C(1) which follows from the last three factorization identities. In other words, we solve the equation ⎛ ⎞⎛ ⎞ ⎞ ⎛ res A(1) ( − 52 )3 ( 52 )2 − 25 D4 (h; − 29 )P2 (dr2 +hr )(1) ⎜ ⎟⎜ ⎟ ⎟ ⎜ P2 (h)D4res (h; 2)(1) 22 2 ⎠ ⎝B(1)⎠ = ⎝ ⎝ 23 ⎠ 1 1 1 P4 (h)D2res (h; 1)(1) C(1) for C(1). We ﬁnd ⎛ ⎛ ⎞ ∗ A(1) 4 ⎜ ⎜ ⎟ ⎝∗ ⎝B(1)⎠ = − 315 4 C(1)

∗ ∗ 175 8

⎞ ⎛ res ⎞ ∗ D4 (h; − 29 )P2 (dr2 +hr )(1) ⎟⎜ ⎟ ∗ ⎠⎝ P2 (h)D4res (h; 2)(1) ⎠. 225 res P4 (h)D2 (h; 1)(1) − 2

284

Chapter 6. Conformally Covariant Families

In particular, Q6 (h) can be written as the linear combination 16 9 D4res h; − − P2 (dr2 +hr )(1) 9·7·5 2 5 10 P2 (h)D4res (h; 2)(1) + P4 (h)D2res (h; 1)(1). (6.11.25) − 9·2 7 The analogous formula for Q4 (h) is (6.11.16). But the families D4res (h; λ) and D2res (h; λ), in turn, can be written as compositions of P4 (h),

P2 (h)

and P2 (dr2 +hr ).

This shows that Q6 (h) can be written as a composition of P4 (h), P2 (h) and P2 (dr2+ hr ) (acting on u = 1). This leads to a recursive formula for Q6 in terms of Q4 , Q2 and the GJMS-operators P4 , P2 (Theorem 6.11.5). The results of the evaluation of the terms in (6.11.25) and the resulting formula for C(h)(1) are listed in Table 6.3. term Δ J Δ(|P|2 ) JΔJ (dJ, dJ) (P, Hess(J)) J3 |P|2 J tr(P3 ) (B, P) 2

P4 D2res (1)(1) 1 0 −5 0 4 3 −2 0 0

P2 D4res (2)(1)

D4res (− 92 )P2 (1)

C(1)

2 −8 −4 0 0 0 16 0 0

− 52 (− 52 )14 (− 52 )(− 31 2 )

1 4 −8 0 8 8 −24 16 8

0 (− 52 )18 (− 52 ) 117 4 (− 52 )(− 263 2 ) 21(− 25 )6 21(− 25 )3

Table 6.3: Evaluation of right-hand sides of factorization identities and C(1) The result for Q6 ﬁts with Table 6.1 for n = 6. Among the calculations that lead to Table 6.3, only those which yield the last two contributions are interesting enough to be given explicitly. These two terms arise from the contribution 1 5 21 − (∂/∂r)4 1r=0 J(dr2 +hr ) 2 in the composition D4res (h; − 92 )P2 (dr2 +hr )(1). Lemma 6.11.9. For (M 6 , h), 1 (∂/∂r)4 1r=0 J(dr2 +hr ) = 3(B, P) + 6 tr(P3 ).

6.11. Factorization identities for residue families. Recursive relations

285

Proof. From the proof of Lemma 6.11.1 we recall that J(dr2 +hr ) = −

1 ˙ tr(h−1 r hr ) 2r

(see (6.11.8)). But the formula for h−1 r in the proof of Lemma 6.11.1 and h˙ r = 2rh(2) + 4r3 h(4) + 6r5 h(6) + · · · imply

1 −1 −1 −1 4 ˙ tr(h−1 r hr ) = · · · + r 6 tr(h0 h(6) ) − 4 tr(h0 h(2) h0 h(4) ) r

−1 −1 3 h ) ) − 2 tr(h h h h ) + ··· . + 2 tr((h−1 (2) (4) 0 (2) 0 0

Hence for (∂/∂r)4 |r=0 (J(dr2 +hr )) we ﬁnd the formula −1 −1 −1 3 4! −3 tr(h−1 0 h(6) ) + 3 tr(h0 h(2) h0 h(4) ) − tr((h0 h(2) ) )

= 4! −3 tr(h(6) ) + 3 tr(h(2) h(4) ) − tr((h(2) )3 ) . Using the relation (6.9.23) and the explicit formulas for h(2) , h(4) (Theorem 6.9.1), the latter sum simpliﬁes to 1 1 1 3 3 (B, P) + tr((P ) ) . 4! tr(h(2) h(4) ) − tr((h(2) ) ) = 4! 2 8 4 The proof is complete.

Thus in C(h)(1) we ﬁnd the contribution 4 5 − 4 − 21 3(B, P) + 6 tr(P3 ) = 8(B, P) + 16 tr(P3 ). 315 2 These terms yield the last two terms in Table 6.3. Next, we use (6.11.25) to derive a recursive formula for Q6 which generalizes Corollary 6.11.1 for Q4 . Theorem 6.11.5 (Recursive formula for Q6 ). For any Riemannian 6-manifold (M, h), 2 5 8 ¯ 2, Q6 = (P2 Q4 + P4 Q2 ) − P22 Q2 + i∗ P¯22 Q 3 3 3 where ¯ 2 (h) = Q2 (dr2 +hr ), P¯2 (h) = P2 (dr2 +hr ), Q and all other quantities are understood with respect to h.

286

Chapter 6. Conformally Covariant Families

Proof. First, we determine a formula for D4res (λ) in terms of Yamabe and Paneitz operators. In order to ﬁnd such a formula, we use the factorization identities D4res (−1) = P4 i∗ ,

D4res (0) = P2 D2res (0),

D4res (−5/2) = D2res (−9/2)P¯2 .

These are special cases of Theorem 6.11.3 for n = 6. We ﬁnd 2 8 ¯ 4 ¯2 2 2 = − P4 + 2P2 − P2 P2 + P2 λ 3 3 3 5 20 ¯ 4 ¯2 2 + − P4 + 7P2 − P2 P2 + P2 λ + 5P22 − 4P2 P¯2 ; 3 3 3

D4res (λ)

here we omit the restrictions i∗ in order to simplify the formulas. It follows that D4res (2) = −6P4 + 27P22 − 28P2 P¯2 + 8P¯22 , D4res (−9/2) = −6P4 + 14P22 − 28P2 P¯2 + 21P¯22 . Now

(6.11.26) (6.11.27)

D2res (λ) = −(2λ+4)i∗P¯2 + (2λ+5)P2 i∗

and (6.11.25) imply Q6 =

15 5 − P23 + P2 P4 + 10P4 P2 (1) 2 3 106 2 ∗ ¯ 4 16 124 P4 + P2 i P2 (1) − P2 i∗ P¯22 (1) − i∗ P¯23 (1). + − 15 15 5 15

But using P2 (1) = −2J,

5 i∗ P¯2 (1) = − J, 2

P4 (1) = Q4 ,

the latter formula yields Q6 =

5 2 8 4 16 P2 Q4 + P4 Q2 − P22 Q2 − P2 i∗ P¯22 (1) − i∗ P¯23 (1). 3 3 3 5 15

Now combining this identity with the relation ¯ 2 = P2 Q2 + 4 i∗ P¯ 2 (1) Q4 = P2 Q2 − 2i∗ P¯2 Q 2 5 (Lemma 6.11.5 for n = 6), we ﬁnd Q6 =

8 5 2 8 ¯ 2. P2 Q4 + P4 Q2 − P22 Q2 + P22 Q2 − P2 Q4 + i∗ P¯22 Q 3 3 3 3

This yields the assertion.

6.11. Factorization identities for residue families. Recursive relations

287

¯ 2 is responsible for the contribution (B, P) In Theorem 6.11.5, the term i∗ P¯2 Q to Q6 . Example 6.11.1. Let n = 6 and h be Einstein. Then hr = (1−cr2 )2 h,

c = J(h)/2

(Theorem 6.16.1) and J(dr2 +hr ) = −

1 2 −1 ˙ tr(h−1 r hr ) = J(1−cr ) 2r

(by (6.11.8)). Moreover, for the Yamabe operator P¯2 we ﬁnd P2 (dr2 +hr ) =

∂ ∂2 5 + (1−cr2 )−2 Δ − J(1−cr2 )−1 . − Jr(1−cr2 )−1 ∂r2 ∂r 2

Now on (M 6 , h), GJMS-operators of order up to 6 are given by P2 = Δ − 2J,

P4 = (Δ − 2J)(Δ − 4/3J),

Hence Q2 = J,

8 Q4 = J2 3

and

P6 = (Δ − 2J)(Δ − 4/3J)Δ.

3 40 3 J J Q6 = 5! = 3 9

(see also (6.16.1)). Some calculations show that 5 P¯22 (1) = (28c − 12c2 r2 )(1−cr2 )−3 2 and hence ¯2 = i∗ P¯22 Q

29 3 J . 6

Thus the recursive formula for Q6 reads 2 16 60 116 3 40 3 16 8 3 5 3 8 29 3 J = − J3 − J3 + J = J . Q6 = − + J − 4J + 3 3 3 3 3 6 9 9 9 9 Next, recall that by Lemma 6.11.5 the formula ¯2 Q4 = P2 Q2 − 2i∗ P¯2 Q (Corollary 6.11.1) for the critical Q-curvature of a Riemannian four-manifold (M, h) continues to hold true in all dimensions n ≥ 4. The following result claims an analogous eﬀect for the recursive formula for Q6 . Theorem 6.11.6. For a Riemannian manifold (M, h) of dimension n ≥ 6, Q6 =

2 5 8 ¯ 2. (P2 Q4 + P4 Q2 ) − P22 Q2 + i∗ P¯22 Q 3 3 3

(6.11.28)

288

Chapter 6. Conformally Covariant Families

term 2

Δ(|P| ) Δ2 J JΔJ (dJ, dJ) (P, Hess(J)) J3 tr(P3 ) |P|2 J (B, P)

contribution in Q6,n 4 1 2−3n 2

6−n 8 n2 −4 4

16 −4n 16 n−4

Table 6.4: The structure of Q6,n We emphasize that the numerical coeﬃcients in (6.11.28) do not depend on the dimension of M . We rephrase this observation by saying that (6.11.28) is universal. Proof. We prove (6.11.28) by direct calculation of the right-hand side and comparison with Table 6.1, i.e., with the equivalent Table 6.4. The right-hand side of (6.11.28) is the sum of

n−2 n 2 J J −2|P|2 −ΔJ Δ− 2 2

n−4 n 2 2 2 2 J −2|P| −ΔJ J + Δ + δ((n−2)Jh−4P)#dJ + 3 2 2 2 n−2 5 J J Δ− − 3 2

2 3

and 8 (4) ¨ − (n−1)J¨J + Δ¨J + Δ2 J − n−1 Δ(J2 ) J − 2J¨J + Δ¨J + ΔJ 3 2 2

n−1 n−1 ¨ n−1 JJ − JΔJ + − J3 . 2 2 2 We evaluate these terms by using 1 1 J(4) = 4! tr(h(2) h(4) ) − tr(h3(2) ) = 3! (B, P) + tr(P3 ) 2 n−4

(6.11.29)

6.11. Factorization identities for residue families. Recursive relations

289

¨ = (dJ, du) + 2(P, Hess(u)). We ﬁnd (extending Lemma 6.11.9), ¨J = |P|2 and Δu that the respective coeﬃcients of JΔJ and (dJ, dJ) are given by 83 3n 2 3n−2 2 3n−8 5 3 − + (n−2) − (n−1) = 1− 3 2 3 2 32 32 2 and

2 2 5 8 n − (n−6) + (n−2) − (n−2) = 6−n. 3 3 3 3 3 2 Next, the respective coeﬃcients of J and |P| J are 2 2 8 n−1 n2 −4 2 n(n−2) 2 n(n−4) 5 n−2 + − + = − 3 4 3 4 3 2 3 2 4

and

2 2 8 3n+1 (n−2) − (n−4) − = −4n. 3 3 3 2 Finally, (6.11.29) yields the correct contributions of tr(P3 ) and (B, P). We omit the details concerning the coeﬃcients of Δ(|P|2 ), Δ2 J and (P, Hess(J)).

Combining Theorem 6.11.6 with (6.11.18), yields a formula for Q6,n which contains the Paneitz operator P¯4 instead of P¯22 . ¯ 2. Corollary 6.11.2. Q6 = 2 P2 Q4 + (P 2 − 2P4 )Q2 + 8 i∗ P¯4 Q 2

3

3

Now we generalize. The method in the proof of Theorem 6.11.5 is expected to generate the following recursive formula for Qn in terms of lower order Qcurvatures, lower order GJMS-operators and powers of the Yamabe operator P¯2 . Conjecture 6.11.1 (Recursive formula for Qn ). For a conformally ﬂat metric, the critical Q-curvature Qn admits a representation Qn =

n

aI P2I (Qn−2|I| ) + (−1) 2 −1 (n)

I

(n−2)(n−4) · · · 2 ∗ ¯ n2 −1 ¯ i P2 (Q2 ) (n−3)(n−5) · · · 1

(6.11.30)

(n)

with rational coeﬃcients aI . In (6.11.30) the sum runs over all ordered partitions I of integers in the interval [1, n2 − 1] as sums of natural numbers. For P2I is deﬁned as the composition P2I1 ◦ · · · ◦ P2Im I = (I1 , . . . , Im ), the operator of GJMS-operators and |I| = i Ii . A number of comments are in order. An additional important feature of the representation (6.11.30) is its conjectural universality. It states that in the critical case, i.e., for the critical Q-curvature Qn in dimension n, the formula is also valid in all higher even dimensions, i.e., represents the subcritical Q-curvature Qn in even dimensions ≥ n + 2. Here the point is that the coeﬃcients aI depend only on the order of the curvature invariant but not on the dimension of the underlying space. This extends the observed universality in Lemma 6.11.5 and Theorem 6.11.6.

290

Chapter 6. Conformally Covariant Families

The proof of Theorem 6.11.5 uses the universality of the recursive formula ¯ 2 (in dimension 6) in order to replace the term P¯ 2 (1). In a Q4 = P2 Q2 − 2i∗ P¯2 Q 2 similar way, universality plays a central role in the derivation of the representations (6.11.30) for all n. Theorem 6.11.6 supports the expectation that ﬁnally the assumption of conformal ﬂatness can be removed from the scene. It appears here since the factorization identities which involve the factor P¯2 are (presently) not known for general metrics. We count the number of terms in (6.11.30). The number of solutions of the equation (6.11.31) I1 + · · · + Ir = m, m ∈ N0 with Ii ∈ N0 is

m+r−1 . r−1

Hence the number of solutions of (6.11.31) with m ≥ r and Ii ∈ N is m−r+r−1 m−1 = . r−1 r−1 In particular, m ∈ [1, n2 −1] can be written in m−1 m−1 m−1 + + ··· + = 2m−1 0 1 m−1 ways as the sum of natural numbers (here we take the order into account). Thus there are 0 n n 2 + 21 + · · · + 2 2 −2 + 1 = 2 2 −1 terms on the right-hand side of (6.11.30). n −1 ¯ The coeﬃcient of the term i∗ P¯22 (Q 2 ) can be determined as follows by n tracing the contribution of tr(P 2 ). On the one hand, using Lemma 6.14.1, we n determine the coeﬃcient of tr(P 2 ) in the holographic formula for Qn . By Newton’s formula, tr(∧m T ) =

1 tr(T )m + · · · + (m−1)!(−1)m−1 tr(T m ) . m!

Thus we ﬁnd " n n n n n

# 1

n n 2 2 −1 ! −1 ! n (−1) 2 −1 −1 ! = (−2) 2 −1 −1 !2 2 2 ( 2 )! 2 2

(6.11.32)

for this coeﬃcient. The latter result is conﬁrmed for n ≤ 6 by the explicit formulas n n −1 ¯ for Qn . On the other hand, P¯22 (Q 2 ) contains a multiple of tr(P 2 ) in i∗ (∂/∂r)n−2 J(dr2 +hr ).

6.11. Factorization identities for residue families. Recursive relations

291

Similarly as in the proof of Lemma 6.11.9, we ﬁnd that this coeﬃcient is (n−2)! . n 2 2 −1 Now n

(−1) 2 −1

n

n (n−2)(n−4) · · · 2 (n−2)! 2 −1 −1 !2 = (−2) n (n−3)(n−5) · · · 1 2 2 −1 2 n

−1 ¯ conﬁrms the coeﬃcient of P¯22 (Q 2 ) in (6.11.30). It is clearly of interest to make the rational coeﬃcients aI more explicit. This problem is discussed in [95]. The fact that the 2N right-hand sides of the system of factorization identin −1 ¯ res (λ) satisfy N −1 linear relations can be used to replace P¯22 (Q ties of D2N 2 ) in (6.11.30) by similar compositions which involve higher order GJMS-operators of dr2 +hr . For Q6 , such a result is given in Corollary 6.11.2. As explained above, the representation (6.11.30) of the critical Q-curvature follows by combining the relation n D˙ nres (0)(1) = −(−1) 2 Qn

with recursive formulas for subcritical Q-curvatures. It is natural to ask for a similar device to derive recursive formulas in the subcritical case. Now the holographic formula in Conjecture 6.9.1 actually may be regarded as pointing to the existence of a relation between the quantities n

res − + N (1). Q2N and D˙ 2N 2 res Although for 2N < n, the value D˙ 2N (− n2 + N )(1) does not coincide with Q2N , Theorem 6.11.7 establishes such relations for N = 2 and N = 3. The proof shows that the relations are equivalent to the respective recursive formulas (see Remark 6.11.3). On the other hand, the relations in Theorem 6.11.7 can be shown to be equivalent to the corresponding holographic formula in Conjecture 6.9.1. Theorem 6.11.7 will be used to derive explicit formulas for the Q-polynomials res Qres 4 (λ) and Q6 (λ) which generalize the holographic formulas for the respective Q-curvatures Q4 and Q6 .

Theorem 6.11.7. For any Riemannian n-manifold (M, h),

n−4 n D˙ 4res h; − + 2 (1) = −Q4,n − (Q4,n + P2 Q2,n ) 2 2 for n ≥ 4, and

n n−6 3 3 D˙ 6res h; − + 3 (1) = Q6,n + Q6,n + P2 Q4,n − 2P4 Q2,n + P22 Q2,n 2 2 2 2 for n ≥ 6.

292

Chapter 6. Conformally Covariant Families

Theorem 6.11.7 shows that for N ≤ 3 the linear term of the polynomial res (h; λ)(1) is a combination of Q2N,n and contributions in terms of lower order D2N Q-curvatures and GJMS-operators. In particular, it conﬁrms that in the respective critical cases D˙ 4res (0)(1) = −Q4 and D˙ 6res (0)(1) = Q6 . The general picture will be discussed below. Proof. The factorization identities

n D4res h; − +2 = P4 (h)i∗ , 2

n n D4res h; − +3 = P2 (h) ◦ D2res h; − +3 , 2 2 n−1 n+3 D4res h; − = D2res h; − ◦ P2 (dr2 +hr ) 2 2 (Theorem 6.11.3) imply that the coeﬃcients of

2

n n D4res (h; λ) = A λ+ −2 + B λ+ −2 + C 2 2 are determined by ⎛ ⎞ ⎛ 4 A 15 4 ⎝B ⎠ = ⎜ ⎝− 15 C 0

2 5 3 5

0

⎞ ⎛ res n+3 ⎞ − 32 D2 − 2 P¯2 ⎟ ⎟ ⎜ − 31 ⎠ ⎝P2 D2res − n2 +3 ⎠ . 1 P4 i∗

Here the arguments are analogous to those which yield (6.11.16). Hence n

D˙ 4res − +2 (1) = B(1) 2 n

4 3 1 n+3 ¯ P2 (1). = − P4 (1) + P2 D2res − +3 (1) − D2res − 3 5 2 15 2 But using D2res

n

− +3 = −4i∗ P¯2 + 5P2 i∗ 2

and

D2res

n+3 − = 5i∗ P¯2 − 4P2 i∗ , 2

we ﬁnd n

1 4 4 D˙ 4res − +2 (1) = − P4 (1) + 3P22 (1) − P¯22 (1) − P2 i∗ P¯2 (1) 2 3 3 3 1 n−4 n−2 4 n−1 ∗ ¯ ¯ 4 n−1 ¯ 2. Q4 − 3 P2 Q2 + i P2 Q2 + P2 i∗ Q =− 3 2 2 3 2 3 2

6.11. Factorization identities for residue families. Recursive relations

293

¯ 2 = Q2 (Lemma 6.11.1), we obtain Thus, in view of i∗ Q n

1 n−4 5n−14 2n−2 ∗ ¯ ¯ Q4 − P2 Q2 + i P2 Q2 . D˙ 4res − +2 (1) = − 2 3 2 6 3 Next, we apply Lemma 6.11.5, i.e., ¯ 2, Q4 = P2 Q2 − 2i∗ P¯2 Q and ﬁnd n

n−4 5n−14 n−1 D˙ 4res − +2 (1) = − Q4 − P2 Q2 + (P2 Q2 − Q4 ) 2 6 6 3 n−4 n−2 Q4 − P2 Q2 . =− 2 2 This proves the ﬁrst assertion. In order to prove the second identity, we ﬁrst observe that

3

2

n n n D6res (λ) = A λ+ −3 + B λ+ −3 + C λ+ −3 + D 2 2 2 with ⎛ ⎞ ⎛ A 2 − 35 4 ⎜B ⎟ ⎜−6 − 105 4 ⎜ ⎟ ⎜ 8 ⎜ ⎟=− ⎜ 175 ⎝C ⎠ 315 ⎝ 4 8 D

0

0

45 2 45 4 − 225 2

0

¯ ⎞ D4res (− n+3 2 )P2 n ⎟ 63 ⎟ ⎜ res 8 ⎟ ⎜P2 D4 − 2 +5⎟ ⎟ ⎟. ⎜ 693 ⎠ ⎝ P4 D2res − n2 +4 ⎠ 8 − 315 P6 i∗ 4 − 63 4

⎞⎛

The latter formula follows from an extension to general dimension of the four factorization identities in Theorem 6.11.4. Hence for n

D˙ 6res − +3 (1) = C(1), 2 we ﬁnd −

16 n+3 ¯ D4res − P2 (1) 9·7·5 2 n n

10 11 5 P2 D4res − +5 (1) + P4 D2res − +4 (1) − P6 (1). − 9·2 2 7 2 10

The same arguments as in the proof of Theorem 6.11.5 simplify the sum to 5 15 − P23 + P2 P4 + 10P4 P2 (1) 2 3 106 2 ∗ ¯ 4 16 11 124 P4 + P2 i P2 (1) − P2 i∗ P¯22 (1) − i∗ P¯23 (1) − P6 (1). + − 15 15 5 15 10

294

Chapter 6. Conformally Covariant Families

Now using P2 (1) = −

n−2 Q2 , 2

i∗ P¯2 (1) = −

n−1 ∗ ¯ n−1 i Q2 = − Q2 , 2 2

P4 (1) =

n−4 Q4 , 2

the latter sum equals 106(n−1) 15(n−2) 5(n−4) 13n−88 P2 Q4 − P4 Q2 − − P22 Q2 6 15 30 4 4 16(n−1) ∗ ¯ 2 ¯ 11 n−6 i P2 Q2 + Q6 . − P2 i∗ P¯22 (1) + 5 30 10 2 But the relation i∗ P¯22 (1) =

n−1 ∗ ¯ ¯ n−1 i P2 Q2 = − (−P2 Q2 + Q4 ) 2 4

(Lemma 6.11.5) shows that the sum can be written in the form n−1 11 n−6 ¯2 Q6 + 2P2 Q4 + 2P4 Q2 − 5P22 Q2 + 8i∗ P¯22 Q 10 2 15 n−6 3 (P2 Q4 − 2P4 Q2 + P22 Q2 ). + 2 2 Now Theorem 6.11.6 completes the proof.

Remark 6.11.3. The proof of Theorem 6.11.7 also shows that the recursive formula ¯2 Q4 = P2 Q2 − 2i∗ P¯2 Q (in any dimension n ≥ 3) follows from the ﬁrst assertion of Theorem 6.11.7 and the factorization identities (used to evaluate D˙ 4res (− n2 + 2)(1)). A similar comment concerns the recursive formula (6.11.28) for Q6 . Now we apply Theorem 6.11.7 in order to derive explicit formulas for the Q-polynomials N res Qres 2N (h; λ) = −(−1) D2N (h; λ)(1) for N = 2 and N = 3 in terms of Q-curvatures and GJMS-operators. We recall that

n

n N 2N Qres N ! − −λ+2N −1 . . . − −λ+N 2N (λ) = −(−1) 2 2 2 ∗ (λ+n−2N ) (v0 ) + · · · + T0∗ (λ+n−2N ) (v2N )] × [T2N (Deﬁnition 6.6.3). In particular, the critical Q-polynomial is given by

n n n Qres ! λ− +1 . . . λ [Tn∗ (λ)(v0 ) + · · · + T0∗ (λ)(vn )] . n (λ) = −2 2 2

6.11. Factorization identities for residue families. Recursive relations

Theorem 6.11.8. For all metrics,

n n Qres 4 (λ) = −λ λ+ −3 Q4 − λ λ+ −2 P2 Q2 2 2

295

(6.11.33)

for even n ≥ 4, and Qres 6 (λ) =

n n 1 λ λ+ −4 λ+ −5 Q6 2 2 2

n n 3 + λ λ+ −3 λ+ −4 P2 Q4 + P2 Q2 2 2 2

n n − λ λ+ −3 λ+ −5 P4 Q2 2 2

(6.11.34)

for even n ≥ 6. Proof. The proof of the ﬁrst assertion rests on the characterizing properties n

n−4 Qres − +2 Q4 , = −P4 (1) = − 4 2 n n2

− +3 = −P2 D2res − +3 (1), Qres 4 2 n n 2

res res Q˙ 4 − +2 = −D˙ 4 − +2 (1) 2 2 of the Q-polynomial. We verify that the polynomial on the right-hand side of (6.11.33), in fact, has these three properties. The ﬁrst identity is obvious. The value of that polynomial at λ = − n2 +3 is n

−3 P2 Q2 . 2 But n

n n−1 n−2 Q2 + 5 Q2 = −3 Q2 . −D2res − +3 (1) = 4i∗ P¯2 (1) − 5P2 (1) = −4 2 2 2 2 This proves the second property. The third property is a consequence of Theorem 6.11.7. Similarly, the proof of the formula for Qres 6 (λ) rests on the characterizing properties n

n−6 − +3 =− Q6 , Qres 6 2 2 n

n − +4 = P4 D2res − +4 (1), Qres 6 2 2 n n

res res Q6 − +5 = P2 D4 − +5 (1), 2

n n 2

res res ˙ ˙ Q6 − +3 = D6 − +3 (1) 2 2

296

Chapter 6. Conformally Covariant Families

of the Q-polynomial. We verify that the polynomial on the right-hand side of (6.11.34), in fact, has these four properties. The ﬁrst property is obvious. The last property follows from Theorem 6.11.7. The remaining two identities assert that

n n −4 Q2 = D2res − +4 (1), − 2 2 n n

3 res − −5 2 Q4 + Q2 = D4 − +5 (1), 2 2 2 respectively. But n

n n−1 ∗ ¯ n−2 D2res − +4 (1) = −6i∗ P¯2 (1)+ 7P2 (1) = 6 i Q2 − 7 Q2 = − −4 Q2 2 2 2 2 and

n n

n

− +5 −5 (1) = − (2Q4 + 3P2 Q2 ) D4res − +5 (1) = −Qres 4 2 2 2

by (6.11.33). The proof is complete. Note that Example 6.6.7 is the special case n = 4 of (6.11.33). The formula for the polynomial Qres 4 (λ) has an interesting consequence. Corollary 6.11.3. In all dimensions, ¯4 = i∗ Q

3 1 Q4 + P2 Q2 2 2

Proof. The identity D4res

and

¯2 = i∗ P¯2 Q

1 1 P2 Q2 − Q4 . 2 2

n−3 − = i∗ P¯4 2

(Theorem 6.11.3) implies n−3 n−3 ∗ ¯ −Qres i Q4 . − = 4 2 2 But (6.11.33) yields n−3 1 n−3 n−3 3 Q4 + P2 Q2 . − −Qres = 4 2 2 2 2 2 Hence

3 1 Q4 + P2 Q2 . 2 2 Similarly, the second assertion is a consequence of the factorization identity n−1 n−3 ¯ D4res − P2 = D2res − 2 2 ¯4 = i∗ Q

6.11. Factorization identities for residue families. Recursive relations

297

(Theorem 6.11.3). In fact, we get n−1 n−3 n−1 ¯ res Qres − − Q2 = D 4 2 2 2 2

and (6.11.33) implies the assertion.

Note that the ﬁrst identity in Corollary 6.11.3 already appeared in the proof of Lemma 6.11.4 in the case n = 4. The second identity coincides with Lemma 6.11.5. For the convenience of later reference, we summarize the critical cases of Theorem 6.11.8. Theorem 6.11.9. For (M 4 , h), the critical Q-polynomial Qres 4 (h; λ) is given by 2 Qres 4 (λ) = −λ(λ−1)Q4 − λ P2 Q2 .

Similarly, for (M 6 , h) the critical Q-polynomial Qres 6 (h; λ) is given by 1 3 res 2 Q6 (λ) = λ(λ−1)(λ−2)Q6 + λ (λ−1)P2 Q4 + P2 Q2 − λ2 (λ−2)P4 Q2 . 2 2 If h is Einstein, the formulas for the critical Q-polynomials Qres n (h; λ) (n ≤ 6) further simplify. Corollary 6.11.4. Let h be Einstein. Then the respective critical Q-polynomials factorize as Qres 2 (h; λ) = λQ2 (h), λ(λ−3) Q4 (h), 3 λ(λ−4)(λ−5) Q6 (h). Qres 6 (h; λ) = 4·5

Qres 4 (h; λ) = −

Proof. The ﬁrst assertion is trivial by Example 6.6.6. For the proof of the second assertion we use 2 J 3 = J2 P2 = Δ − J, Q4 = 3! 2 2 (Theorem 3.2.3). Hence 3 1 2 2 2 2 Qres 4 (λ) = − λ(λ−1)J + λ J = − λ(λ−3)J . 2 2 Similarly, for the proof of the last assertion we apply P2 = Δ−2J,

P4 = (Δ−2J)(Δ−4/3J)

298

Chapter 6. Conformally Covariant Families

and Q4 =

8 2 J , 3

Q6 = 5!

3 J 40 3 J . = 3 9

Hence 20 2 8 λ(λ−1)(λ−2)J3 + λ2 (λ−1)J3 − λ2 (λ−2)J3 9 3 3 2 = λ(λ−4)(λ−5)J3 . 9

Qres 6 (λ) =

The proof is complete.

It is open whether Corollary 6.11.4 extends to general even dimensions. Moreover, the above techniques yield the following generalization of Theorem 6.11.9 for conformally ﬂat metrics. Theorem 6.11.10. Let (M, h) be conformally ﬂat of even dimension n. Assume that n −1, (6.11.35) Qres 2j (0) = 0 for j = 1, . . . , 2 and deﬁne the polynomials Qres 2j (λ) by res Qres 2j (λ) = λQ2j (λ).

Then Qres n (λ)

= (−1)

n 2 −1

λ −

n 2 −1

λ

n 2 −1

+λ

j=1

Proof. The

and

+k

k

k=1

n 2

(−1)

Qn

2 λ− n

j

k=1 k=j

n 2

+k k−j

P2j Qres n−2j

n 2

−j .

(6.11.36)

n 2 +1

relations n

n

j res Qres P Q −j = (−1) − j , 2j n−2j n 2 2

j = 1, . . . ,

n 2

n 2 ˙ res Q˙ res n (0) = −(−1) Dn (0)(1)

characterize the Q-polynomial Qres n (λ). We verify that the given polynomial of degree n2 has these properties. The deﬁnition immediately implies that for 1 ≤ N ≤ n2 its value at λ = n2 −N is

n n n N res −N (−1)N P2N Qres −N = (−1) −N . P Q 2N n−2N n−2N 2 2 2 Next, the value at λ = 0 of its derivative is Qn . Thus it suﬃces to apply the n holographic formula Qn = −(−1) 2 D˙ nres (0)(1).

6.11. Factorization identities for residue families. Recursive relations

299

Remark 6.11.4. Theorem 6.11.10 can be regarded as a generalization of the holon graphic formula Qn = −(−1) 2 D˙ nres (0)(1). The latter identity is valid for all metrics. Since the proof of Theorem 6.11.10 rests only on that system of factorization identities which extends to general metrics (we refer to the comments at the end of the present section), (6.11.36) also continues to hold true for all metrics (under the assumption (6.11.35)). Remark 6.11.5. The condition Qres 2N (0) = 0 means that the polynomial

n

n

− −λ+2N −1 · · · − −λ+N 2 2 ∗ (λ+n−2N )(v0) + · · · + T0∗ (λ+n−2N )(v2N )] × [T2N

(6.11.37)

vanishes at λ = 0. For N = n2 , this is equivalent to the condition Pn∗ (0)(1) = 0 which is satisﬁed since Pn is self-adjoint and annihilates constants. Moreover, by Theorem 6.11.8, the vanishing holds true for N = 1, 2, 3. We expect that the vanishing holds true in general. In fact, this would be a consequence of the general formula (6.11.38). We work out the meaning of the condition for N = 1 and N = 2. For N = 1, it is equivalent to the fact that −2(n−2+2λ) (T2∗ (λ+n−2)(1) + v2 ) = (Δ−(λ+n−2)J)(1) − 2(n−2+2λ)v2 vanishes at λ = 0. For N = 2, the condition states that (2λ+n−6)(2λ+n−4) [T4∗ (λ+n−4)(v0 ) + T2∗ (λ+n−4)(v2 ) + v4 ] vanishes at λ = 0. Using Remark 6.6.5, the latter assertion is equivalent to the (obvious) relation − (Δ−(n−4)J)(n−2)J + (n−4)(n−6)|P|2 − (n−6)ΔJ − 4(n−4)(Δ−(n−4)J)(v2) + 8(n−6)(n−4)v4 = 0. In the recursive formula (6.11.36), the lower order Q-polynomials play an important role. For these polynomials, the following analogous formula would generalize (6.11.36): N −1

− 2N + k Q2N k k=1 N −1 N n

λ + n2 − 2N + k − +2N −j . +λ (−1)j P2j Qres 2N −2j k−j 2 j=1

N −1 λ Qres 2N (λ) = (−1)

λ+

n 2

(6.11.38)

k=1 k=j

Note that (6.11.38) implies Qres 2N (0) = 0, i.e., shows that the assumption (6.11.35) in Theorem 6.11.10 is always satisﬁed.

300

Chapter 6. Conformally Covariant Families

Now Theorem 6.11.10, together with (6.11.38), yields a formula of the form Qres cI (λ)P2I Qn−2|I| , (6.11.39) n (λ) = I

where the sum runs over all partitions I of integers in [0, n2 −1] and the coeﬃcients cI (λ) are polynomials of degree n2 . Theorem 6.11.8 covers the special cases n = 4 and n = 6. The value of the polynomial on the right-hand side of (6.11.38) at λ = − n2 +2N −L (1 ≤ L ≤ N −1) is given by n n

− +2N −L (−1)L P2L Qres 2N −2L − +2N −L 2 2

n +2N −L . = (−1)L P2L Qres − 2N −2L 2

Similarly, for λ = − n2 +N , we ﬁnd the value N −1 n

n −N +k − +N (−1)N −1 −N Q2N . Q2N = − 2 k 2 k=1

These observations show that the polynomial on the right hand side of (6.11.38) satisﬁes the same N factorization identities as those which follow for Qres 2N (λ) from the ﬁrst N factorization identities in Theorem 6.11.1. Moreover, the validity of implies a formula for

n n N ˙ res D +N = −(−1) +N (1), Q˙ res − − 2N 2N 2 2 and, by the above arguments, an independent proof of that formula would be enough to prove (6.11.38). The following result makes the missing identity explicit. Theorem 6.11.11 (Holographic formula for Q2N ). The identity (6.11.38) implies <

= −1 n

N

1 − −N Q2N = Q˙ res +N 1+ 2N 2 k 2 k=1 −1 n

N

1 N −1 N n − −N +2N −j . − (−1) P2j Qres 2N −2j 2 N −j j −1 2 j=1 n

(6.11.40)

For 2N = n, (6.11.40) specializes to the holographic formula n Qn = −(−1) 2 D˙ nres (0)(1) = Q˙ res n (0).

All terms on the right-hand side of (6.11.40) are given by holographic data, i.e., holographic coeﬃcients and the asymptotics of eigenfunctions of the Laplacian of the Poincar´e-Einstein metric. Theorem 6.11.11 emphasizes the role of the holographic formula (6.11.40) for the subcritical Q-curvatures in the proof of the for-

6.11. Factorization identities for residue families. Recursive relations

301

mula (6.11.38) for the subcritical Q-polynomials. It remains to ﬁnd an independent proof of (6.11.40). Example 6.11.2. The cases N = 2 and N = 3 of (6.11.40) are covered by Theorem 6.11.7. In fact, for N = 2, (6.11.40) reads n n

n

n res − − +2 = 1+ −2 Q −2 P +3 . + Q Q˙ res 4 2 4 2 2 2 2 2 In view of Qres 2 (λ) = Q2 , this formula coincides with the ﬁrst identity of Theorem 6.11.7. Similarly, for N = 3, (6.11.40) states

n n−6 3 +3 = 1+ · Q˙ res − Q6 6 2 2 2 n n

1

n res − − −3 P2 Qres +5 +4 + 2P − Q . 4 2 4 2 2 2 2 Now using Qres 2 (λ) = Q2 and n

Qres − +5 = −2Q4 − 3P2 Q2 , 4 2 the latter formula reads

n n n−6 3 3 2 res ˙ Q6 − +3 = 1+ · −3 P2 Q4 + P2 Q2 − 2P4 Q2 . Q6 + 2 2 2 2 2 This formula coincides with the second identity of Theorem 6.11.7. Proof of Theorem 6.11.11. Diﬀerentiating the ﬁrst term in (6.11.38) at λ = − n2 + N , yields N −1

(−1)

N −1 k=1

−N +k k

Q2N

N −1 N −1

n 1 −N +k N −1 + − +N (−1) Q2N 2 r k r=1 k=1 k=r

= Q2N −

n 2

−N

−1

N r=1

1 Q2N . −N +r

Next, by diﬀerentiation of the second term, we obtain −1 N −1 n

N 1 −N +k j − +N (−1) P2j Qres 2N −2j (· · ·). 2 N −j k−j j=1 k=1 k=j

But since

N −1 k=1 k=j

−N +k k−j

N −j

= −(−1)

N −1 , j −1

302

Chapter 6. Conformally Covariant Families

the sum equals n 2

N −1

−N (−1)N j=1

1 N −j

N −1 P2j Qres 2N −2j (· · ·). j −1

The proof is complete.

The polynomial identity (6.11.38) is equivalent to a set of N identities for Q2N which arise by comparing coeﬃcients of powers of λ on both sides. Theorem 6.11.11 deals with the coeﬃcient of λ. In the critical case 2N = n, it leads to the holographic formula for Qn . A second case of substantial interest concerns the coeﬃcients of the leading power. In fact, comparing coeﬃcients of λN in (6.11.38) implies top (N −1)!(Qres = (−1)N −1 Q2N 2N ) N −1

n N −1 +2N −j . − − P2j Qres 2N −2j j −1 2

(6.11.41)

j=1

top Here (Qres denotes the coeﬃcient of λN in the Q-polynomial Qres 2N ) 2N (λ). In the critical case 2N = n, we ﬁnd the following result.

Corollary 6.11.5. Under the assumption (6.11.35), n n 2 −1 n

n res top −1 2 −1 P Qres −1 !(Qn ) = (−1) 2 Qn − −j . 2j n−2j j −1 2 2

n

(6.11.42)

j=1

As described above, the terms which involve the values of the polynomials Q again can be written in terms of GJMS-operators and Q-curvatures. Since the lefthand side is deﬁned in terms of holographic data, (6.11.42) is a second holographic formula for Qn (see also Remark 6.11.6). The main diﬀerence between that formula top and the holographic formula of Section 6.6 is that the structure of (Qres is much n ) res simpler than that of Q˙ n (0). We illustrate the relations between both holographic formulas by two examples. Example 6.11.3. Let n = 4. The following discussion is valid for all metrics. (6.11.42) states that − (Q4 + P2 Q2 (1)) = − (Q4 + P2 Q2 ) res coincides with the coeﬃcient of λ2 in Qres 4 (λ) = −D4 (λ)(1), i.e., in

− 24 2!(λ−1)λ [T4∗ (λ)(v0 ) + T2∗ (λ)(v2 ) + v4 ] = −P4∗ (λ)(v0 ) + 8λP2∗ (λ)(v2 ) − 32λ(λ − 1)v4 . Now we observe

6.11. Factorization identities for residue families. Recursive relations

Lemma 6.11.10.

303

P4∗ (λ)(1) = 6λP2∗ (λ)(v2 ) − 16λ(λ−1)v4 .

Proof. We recall that P4∗ (λ)(1) = λ2 (J2 + 2|P|2 ) + λ(−3ΔJ + 2J2 − 2|P|2 ) (for the discussion of the linear term see Example 6.6.4). For the right-hand side we ﬁnd the formula 1 6λ(Δ−λJ) − J − 2λ(λ−1)(J2 − |P|2 ) 2

and the assertion follows by an easy calculation. Hence (6.11.42) reads − (Q4 + P2 Q2 ) = (−P4∗ (λ)(1) + 8λP2∗ (λ)(v2 ) − 32λ(λ−1)v4 )

top

= (2λP2∗ (λ)(v2 ) − 16λ(λ−1)v4 )top = −2Jv2 − 16v4

(6.11.43)

or, equivalently, Q4 = −P2 (Q2 ) − Q22 + 16v4 .

(6.11.44)

The latter formula will have an analog for Q6 (see (6.11.48)) and Q8 (see Theorem 6.13.1). (6.11.44) should be compared with the holographic formula Q4 = 16v4 + 2Δv2 (Theorem 6.6.6, see also Example 6.6.4). Both formulas relate Q4 to 16v4 and are easily seen to be equivalent. Note that Lemma 6.11.10 shows ∗ ∗ Qres 4 (λ) = −P4 (λ)(v0 ) + 8λP2 (λ)(v2 ) − 32λ(λ−1)v4 ∗ = 2λP2 (λ)(v2 ) − 16λ(λ−1)v4 .

We continue with a detailed discussion of the case n = 6. Example 6.11.4. Let n = 6. The following discussion is valid for all metrics. In view of Theorem 6.11.8, the relation (6.11.42) states that res Q6 − P2 Qres 4 (2) − 2P4 Q2 (1)

= Q6 − P2 (−2Q4 − 3P2 Q2 ) − 2P4 Q2 = Q6 + 2P2 Q4 + 3P22 Q2 − 2P4 Q2 coincides with the coeﬃcient of λ3 in the polynomial res 2Qres 6 (λ) = 2D6 (λ)(1).

304

Chapter 6. Conformally Covariant Families

Here D6res (λ)(1) is given by 26 3!(−λ+2)(−λ+1)(−λ) [T6∗ (λ)(v0 ) + T4∗ (λ)(v2 ) + T2∗ (λ)(v4 ) + v6 ] , i.e., equals P6∗ (λ)(1) − 21 3!λP4∗ (λ)(v2 ) + 24 3!λ(λ−1)P2∗ (λ)(v4 ) − 26 3!λ(λ−1)(λ−2)v6 . Similarly as in Example 6.11.3, we observe that in the latter sum the ﬁrst term can be written as a linear combination of the other terms. Lemma 6.11.11. P6∗ (λ)(1) = 10λP4∗ (λ)(v2 ) − 26 λ(λ−1)P2∗ (λ)(v4 ) + 26 3λ(λ−1)(λ−2)v6 . Proof. We compare coeﬃcients of powers of λ. By the self-adjointness of P6 = P6 (0), we have P6∗ (0)(1) = P6 (0)(1) = 0. This implies that the constant terms of the polynomials on both sides vanish. For the coeﬃcients of λ the claim is that P˙ 6∗ (0)(1) = 10P4∗ (0)(v2 ) + 26 P2∗ (0)(v4 ) + 26 3!v6 .

(6.11.45)

But using P˙ 6 (0)(1) = −Q6 , Theorem 6.10.2 yields P˙6∗ (0)(1) = −Q6 + 32P2∗ (0)(v4 ) + 8P2∗ (0)(v2 ). Now substituting Q6 = −8 · 48v6 − 32P2∗ (0)(v4 ) − 2P4∗ (0)(v2 ) (Corollary 6.10.1), completes the proof of (6.11.45). In order to compare the coeﬃcients of λ3 , we apply (6.6.41) and (6.10.2). We ﬁnd that on the left-hand side the coeﬃcient is given by 24 (log det h) + 23 (log det h) J − J(J2 + 2|P|2 ). In order to simplify the sum, we observe that 1 1 1 (log det h) = − (B, P) − tr(P3 ) and (log det h) = − |P|2 . 4 2 2 The latter formula has been derived in the proof of Theorem 6.10.1. Similarly, the ﬁrst formula can be proved on the basis of the formulas established in the proof of Theorem 6.10.1. We omit the details. Hence 24 (log det h) + 23 (log det h) J − J(J2 + 2|P|2 ) = −4(B, P) − 8 tr(P3 ) − 6J|P|2 − J3 . But using 1 3 1 1 1 J − 3J|P|2 + 2 tr(P3 ) − (B, P) v6 = − tr(∧3 P) − (B, P) = − 8 48 48 48

6.11. Factorization identities for residue families. Recursive relations

305

(Theorem 6.9.2), it follows that the sum can be written in the form 26 3v6 + 3J3 − 18|P|2 J. Now it only remains to verify that (10λP4∗ (λ)(v2 ) − 26 λ(λ−1)P2∗ (λ)(v4 ))top = 3J3 − 18J|P|2 . But it is easy to see that the left-hand side equals −5(J2 + 2|P|2 )J + 8J(J2 − |P|2 ). We omit the detailed discussion of the quadratic term and indicate only, where the contribution 12((B, P)+2 tr(P3 )) appears on both sides. By (6.10.2), the coeﬃcient of λ2 on the left-hand side contains the term −24 3(log det h) , i.e., 12((B, P) + 2 tr(P3 )). On the other hand, the coeﬃcient of λ2 on the right-hand side contains the term −26 9v6 , i.e., in particular, the term 12((B, P) + 2 tr(P3 )). The discussion of the remaining terms can be done on the basis of the results of Section 6.10. The proof is complete. Lemma 6.11.11 shows that the Q-polynomial is given by the formula ∗ 5 ∗ 6 Qres 6 (λ) = −2λP4 (λ)(v2 ) + 2 λ(λ−1)P2 (λ)(v4 ) − 2 3λ(λ−1)(λ−2)v6 ,

and that (6.11.42) is equivalent to Q6 + 2P2 Q4 + 3P22 Q2 − 2P4 Q2 top − 26 3!v6 . = −4λP4∗ (λ)(v2 ) + 26 λ(λ−1)P2∗ (λ)(v4 )

(6.11.46)

Thus we obtain the formula Q6 + 2P2 Q4 + 3P22 Q2 − 2P4 Q2 = −4(J2 + 2|P|2 )(v2 ) − 64Jv4 − 26 3!v6 = −6J3 + 12|P|2 J − 26 3!v6 .

(6.11.47)

Now using Q4 + P2 Q2 = J2 − 2|P|2 , we can write (6.11.47) in the form res 6 Q6 = [P2 (Qres 4 (2)) + 2P4 (Q2 (1))] − 6 [Q4 + P2 (Q2 )] Q2 − 2 3!v6 .

(6.11.48)

This is an analog of (6.11.44). The structure of (6.11.48) substantially diﬀers from that of the holographic formula Q6 = −26 3!v6 − 32P2∗ (0)(v4 ) − 2P4∗ (0)(v2 )

(6.11.49)

306

Chapter 6. Conformally Covariant Families

(see (6.10.4)) although both formulas express a relation between the quantities Q6 and −26 6v6 . The main diﬀerence concerns the way in which both formulas explain the divergence terms. Whereas in (6.11.49) these contributions are formu∗ lated in terms of the operators P2j (0) (which describe the asymptotics of harmonic functions of the Poincar´e-Einstein metric) acting on holographic coeﬃcients, in (6.11.48) these are covered by lower order GJMS-operators acting on lower order Q-curvature. In particular, (6.11.48) is recursive. Remark 6.11.6. Comparing coeﬃcients of powers of λ in (6.11.36) actually implies a set of n2 identities. The respective coeﬃcients of vn (on the left-hand side) and Qn (on the right-hand side) are n

2 n

n λ− +k ! −2 2 2

n

(6.11.50)

k=1

and

n

(−1)

n 2 −1

2

n 1 λ− +k . ( n2 −1)! 2

(6.11.51)

k=1

The polynomial Pn∗ (λ)(1) on the left-hand side plays a special role. It is conjectured to be a linear combination of the other terms so that vn contributes again with the coeﬃcient n 2 n

n n −1 n−1 λ− +k ! (−1) 2 2 2 2 k=1

(see (6.11.56)); the cases n = 4 and n = 6 are covered by Lemma 6.11.10 and Lemma 6.11.11. On the left-hand side of (6.11.36) this yields the additional contribution n 2

n n−1 n λ− +k vn . ! 2 2 2 k=1

Together with (6.11.50) we ﬁnd n

−2

n−1

2 n

n λ− +k vn ! 2 2

k=1

on the left-hand side of (6.11.36). Using (6.11.51), it follows that all identities relate n n

n ! −1 !vn to Qn . (−1) 2 2n−1 2 2 These identities contain additional holographic terms and lower order GJMSoperators and Q-curvatures.

6.11. Factorization identities for residue families. Recursive relations

307

Lemma 6.11.10 and Lemma 6.11.11 state that for n = 4 and n = 6 the respective leading term Pn∗ (λ)(1) in Qres n (λ) can be written as a linear combination of the other terms which enter into Qres n (λ). The following result and the accompanying conjecture provide a uniform formulation of these relations for general n. Theorem 6.11.12. For a Riemannian metric h on a manifold of even dimension n, the function n

2

n Vn (h; λ) = λ (λ−1) · · · λ− +1 (n+2j)T2j∗ (h; λ)(vn−2j ) 2 j=0

def

(6.11.52)

is a polynomial with vanishing constant and linear terms. Proof. The families T2j (λ), j = 1, . . . , n2 − 1 are regular at λ = 0. Hence the relation n

n n !(−1) 2 λ (λ−1) · · · λ− +1 Tn (λ) (6.11.53) Pn (λ) = 2n 2 2 (see (6.6.3)) implies that the constant term of Vn (λ) is a multiple of Pn∗ (0)(1) = Pn∗ (1) = Pn (1) = 0. In order to determine the coeﬃcient of λ, we apply (6.11.53). We ﬁnd (−1)

n 2 −1

n

2 −1

n −1 n −1 ! ! (n+2j)T2j∗ (0)(vn−2j ) + 2n 2n (−1) 2 P˙ n∗ (0)(1), 2 2 j=0

n

n

i.e., the product of (−1) 2 2−(n−1)

n −1 2

and

n

2 −1

n

n ∗ n−1 ˙ −1 ! 2 nPn (0)(1) − (n+2j)T2j∗ (0)(vn−2j ). 2 2 j=0

(6.11.54)

In order to prove that (6.11.54) vanishes, we recall that by Theorem 6.6.4, n

2 −1 n

∗ n n ˙ ˙ 2jT2j∗ (0)(vn−2j ). ! −1 ! nPn (0)(1) = nPn (0)(1) + 2 2 2 j=0 n But P˙n (0)(1) = (−1) 2 Qn and Theorem 6.6.6 imply n

nP˙ n∗ (0)(1)

=2

n−1

2 −1 n n

(n−2j)T2j∗ (0)(vn−2j ) ! −1 ! 2 2 j=0 n

2 −1 n n

! −1 ! 2jT2j∗ (0)(vn−2j ) +2 2 2 j=0

n

n

2 −1 n

n−1 n ! −1 ! =2 (n+2j)T2j∗ (0)(vn−2j ). 2 2 j=0

The latter identity means that (6.11.54) vanishes.

308

Chapter 6. Conformally Covariant Families

Conjecture 6.11.2. For any Riemannian manifold (M, h) of dimension n, the polynomial Vn (h; λ) vanishes identically. Of course, Conjecture 6.11.2 is equivalent to the vanishing of the rational function n 2 (n+2j)T2j∗ (h; λ)(vn−2j ). (6.11.55) j=0

Lemma 6.11.10 and Lemma 6.11.11 are the special cases n = 4 and n = 6 of Conjecture 6.11.2. In fact, Vn (λ) ≡ 0 is equivalent to n

Pn∗ (λ)(1) = (−1) 2 −1

n 2

−1 !2n−2

n 2 −1

×

(−1)j (n+2j)

j=0

λ(λ−1) . . . (λ− n2 +j +1) ∗ P2j (λ)(vn−2j ). 22j j!

(6.11.56)

It suﬃces to specialize that formula to n = 4 and n = 6. Conjecture 6.11.2 yields an alternative formula for the Q-polynomial. Theorem 6.11.13. Vn ≡ 0 is equivalent to n

nQres n (λ)

2 n

n ! λ(λ−1) . . . λ− +1 =2 2jT2j∗ (λ)(vn−2j ). 2 2 j=0

n

(6.11.57)

Proof. It suﬃces to notice that n

2 n

n ! λ(λ−1) . . . λ− +1 2 T2j∗ (λ)(vn−2j ) 2 2 j=0

n

n

= (−1) 2 Dnres (λ)(1) = −Qres n (λ). Example 6.11.5. For n = 4, (6.11.57) states that 5 ∗ ∗ 4Qres 4 (λ) = 2 λ(λ−1) [2T2 (λ)(v2 ) + 4T4 (λ)(v0 )] .

Using T2 (λ) = −

1 P2 (λ) 4(λ−1)

and

T4 (λ) =

1 P4 (λ), 32λ(λ−1)

this formula is equivalent to ∗ ∗ Qres 4 (λ) = −4λP2 (λ)(v2 ) + P4 (λ)(v0 ).

6.11. Factorization identities for residue families. Recursive relations

309

The latter identity can be checked directly using the explicit formulas for the terms involved. For the right-hand side we ﬁnd 2λ(Δ−λJ)J + (Δ−λJ)(Δ−(λ+2)J)(1) + 2λ(λ−1)|P|2 − 2(λ−1)ΔJ = λ2 (2|P|2 − J2 ) + λ(−ΔJ − 2|P|2 + 2J2 ) = −λ2 (Q4 + P2 Q2 ) + λQ4 . By Theorem 6.11.9, this is Qres 4 (λ). Next, we discuss a consequence of (6.11.57) for global conformally compact Einstein metrics. The setting is the following. Let X n+1 be a compact Riemannian manifold with boundary ∂X = M . Assume that gE is a global conformally compact Einstein metric on X \ ∂X with conformal inﬁnity [h]. A choice of a representing metric h on M gives rise to a deﬁning function r such that |dr|2r2 gE = 1 near ∂X. Near the boundary gE can be written in the normal form r−2 (dr2 +hr ), and for even n we have the asymptotics vol(gE ) = c0 ε−n + c2 ε−(n−2) + · · · + cn−2 ε−2 − L log ε + V + o(1) (6.11.58) r≥ε

for ε → 0. Here V = V (gE , h) is the asymptotic or renormalized volume of gE with respect to h. Lemma 6.11.12. V¨n (0) = 0 (i.e., Conjecture 6.11.2 for the quadratic coeﬃcient ) implies n 2

−1

(−1) c n 2

1 n

n

2

2j T˙2j∗ (0)(vn−2j ) vol = −

M j=1

1 2

¨ res (0) vol −α Q n

M

Qn vol, M

−1 n/2−1 where c n2 = 2n n2 ! n2 −1 ! and α = k=1 k1 . Proof. By Theorem 6.11.13, the condition V¨n (0) = 0 is equivalent to n

2

n n

1 ¨ res n 2j −αT2j∗ (0) + T˙2j∗ (0) (vn−2j ); nQn (0) = 2n ! −1 !(−1) 2 −1 2 2 2 j=0

recall that Tn (λ)(1) is holomorphic at λ = 0. But ∗ T2j (0)(vn−2j ) vol = vn−2j T2j (0)(1) vol = 0 M

M

for j = 1, . . . , n2 − 1. Moreover, (6.11.53) implies n n

! −1 !Tn∗ (0)(1) = P˙n∗ (0)(1). −2n 2 2

310

Chapter 6. Conformally Covariant Families

Hence n

n n n −1 2 2 ! −1 !(−1) Tn∗ (0)(1) vol 2 2 M n n P˙ ∗ (0)(1) vol = (−1) 2 = (−1) 2

P˙n (0)(1) vol =

n

M

M

Qn vol M

and thus we obtain 1 ¨ res (0) vol Q n 2 M n n 2 n

n n n ! −1 !(−1) 2 Qn vol −2 2j T˙2j∗ (0)(vn−2j ) vol, = −αn 2 2 M M j=0 i.e., n 2

−1

(−1) c n 2

1 n

n

2

2j T˙2j∗ (0)(vn−2j ) vol = −α

M j=0

Qn vol − M

1 2

¨ res Q n (λ) vol . M

The proof is complete. ˙ res Combined with Qres n (0) = 0 and Qn (0) = Qn , we get Qres (λ) vol = λ Qn vol n M M ⎛ ⎞ n 2 n 1 Qn vol +(−1) 2 c−1 2j T˙2j∗ (0)(vn−2j ) vol⎠ + · · · . − λ2 ⎝α n 2 n M j=0

(6.11.59)

Now we need Theorem 6.11.14 ([70]). Let n ≥ 2 be even. For a global conformally compact Einstein metric gE and a representing metric h in its conformal inﬁnity, n 2 1 ˙ S(n)(1) vol + V =− 2j T˙2j∗ (0)(vn−2j ) vol . n M j=1 M Here V denotes the renormalized volume as in (6.11.58), S is the scattering operator and the local terms on the right-hand side are to be understood with respect to the metric h. ˙ The quantity S(n)(1) is well deﬁned since S(λ)(1) is holomorphic near λ = n. ˙ V and S(n)(1) both depend on the global metric gE . For odd n, Feﬀerman and Graham ([97]) proved that ˙ S(n)(1) vol . V = M

Theorem 6.11.14 is an analog of that relation for even n.

6.11. Factorization identities for residue families. Recursive relations

311

Example 6.11.6. For n = 2, we have T2 (λ)(1) = 14 J and Theorem 6.11.14 states that ˙ S(2)(1) vol = 0. V + M

In dimension n = 4, Theorem 6.11.14 states that 2 1 ˙ S(4)(1) vol = 2jT2j∗ (0)(v4−2j ) vol . V + 4 M j=1 M By Example 6.11.5, the right-hand side coincides with res Q4 (λ) 11 1 vol . 1 32 M λ(λ−1) λ=0 But Theorem 6.11.9 implies that the latter integral equals 1 1 P2 Q2 vol = − J2 vol . 32 M 32 M

Hence

1 ˙ S(4)(1) vol = − 32 M

V +

J2 vol . M

We use Theorem 6.11.14 to rewrite the expansion (6.11.59) in the form n −1 2 2 ˙ S(n)(1) vol +V + · · · . Qn vol −λ α Qn vol −λ (−1) 2 c n λ M

2

M n 2

(−1) c

But n 2

M

S(n)(1) vol

Qn vol = M

M

(Theorem 3.2.2) yields the equivalent formula n S(n)(1) vol λ(−1) 2 c−1 n 2 M n n −1 2 ˙ 2 S(n)(1) vol − λ2 (−1) 2 c−1 − λ (−1) c n α S(n)(1) vol + n V + ··· . 2

M

2

M

Since for the polynomial n

α(λ) = (−1) 2

(λ−1) . . . (λ− n2 +1) ( n2 −1)!

we have α(0) = −1 and α(0) ˙ = α, we can write the expansion in the more compressed form n n −1 2 S(n−λ)(1) vol −λ2 (−1) 2 c−1 −λα(λ)(−1) c n n V + ··· . 2

Thus we have proved

M

2

312

Chapter 6. Conformally Covariant Families

Theorem 6.11.15. In the situation of Theorem 6.11.14, the relation V¨n (0) = 0 implies n

c n2 (−1) 2

n

2 Qres n (λ) vol +(−1)

M

λ(λ−1) . . . (λ− n2 +1) ( n2 −1)!

S(n−λ)(1) vol M

= −λ2 V + O(λ3 ). Theorem 6.11.15 relates the quadratic coeﬃcient in the total Q-polynomial res Q n (λ) vol to the scattering operator and the asymptotic volume. That relaM tion, in particular, allows us to determine its inﬁnitesimal conformal variation. Theorem 6.11.16. In the situation of Theorem 6.11.15, •

n 1 res ¨ Qn (0) vol [ϕ] = ϕ 2Qn − (−1) 2 c−1 n vn vol . 2 2 M M Here the bullet denotes the inﬁnitesimal conformal variation F • (h)[ϕ] = d/dt|t=0 (F (e2tϕ h)) of the functional F . Proof. Lemma 6.11.12 and Theorem 6.11.14 imply 1 n n res ˙ ¨ 2 2 n n V + S(n)(1) vol = − (−1) c 2 Qn (0) vol −α(−1) c 2 Qn vol . 2 M M M Now we have

• Qn vol [ϕ] = 0,

M

and

M

V • [ϕ] =

ϕvn vol M

• n ˙ 2 S(n)(1) vol [ϕ] = −2(−1) c n2

ϕQn vol

M

(by Theorem 4.3 in [70]). The assertion follows by combining these results.

Note that, by Theorem 6.6.6, the conformal anomaly on the right-hand side is the divergence of a natural one-form. It follows that ¨ res (th; 0) vol(th) = 0. Q d/dt|t=0 n M

The latter observation actually is a consequence of the scaling property 2 −n res Qres Qn (h; λ), c ∈ R+ n (c h; λ) = c

(6.11.60)

of the Q-polynomial. In order to prove (6.11.60), it suﬃces to observe that T2j (c2 h; λ) = c−2j T2j (h; λ)

and v2j (c2 h) = c−2j v2j (h).

6.11. Factorization identities for residue families. Recursive relations

313

The proof of Theorem 6.11.16 uses global arguments (involving V and S) to prove a local result. Therefore, it is natural to ask for a local proof. In the following local proof, we do not use the relation V¨n (0) = 0. Local proof of Theorem 6.11.16. The transformation formula ˆ λ) = e(λ−n)ϕ ◦ Dres (h; λ) ◦ κ∗ ◦ (κ∗ (r)/r)λ , h ˆ = e2ϕ h Dnres (h; n (see Theorem 6.6.3) implies

ˆ λ)(1) = e(λ−n)ϕ Dres (h; λ) κ∗ (κ∗ (r)/r)λ . Dnres (h; n

In order to determine the quantity n 1 ¨ res (h; 0), −(−1) 2 Q 2 n we diﬀerentiate this identity twice at λ = 0 and ﬁnd the formula ¨ nres (h; 0)(1) + e−nϕ Dnres (h; 0) κ∗ log2 (κ∗ (r)/r) ϕ2 e−nϕ Dnres (h; 0)(1) + e−nϕ D + ϕe−nϕ D˙ res (h; 0)(1) + e−nϕ D˙ res (κ∗ log(κ∗ (r)/r)).

n

n

Now using Dnres (h; 0) = Pn (h)i∗ and i∗ log(κ∗ (r)/r) = −ϕ, the ﬁrst three items simplify to ¨ res (h; 0)(1) + e−nϕ Pn (h)(ϕ2 ). ϕ2 e−nϕ Pn (h) + e−nϕ D n The remaining two items give n −nϕ ˙ res Dn (h; 0)(κ∗ log(κ∗ (r)/r)). −(−1) 2 ϕe−nϕ Q˙ res n (h; 0) + e

It follows that 1 2

• ¨ res Q (h; 0) vol(h) [ϕ] n

M

equals the sum of

ϕQn (h) vol(h) M

(Theorem 6.6.1) and n 2 −(−1) D˙ nres (h; 0) ((κ∗ log(κ∗ (r)/r))• [ϕ]) vol(h).

(6.11.61)

M

Now we observe that in the latter contribution the conformal variations of all terms which involve normal derivatives vanish. Thus using i∗ log(κ∗ (r)/r) = −ϕ, (6.11.61) equals n 2 P˙n∗ (h; 0)(v0 ϕ) vol(h) (−1) M

2 −1 n n

− (−1) 2 ! −1 ! T2j∗ (h; 0)(vn−2j ϕ) vol(h), 2 2 M j=0 n

n 2

n

314

Chapter 6. Conformally Covariant Families

i.e., (−1)

n 2

ϕP˙n (h; 0)(1) vol(h) M

2 −1 n n

− (−1) 2 ! −1 ! ϕ T2j (h; 0)(1)vn−2j vol(h) 2 2 M j=0

n n n n −1 ! ϕP˙n (h; 0)(1) vol(h) − (−1) 2 2n ϕvn vol(h). = (−1) 2 ! 2 2 M M n

n 2

n

Here we have used that T2j (0)(1) = 0 for j = 1, . . . , n2 − 1. Now P˙n (0)(1) = n (−1) 2 Qn implies that (6.11.61) is given by

n ϕ Qn − (−1) 2 c−1 n vn vol . 2

M

This completes the proof. Example 6.11.7. For n = 4, we have 1 ¨ res Q (0) = −2(Q4 + P2 Q2 ) = −16v4 − 2Jv2 2 4 (see Example 6.11.3) and Q4 − 16v4 = −ΔJ. Thus Theorem 6.11.16 states that • (16v4 + 2Jv2 ) vol [ϕ] = −2

− M

ϕΔJ vol .

M

By the conformal invariance of M

M

v4 vol, this is equivalent to

• J2 vol [ϕ] = −2

ϕΔJ vol .

M

The latter identity is a consequence of the transformation formula (2.5.7) for J. We continue with a derivation of a recursive formula for the GJMS-operator P6 for conformally ﬂat metrics. An alternative formula will be derived in Section 6.12. Theorem 6.11.17. On conformally ﬂat manifolds of dimension 6, the critical GJMS-operator P6 can be written in the form P6 i∗ = 6P4 P2 − 2P2 P4 − 3P23 i∗ + 4i∗ P¯2 , P¯4 = 2 (P4 P2 + P2 P4 ) i∗ − 3P23 i∗ + 4 i∗ P¯2 , P¯4 − [P2 , P4 ] i∗ .

(6.11.62)

6.11. Factorization identities for residue families. Recursive relations

315

Proof. We combine the presentation (6.11.24), i.e., P6 i∗ = αP4 D2res (1) + βP2 D4res (2) + γD4res (−9/2)P¯2 + δD2res (−11/2)P¯4 , where (α, β, γ, δ) = with

1 (18, −5, −4, 12) , 21

D2res (1) = −6i∗ P¯2 + 7P2 i∗ ,

D2res (−11/2) = 7i∗ P¯2 − 6P2 i∗

and (6.11.26), (6.11.27). We ﬁnd (omitting i∗ ) P6 = αP4 −6P¯2 + 7P2 + βP2 −6P4 + 27P22 − 28P2 P¯2 + 8P¯22 + γ −6P4 + 14P22 − 28P2 P¯2 + 21P¯22 P¯2 + δ 7P¯2 − 6P2 P¯4 . Hence 21P6 i∗ = 126P4 P2 + 30P2 P4 − 135P23 i∗ + 84 P22 − P4 i∗ P¯2 + 72P2 i∗ P¯22 − P¯4 + 84i∗ P¯2 P¯4 − P¯23 . Now we use the identity P22 − P4 i∗ = i∗ P¯22 − P¯4 (Lemma 6.11.6 for n = 6) twice and ﬁnd 21P6 i∗ = 126P4 P2 + 30P2 P4 − 135P23 i∗ + 84i∗ P¯22 − P¯4 P¯2 + 72P2 P22 − P4 i∗ + 84i∗ P¯2 P¯4 − P¯23 = 126P4 P2 − 42P2 P4 − 63P23 i∗ + 84i∗ P¯2 , P¯4 . This implies the ﬁrst assertion. The second assertion is a direct consequence.

Remark 6.11.7. Analogous arguments show that Theorem 6.11.17 holds true for conformally ﬂat metrics in any dimension n ≥ 3, i.e., (6.11.62) is universal. (6.11.62) implies that the operator i∗ P¯2 , P¯4 is tangential, i.e., can be written in the form i∗ P¯2 , P¯4 = Li∗ with a natural operator L. In these terms, we ﬁnd

316

Chapter 6. Conformally Covariant Families

Corollary 6.11.6. On conformally ﬂat manifolds of dimension n = 6, P6 = 2(P4 P2 + P2 P4 ) − 3P23 + 2(L + L∗ ).

(6.11.63)

Proof. We apply the fact that the operators P6 , P4 and P2 are selfadjoint. Hence (6.11.62) implies L∗ − [P2 , P4 ]∗ = L − [P2 , P4 ]. In particular, L − L∗ = 2[P2 , P4 ] and L − [P2 , P4 ] =

1 1 ((L − [P2 , P4 ]) + (L − [P2 , P4 ])∗ ) = (L + L∗ ). 2 2

This implies the assertion.

Remark 6.11.8. On the round sphere S n , the operator L is given by L = 3P2 . Similarly, for Einstein metrics, L = 3(4c)2 P2 , where 4c = τ /n(n−1) (see (6.16.13)). In these cases, Theorem 6.11.17 can be veriﬁed directly using (6.16.13). For general metrics, P6 diﬀers from 2(P4 P2 + P2 P4 ) − 3P23 by a second-order operator (Corollary 6.12.1). In the following result, the terms with at least three derivatives in Corollary 6.11.6 (and its generalization to general dimensions) are made explicit. Corollary 6.11.7. On conformally ﬂat manifolds of dimension n ≥ 3, 3 P6 = Δ3 − (n−2)JΔ2 + 16(P, Hess Δ) + 16(∇P, ∇ Hess) + (22−3n)(dJ, dΔ), 2 up to terms of order ≤ 2. Alternatively, Corollary 6.11.7 follows from (6.10.2). We omit the details. Corollary 6.11.7 ﬁts with Lemma 2.2 in [118] describing the structure of any conformally covariant cube of the Laplacian. The structure of the suppressed terms is more subtle and will be discussed in Section 6.12. For the convenience of the reader, we ﬁnally summarize the main formulas for Q6 . • Holographic formula. For n ≥ 6, Q6 = −26 3!v6 − 32P2∗

n

n

−3 (v4 ) − 2P4∗ −3 (v2 ) 2 2

(Corollary 6.10.1 for n = 6, Theorem 6.10.4 in general). • Universal recursive formula. For n ≥ 6, Q6 = (Theorem 6.11.6).

2 2 5 8 ¯ 2) P2 (Q4 ) + P4 (Q2 ) − P22 (Q2 ) + i∗ P¯22 (Q 3 3 3 3

6.11. Factorization identities for residue families. Recursive relations

317

• For n ≥ 6, Q6 = − 3P22 (Q2 ) + 2P2 (Q4 ) − 2P4 (Q2 ) − 6[Q4 + P2 (Q2 )]Q2 − 26 3!v6 ((6.11.48) for n = 6). These formulas are the analogs of the following results for Q4 . • Holographic formula. For n ≥ 4, Q4 = 16v4 + 2P2∗

n 2

−2 (v2 )

(Example 6.6.4 for n = 4, (6.10.15) in general). • Universal recursive formula. For n ≥ 4, ¯ 2) Q4 = P2 (Q2 ) − 2i∗ P¯2 (Q (Lemma 6.11.1). • For n ≥ 4,

Q4 = −P2 (Q2 ) − Q22 + 16v4

((6.11.44) for n = 4). The respective last formulas for Q4 and Q6 were derived above in the respective critical dimensions n = 4 and n = 6. For more details concerning the extension of this formula for Q6 to general dimensions we refer to Section 6.12, where it is used to derive a universal recursive formula for the GJMS-operator P6 . We ﬁnish the present section with some comments concerning the extension of the methods beyond the conformally ﬂat case. In that connection, it is crucial to establish the factorization identities for residue families in full generality. For a conformally ﬂat metric h, the set of 2N factorization identities for res (h; λ) consists of two subsets of N identities of diﬀerent nature. The N idenD2N res tities in the ﬁrst set (see (6.11.2)) factorize D2N (h; λ), for certain values of λ, into products of lower order residue families and GJMS-operators on M . In the proofs of Theorem 6.11.2 and Theorem 6.11.4, we have seen that the identities in this set follow from factorizations of the families P2N (λ) (Lemma 6.11.7). In fact, this is a general phenomenon. The following result actually suﬃces to prove the ﬁrst system of factorization identities for general metrics. Theorem 6.11.18. For (M n , h), k = 0, . . . , N and 2N ≤ n, n n

P2N h; −k = P2N −2k h; +k ◦ P2k (h). 2 2

(6.11.64)

For the proof of Theorem 6.11.18, we recall the identiﬁcation of GJMSoperators as residues of the scattering operator (Chapter 3). Now the assertion follows from the fact that the scattering operator relates the leading terms of both

318

Chapter 6. Conformally Covariant Families

ladders in the asymptotics of an eigenfunction for the Laplacian of the corresponding Poincar´e-Einstein metric. In the ﬂat case, this argument was used in the proof of Theorem 5.2.7. res (h; λ) (see (6.11.3)) are of The remaining N factorization identities for D2N diﬀerent nature. For N = 2 and N = 3, the diﬀerence is clearly indicated by the corresponding proofs of Theorem 6.11.2 and Theorem 6.11.4. For the proof of Conjecture 6.11.1, it would suﬃce to establish the factorization identities with the factor P¯2 . But even for conformally ﬂat metrics, these identities only follow by using the full power of the arguments of Theorem 6.11.1. The remaining identities in the second set are even more mysterious. However, the situation is diﬀerent in connection with the discussion of the structure of the critical Q-polynomial. The proof of Theorem 6.11.10 uses only the ﬁrst set of the factorization identities. Therefore, by the above remarks, it extends to the general case (see Remark 6.11.4). In particular, Corollary 6.11.5 holds true in the general case. This will be used in Section 6.12 and Section 6.13.

6.12 A recursive formula for P6 . Universality In Section 6.11, we have seen that the known structure of the Q-polynomial Qres 6 (λ) (see Theorem 6.11.9) implies the formula Q6 = − 2P2 (Q4 ) − 2P4 (Q2 ) + 3P22 (Q2 ) − 6[Q4 + P2 (Q2 )]Q2 − 26 3!v6 (6.12.1) (see (6.11.48)) for the critical Q6 . Now inﬁnitesimal conformal variation of (6.12.1) yields the following recursive formula for the critical GJMS-operator P6 . Theorem 6.12.1. The critical GJMS-operator P6 can be written in the form P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 24(J2 −|P|2 )Δu + 48J(P, Hess(u)) + 24(d|P|2 , du) + 48(P#dJ, du) + 26 3![v6• [u] + 6uv6 ]. (6.12.2) Here [·]0 denotes the non-constant part of the respective operator. Some comments are in order. By the discussion in Section 6.11, the relation (6.12.1) holds true for all metrics. The same is true for Theorem 6.12.1. The terms in the second and the third line of (6.12.2) deﬁne a second-order linear diﬀerential operator with vanishing constant term. This shows that the operator L + L∗ in (6.11.63) is of second order (see (6.12.5)). Proof. We combine the identity −P6 (h)(ϕ) = 6Q6 (h)ϕ + Q•6 (h)[ϕ] (see (3.1.7)) with (6.12.1). The calculation rests on the inﬁnitesimal conformal transformation

6.12. A recursive formula for P6 . Universality

laws

319

P40 (h)(ϕ) − 4Q4 (h)ϕ = (d/dt)|0 Q4 (e2tϕ h) = Q•4 (h)[ϕ], −P20 (h)(ϕ) − 2Q2 (h)ϕ = (d/dt)|0 Q2 (e2tϕ h) = Q•2 (h)[ϕ]

(see (3.1.7)), and the conformal covariance of P4 and P2 . Since e6tϕ P2 (e2tϕ h)(Q4 (e2tϕ h)) = e2tϕ P2 (h)(e2tϕ Q4 (e2tϕ h)), we ﬁnd

P2 (Q4 )• [ϕ] + 6ϕP2 (Q4 ) = −2[P20 , ϕ](Q4 ) + P2 P40 (ϕ).

Similarly, P4 (Q2 )• [ϕ] + 6ϕP4 (Q2 ) = −[P40 , ϕ](Q2 ) − P4 P20 (ϕ), P22 (Q2 )• [ϕ] + 6ϕP22 (Q2 ) = −2[P20 , ϕ](P2 (Q2 )) − P22 P20 (ϕ) and (Q4 Q2 )• [ϕ] + 6ϕQ4 Q2 = P40 (ϕ)Q2 − P20 (ϕ)Q4 , (P2 (Q2 )Q2 )• [ϕ] + 6ϕP2 (Q2 )Q2 = −P2 (P20 (ϕ))Q2 − P2 (Q2 )P20 (ϕ). These results yield P6 (ϕ) = −4[P20 , ϕ](Q4 ) + 2P2 P40 (ϕ) + 2[P40 , ϕ](Q2 ) + 2P4 P20 (ϕ) − 6[P20 , ϕ](P2 (Q2 )) + 3P22 P20 (ϕ) + 6P40 (ϕ)Q2 − 6P20 (ϕ)Q4 − 6P2 (P20 (ϕ))Q2 + 6P2 (Q2 )P20 (ϕ) + 26 3! (v6• [ϕ] + 6ϕv6 ) . Now reordering gives P6 (u) = 2P2 P4 + 2P4 P2 − 3P23 (u) − 2P2 (Q4 u) + 4P4 (Q2 u) − 6P22 (Q2 u) − [P20 , u] (4Q4 + 6P2 (Q2 )) + 2[P40 , u](Q2 ) − 6(P2 P20 − P40 )(u)Q2 − 6P20 (u)(Q4 + P2 (Q2 )) + 26 3! (v6• [u] + 6uv6 ) by using P2 = P20 − 2Q2 and P4 = P40 + Q4 . In order to evaluate the last sum, we apply the explicit formulas P2 = Δ−2J and P4 = Δ2 +δ(4J−4P#)d+(3J2 −2|P|2 −ΔJ). Since the constant term of P6 vanishes, it suﬃces to determine the constant term of the sum. A calculation yields the result P6 (u) = [2(P2 P4 + P4 P2 ) − 3P23 ]0 (u) − 24(d(J2 −|P|2 ), du) − 24(J2 −|P|2 )Δ(u) − 24δ(P#d)(uJ) + 24δ(P#d)(J)u − 24δ(P#d)(u)J + 26 3! (v6• [u] + 6uv6 ) . The assertion follows from here using δ(P) = −dJ.

320

Chapter 6. Conformally Covariant Families

Using Theorem 6.9.2 and the inﬁnitesimal conformal transformation laws of P and B, the formula (6.12.2) can be simpliﬁed. Corollary 6.12.1. The critical P6 can be written in the form P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 48δ(P2 #du) − 8δ(B#du).

(6.12.3)

Proof. By Theorem 6.9.2 and Newton’s identity tr(∧3 P) = 16 J3 − 12 J|P|2 + 13 tr(P3 ), v6 = −

1 3 1 1 1 J + J|P|2 − tr(P3 ) − (B, P). 48 16 24 48

Hence by the conformal transformation law of P, the term v6• [u] + 6uv6 equals 1 2 1 1 1 1 • J Δu − |P|2 Δu − J(P, Hess(u)) + tr(P2 Hess(u)) − [b [u] + 6ub] , 16 16 8 8 48 where b = (B, P). It follows that (6.12.2) simpliﬁes to P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u + 24(d|P|2 , du) + 48(P#dJ, du) + 48(P2 , Hess(u)) − 8 [b• [u] + 6ub] . (6.12.4) Now δ(P2 #du) = (δ(P2 ), du) − (P2 , Hess(u)) 1 = −(P#dJ, du) − Cjik Pjk ui − (d|P|2 , du) − (P2 , Hess(u)) 2 and

[b• [u] + 6ub] = −2(n−4)Cikj Pij uk − (B, Hess(u))

(see (2.5.9), (6.9.6) and (6.9.5)). Hence the last two lines of (6.12.4) yield 24(d|P|2 , du) + 48(P#dJ, du) + 48(P2 , Hess(u)) + 32Cikj Pij uk + 8(B, Hess(u)) = −48δ(P2 #du) − 16Cikj Pij uk + 8(B, Hess(u)) = −48δ(P2 #du) − 8δ(B#du) − 16Cikj Pij uk + 8(δ(B), du). Now it suﬃces to apply the identity δ(B)k = (n − 4)Cikj Pij . For n = 6, the latter relation is a consequence of the self-adjointness of the GJMS-operators. In fact, combining their self-adjointness with that of the second-order operators δ(P2 #du), δ(B#du), the above calculation shows that the ﬁrst-order diﬀerential operator 2Cikj Pij uk − 8(δ(B), du) is self-adjoint, i.e., vanishes. Note that (6.12.3) makes the self-adjointness of P6 obvious. The Bach tensor B vanishes on the conformal class of ﬂat metrics (although for n = 4 it is not conformally invariant). Hence for conformally ﬂat metrics,

6.12. A recursive formula for P6 . Universality

321

the contribution δ(B#du) in (6.12.3) vanishes, and by comparing Corollary 6.12.1 with Corollary 6.11.6, we ﬁnd the following explicit formula for the self-adjoint part of L: 1 (L + L∗ )0 u = −12δ(P2 #du). (6.12.5) 2 A closely related formula for a conformally covariant operator of the form Δ3 + LOT can be found in [30] (Theorem 2.8). Example 6.12.1. For the round sphere S 6 , we have P = 12 gc and C = 0. Therefore, (6.12.3) reads (6.12.6) P6 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u + 12Δu. This formula also follows from the product formulas for the GJMS-operators on the sphere. In fact, by (1.4.1), the right-hand side of (6.12.6) equals [4(Δ−6)(Δ−6)(Δ−4) − 3(Δ−6)3 ]0 + 12Δ = (Δ3 − 10Δ2 + 12Δ) + 12Δ = (Δ−6)(Δ−4)Δ = P6 . Next we discuss universality. The presentation (6.12.1) has the distinguished property of being universal. More precisely, Theorem 6.12.2. (6.12.1) holds true for all n ≥ 6. Proof. We recall the formula Q6 = Δ2 J − 8δ(P#dJ) − 4Δ(J2 −|P|2 ) n−6 −6JΔJ − 4(dJ, dJ) − 16J|P|2 + (n+6)J3 − 26 3!v6 + 4

(6.12.7)

(see (6.10.13)). Now n

n −2P2 (Q4 ) = −2 Δ − −1 J J2 − 2|P|2 − ΔJ , 2

2 n

n 2 −3P2 (Q2 ) = −3 Δ − −1 J Δ− − 1 J J, 2 2

n−4 n 2 J − 2|P|2 − ΔJ 2P4 (Q2 ) = 2 Δ2 + δ((n−2)J2 − 4P#)d + J 2 2 and

n J2 − 2|P|2 − ΔJ + Δ − −1 J J J 2 2 = −6J3 − 12|P|2 J.

−6(Q4 + P2 (Q2 ))Q2 = −6

n

A direct calculation shows that the sum of these four terms coincides with the right-hand side of (6.12.7), up to the v6 -term.

322

Chapter 6. Conformally Covariant Families

In all cases, inﬁnitesimal conformal variation of Q6 yields P60 (see (3.1.7)). Using (6.12.1) in the subcritical cases, we ﬁnd the following result. It states that the formula for P6 in Theorem 6.12.1 is universal, too. Theorem 6.12.3 (Universal recursive formula for P6 ). On manifolds of dimension n ≥ 6, the GJMS-operator P6 satisﬁes P60 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 24(J2 −|P|2 )Δu + 48J(P, Hess(u)) + 24(d|P|2 , du) + 48(P#dJ, du) + 26 3![v6• [u] + 6uv6 ]. (6.12.8) The central point is that the coeﬃcients in (6.12.8) do not depend on the dimension. Proof. The calculation rests on the inﬁnitesimal conformal transformation laws

n − 5 [P20 , ϕ](Q4 ) + P2 P40 (ϕ), P2 (Q4 )• [ϕ] + 6ϕP2 (Q4 ) =

2n P4 (Q2 )• [ϕ] + 6ϕP4 (Q2 ) = − 4 [P40 , ϕ](Q2 ) − P4 P20 (ϕ), 2 P22 (Q2 )• [ϕ] + 6ϕP22 (Q2 )

n n − 5 [P20 , ϕ](P2 (Q2 )) + − 3 P2 [P20 , ϕ](Q2 ) − P22 P20 (ϕ) = 2 2 and (P2 (Q2 )Q2 )• [ϕ] + 6ϕP2 (Q2 )Q2

n − 3 [P20 , ϕ](Q2 )Q2 − P2 (P20 (ϕ))Q2 − P2 (Q2 )P20 (ϕ). = 2 The calculations are analogous to those in the proof of Theorem 6.12.1. We omit the details. As above, we simplify (6.12.8) by using the known structure of v6 . The arguments in the proof of Corollary 6.12.1 yield the following result. Corollary 6.12.2 (Explicit recursive formula for P6 ). On manifolds of dimension n ≥ 6, the GJMS-operator P6 is given by P60 u = [2(P2 P4 + P4 P2 ) − 3P23 ]0 u − 48δ(P2 #du) − and P6 1 = − with Q6 as in (6.12.1).

n 2

−3 Q6

16 δ(B#du) n−4

(6.12.9)

6.12. A recursive formula for P6 . Universality

323

In dimension n = 4, the right-hand side of (6.12.9) is well deﬁned only for Bach-ﬂat metrics. This reﬂects the non-existence ([118]) of a conformally covariant operator of the form Δ3 + LOT in dimension 4 and, more precisely, shows the role of the Bach tensor as an obstruction. In [118], the non-existence is derived from the fact that the weight of the second-order operator (ΔP, Hess) in any conformally covariant cube of the Laplacian is given by 8(n−2) n−4 (see the calculations on p. 574). It is easy to see that the weight of the corresponding contribution in (6.12.9) is given by the sum 8(n−2) 16 = . 8+ n−4 n−4 The method of curved Casimir operators ([56]) yields an alternative construction of a conformally covariant cube of the Laplacian in all dimensions n ≥ 5 (n = 6). In dimension n = 4 it works only for conformally ﬂat metrics. However, no explicit formulas are given in [56]. Using Theorem 6.10.3 and analytic continuation in dimension, (6.12.9) suggests the following result. Lemma 6.12.1. On four-manifolds, the second-order operator R = −δ(B#d) + (B, P) is conformally covariant:

ˆ = R ◦ e−ϕ . e5ϕ ◦ R

Proof. In dimension 4, the Bach tensor is divergence free. Hence R(u) = (B, Hess(u)) + (B, P)u. The assertion follows by direct calculation using the conformal invariance of B (see (6.9.7)) and the conformal transformation laws for Hess and P. We omit the details. Note that even if the obstruction operator R vanishes in dimension n = 4, i.e., iﬀ the Bach tensor vanishes, deriving a conformally covariant cube of the Laplacian in n = 4 from (6.12.9) is more subtle. In fact, this requires that e2ϕ Bˆ − B = 0. n→4 n−4 lim

By the conformal transformation law (6.9.5) of B, this condition is equivalent to C = 0. This observation is parallel to the observation that the curved Casimir construction of [56] works in n = 4 only if C = 0. Corollary 6.12.2 is an analog of the universal formula P40 u = (P22 )0 u + 4JΔu + 4(dJ, du) + 16[v4• [u] + 4uv4 ] = (P22 )0 u + 4(dJ, du) + 4(P, Hess(u)) = (P22 )0 u − 4δ(P#du)

324

Chapter 6. Conformally Covariant Families

for P4 , which follows from Q4 = −P2 (Q2 ) − Q22 + 16v4 by inﬁnitesimal conformal variation (see also (4.3.2)). Example 6.12.2. On the round sphere S n , we have P = (6.12.9) states that

1 2 gc

and B = 0. Thus

P60 = [2(P2 P4 + P4 P2 ) − 3P23 ]0 + 12Δ. By the product formula (1.4.1), the latter identity follows from the polynomial identity [(x−a(a−1))(x−(a+1)(a−2))(x−(a+2)(a+3))] = [4(x−a(a−1))(x−a(a−1))(x−(a+1)(a−2)) − 3(x−a(a−1))3] + 12(x−a(a−1)). In fact, this argument proves the stronger relation P6 = 2(P2 P4 + P4 P2 ) − 3P23 + 12P2

(6.12.10)

which generalizes P4 = P22 + 2P2 . We recall that the recursive formula P6 = 2(P2 P4 + P4 P2 ) − 3P23 + 2(L + L∗ )

(6.12.11)

(see Corollary 6.11.6) is universal, too. This formula was derived in Section 6.11 under the assumption that h is conformally ﬂat. The conformal ﬂatness of the metric is needed here in order to apply the factorization identity of D6res (λ) which contains the factor P¯4 . We ﬁnish the present section with an observation which supports the conjecture that (6.12.11) holds true without additional assumptions on the metric. In turn, this supports the conjecture that the mentioned factorization identity is valid for general metrics. In the following we conﬁrm the coincidence of the contributions of the Bach tensor B in (6.12.9) and (6.12.11). In order to determine the contribution of B to the commutator i∗ [P¯2 , P¯4 ], we use the explicit formulas ¯ − n−1 ¯J P¯2 = Δ 2 and

n−3 ¯ ¯J − 4P#)d ¯ ¯ 2 + δ((n−1) P¯4 = Δ + 2

n+1 ¯2 2 ¯ ¯ ¯ J − 2|P| − Δ(J) 2

(with obvious notations). It follows that all contributions of B are contained in ¨ −4δ(P#d) −

n−3 (4) n−1 (4) ¨ 2, J + J − (n−3)|P| 2 2

(6.12.12)

6.13. Recursive formulas for Q8 and P8

325

¨ = i∗ (∂/∂r)2 |0 (P) ¯ and J(k) = i∗ (∂/∂r)k (¯J). Now (6.11.9) implies where P 2 1 B 1 ˙ ¨=−∂ 1 h = −4h(4) = −P2 + P r ∂r2 0 2r n−4 by Theorem 6.9.1. Hence the ﬁrst term in (6.12.12) contributes −

4 δ(B#d). n−4

Next, the generalization 6 (B, P) + · · · n−4 of (6.11.9) implies that the remaining terms in (6.12.12) are given by J(4) =

6 ¨ (B, P) − 2(n−3)(P, P). n−4 This sum contributes

6 2(n−3) n−6 − (B, P). (B, P) = −2 n−4 n−4 n−4

These results show that B contributes to 2(L + L∗ ) by −

16 n−6 16 δ(B#d) − (B, P). n−4 2 n−4

(6.12.13)

The latter sum coincides with the contributions of B in (6.12.9) (see Theorem 6.10.3). Finally, we note that an explicit, but less structured, formula for a conformally covariant cube of the Laplacian can be extracted from the work [176] concerning the construction of actions of the form u(Δkg u + · · ·) vol(g) S[g, u] = M

with the conformal invariance n

S[e2ϕ g, e(k− 2 )ϕ u] = S[g, u].

6.13 Recursive formulas for Q8 and P8 We consider a manifold of dimension n = 8. The following identity is the explicit version of the recursive formula (6.11.1) for the critical Q-curvature Q8 . 3 17 Q8 = P2 (Q6 ) + −4P22 + P4 (Q4 ) 5 5 8 28 9 16 22 ¯ 2 ). (6.13.1) + − P23 + P2 P4 + P4 P2 − P6 (Q2 ) − i∗ P¯23 (Q 5 5 5 5 5

326

Chapter 6. Conformally Covariant Families

As described in Section 6.11, its proof rests on the identity −Q8 = D˙ 8res (0)(1), res (λ) for N ≤ 4, and the unicertain factorization identities for the families D2N versality of the recursive formulas for Q6 and Q4 . Since the factorization identity with the second factor P¯2 is only known for conformally ﬂat metrics, (6.13.1) is also only known for such metrics (see the comments after Conjecture 6.11.1). For more details we refer to [95]. ¯ 2 ) is less unAn unfortunate aspect of (6.13.1) is that the contribution i∗ P¯23 (Q derstood. The following alternative recursive formula for Q8 generalizes (6.11.44) and (6.11.48), and holds in full generality. The result stresses the signiﬁcance of the Q-polynomials. Theorem 6.13.1. Assume that V8 (λ) ≡ 0 (Conjecture 6.11.2). Then the critical Q8 admits the presentation res Q8 = − P2 (Qres 6 (3)) + 3P4 (Q4 (2)) + 3P6 (Q2 ) 2 − 12 Q6 − P2 (Qres 4 (1)) − 2P4 (Q2 ) Q2 − 18 Q4 + P2 (Q2 ) + 4!3!27v8 . (6.13.2) Formula (6.13.2) yields an explicit presentation of Q8 when combined with the following results for the individual terms. By Theorem 6.11.8, the coeﬃcients res of the Q-polynomials Qres 4 (λ) and Q6 (λ) can be expressed in terms of GJMSoperators and Q-curvatures (see (6.13.7)). Corollary 6.12.2 gives a recursive formula for P6 in dimension n = 8. Lemma 6.14.1 provides a formula for v8 for (locally) conformally ﬂat metrics. For general metrics, the holographic coeﬃcient v8 is more complicated. By [117], it can be written in terms of P and the extended obstruction tensors Ω(1) and Ω(2) . Proof. We start from the identity top res res = − Q8 + P2 Qres 3!(Qres 8 ) 6 (3) + 3P4 Q4 (2) + 3P6 Q2 (1)

(6.13.3)

(see Corollary 6.11.5 and the comments at the end of Section 6.11); the vanishing res res of Qres 6 (0), Q4 (0) and Q2 (0) follows from Theorem 6.11.8. The left-hand side of (6.13.3) is given by the coeﬃcient of λ4 in D8res (λ)(1) = P8∗ (λ)(1) − 16λP6∗ (λ)(v2 ) + 26 3λ(λ−1)P4∗ (λ)(v4 ) − 29 3λ(λ−1)(λ−2)P2∗(λ)(v6 ) + 210 3λ(λ−1)(λ−2)(λ−3)v8. Now V8 (λ) ≡ 0 is equivalent to P8∗ (λ)(1) = 210 3λ(λ−1)(λ−2)(λ−3)v8 + 26 15λ(λ−1)(λ−2)P2∗(λ)(v6 ) − 24 9λ(λ−1)P4∗ (λ)(v4 ) + 14λP6∗ (λ)(v2 )

6.13. Recursive formulas for Q8 and P8

327

(see (6.11.56)). Hence D8res (λ)(1) = −2λP6∗ (λ)(v2 ) + 48λ(λ−1)P4∗ (λ)(v4 ) − 26 9λ(λ−1)(λ−2)P2∗(λ)(v6 ) + 210 3λ(λ−1)(λ−2)(λ−3)v8. It follows that the left-hand side of (6.13.3) is given by

6 −2P6∗ (λ)[3] (v2 ) + 48P4∗ (λ)[2] (v4 ) − 26 9P2∗ (λ)[1] (v6 ) + 210 3v8 ,

(6.13.4)

where the superscripts indicate the coeﬃcients of the respective powers of λ. Now we have P2∗ (λ)[2] = −J, P4∗ (λ)[2] = J2 + 2|P|2 and

P6∗ (λ)[3] = 16(log det h) + 8(log det h) − J(J2 + 2|P|2 )

by (6.10.2). A calculation shows that J2 + 2|P|2 = 12v6 − 12v2 v4 + 4v23 Hence

1 and (log det h) = − |P|2 . 2

−2P6∗ (λ)[3] (v2 ) = 48(−8v2 v6 + 12v22 v4 − 5v24 ),

and it follows that (6.13.4) is given by the sum of 6 · 48 −32v2 v6 + 24v22 v4 − 5v24 − 16v42

(6.13.5)

and 4!3!27v8 . Now we write (6.13.5) in terms of GJMS-operators and Q-curvatures. For that purpose, we use the relations Q2 = −2v2 , and

Q4 = −P2 Q2 − Q22 + 16v4

(6.13.6)

Q6 = − 3P22 (Q2 ) − 2P4 (Q2 ) + 2P2 (Q4 ) − 6 [Q4 + P2 (Q2 )] Q2 − 16 · 24v6 = [P2 (Qres 4 (1) + 2P4 (Q2 )] − 6 [Q4 +P2 (Q2 )] Q2 − 16 · 24v6 .

The version of the second identity in (6.13.6) in dimension n = 4 was proved in (6.11.44). A direct calculation shows that it is universal. Similarly, the version of the ﬁrst identity for Q6 in n = 6 was proved in (6.11.48). A direct calculation shows that it is universal. Thus (6.13.5) equals 6 1 5 4 2 2 2 2 6 · 48 16v6 Q2 + (Q4 + P2 Q2 + Q2 )Q2 − (Q4 + P2 Q2 + Q2 ) − Q2 16 16 16 = 12 · 24 · 16v6 Q2 + 18 −(Q4 + P2 Q2 + Q22 )2 + 6(Q4 + P2 Q2 + Q22 )Q22 − 5Q42 = 12 [−Q6 + (P2 Qres 4 (1) + 2P4 Q2 ) − 6(Q4 + P2 Q2 )Q2 ] Q2 + 18 −(Q4 + P2 Q2 + Q22 )(Q4 + P2 Q2 − 5Q22 ) − 5Q42 2 = 12 [−Q6 + (P2 Qres 4 (1) + 2P4 Q2 )] − 18(Q4 + P2 Q2 ) .

The proof is complete.

328

Chapter 6. Conformally Covariant Families

Combining the formula in Theorem 6.13.1 with the holographic formula for Q8 yields a formula for the divergence term 2T6∗ (0)(v2 ) + 4T4∗ (0)(v4 ) + 6T2∗ (0)(v6 ) in terms of GJMS-operators and Q-curvatures. It would be of interest to establish such a result in full generality. Inﬁnitesimal conformal variation of (6.13.2) yields a recursive formula for the critical GJMS-operator P8 . We ﬁnish with some comments on that formula. In more explicit terms, the ﬁrst bracket in (6.13.2) equals 3 P2 3Q6 + 12P2 Q4 + P2 Q2 − 8P4 Q2 2 + 3P4 (−3Q4 − 4P2 Q2 ) + 3P6 Q2 . (6.13.7) The remaining terms are non-linear in lower order Q-curvatures. It follows that the resulting recursive formula for the critical P8 expresses the operator as the sum of the main part (6.13.8) 3(P2 P6 + P6 P2 ) − 12(P22 P4 + P4 P22 ) − 8P2 P4 P2 + 9P42 + 18P24 and some low order operators. The following result shows that, on the round sphere S 8 , the low order correction terms in that formula are given by a multiple of the second-order operator P2 . In that case, the GJMS-operators commute, and the main part (6.13.8) coincides with the main part in Lemma 6.13.1. On S 8 , we have P8 = 6P2 P6 − 32P22 P4 + 9P42 + 18P24 + 3!4!P2 .

(6.13.9)

Lemma 6.13.1 follows from the product formulas for GJMS-operators on spheres. It is the analog of (6.12.6). Moreover, (6.13.9) is universal, i.e., holds true on all spheres S n , n ≥ 8. This generalizes (6.12.10). In view of the latter observation, it is natural to ask whether in the curved case the low order correction terms are only of second order, too. In fact, this is what happens for P6 : formula (6.12.9) is the curved analog of (6.12.10). We recall that Lemma 6.12.1 relates the non-existence of P6 in dimension 4 to the conformally covariant second-order operator R which is deﬁned by the Bach tensor. In a similar way, it would be natural to relate the non-existence of P8 in dimension 4 and 6 ([112]) explicitly to the ﬁrst two (extended) FeﬀermanGraham obstruction tensors (in the sense of [117]). In particular, we expect that the existence of P8 in dimension 6 is obstructed by the conformally covariant second-order operator −δ(O#d) + (O, P) deﬁned by the Feﬀerman-Graham obstruction tensor O in dimension 6. Similarly, the existence of P8 in dimension 4 should be obstructed by a fourth-order conformally covariant operator. Finally, it would be of interest to derive a presentation of P8 with the main part (6.13.8) from the factorization identities (generalizing Theorem 6.11.17).

6.14. Holographic formula for conformally ﬂat metrics

329

6.14 The holographic formula for conformally ﬂat metrics and tube formulas res The theory of the residue families D2N (λ) emphasizes the signiﬁcance of all holographic coeﬃcients v2j , 2j ≤ n. In the present section, we determine the holographic coeﬃcients v2j for conformally ﬂat metrics. The result suggests that we compare the situation with Weyl’s tube formula ([130]). Moreover, we discuss the consequences for the holographic formulas for Q-curvatures.

Lemma 6.14.1. The holographic coeﬃcients of a conformally ﬂat metric are given by n (6.14.1) v2j = (−2)−j tr ∧j (P) , j = 1, . . . , . 2 Proof. The result follows from the formula (see (6.9.8)) g = r−2 (dr2 +hr ) = with

2 ρ hρ = 1 − P(h) , 2

1 1 dρ2 + hρ 4ρ2 ρ

2 r2 hr = 1 − P(h) 2

(6.14.2)

for a conformally ﬂat metric h. For the ﬂat metric h0 = n the2 Poincar´e metric of −2 dx , the metric g = r (dr2 +h0 ) satisﬁes Ric(g0 ) + ng0 = 0 and i∗ (r2 g0 ) = 0 i 1 h0 . g0 is conformally ﬂat. For h = e2ϕ h0 ∈ [h0 ], the solution g = r−2 (dr2 + hr ) of Ric(g) + ng = 0 with h(0) = h is the pull-back κ∗ (g0 ) under a (local) diﬀeomorphism κ (ﬁxing the boundary r = 0). In particular, g is conformally ﬂat. It follows that 1 R(g) = −P(g) g = (g g). 2 By Lemma 6.14.2, the latter relation has the consequence ¨ = 1 hh ˙ −1 h˙ h 2

(6.14.3)

for small ρ; here derivatives are taken with respect to ρ (compare also with (6.9.11)). We claim that (6.14.3), in turn, yields ... h =0

(6.14.4)

for small ρ. Since h(2) and h(4) are known by Theorem 6.9.1, formula (6.14.2) follows. It remains to verify (6.14.4). We diﬀerentiate (6.14.3) and ﬁnd ... 1 ¨ −1 ˙ 1 ˙ )h˙ + 1 hh ˙ −1 ¨h. h = hh h + h˙ (h−1 2 2 2

330

Chapter 6. Conformally Covariant Families

˙ ) = −h−1 hh ˙ −1 in the latter formula, gives Using (h−1 ... 1 ¨ −1 ˙ 1 ˙ −1 ˙ −1 ˙ 1 ˙ −1 ¨ hh h + hh h. h = hh h − hh 2 2 2 But using (6.14.3) again, yields ... 1 ˙ −1 ˙ −1 ˙ 1 ˙ −1 ˙ −1 ˙ 1 ˙ −1 ˙ −1 ˙ hh h + hh hh h = 0. h = hh hh h − hh 4 2 4

This proves (6.14.4). Lemma 6.14.2. If the metric g =

1 2 4ρ2 dρ

+ ρ1 hρ satisﬁes R(g) = 12 (g g), then

¨ = 1 hh ˙ −1 h. ˙ h 2 Proof. The assumption implies RiN N j = R(ei , N, N, ej ) = −gij gN N = −

1 hij , N = ∂/∂ρ. 4ρ3

On the other hand, the curvature is given by Rijkl =

1 ((gik )jl + (gjl )ik − (gjk )il − (gil )jk ) + gpq Γpik Γqjl − Γpil Γqjk 2 p,q

with Γkij =

1 ((gjr )i + (gri )j − (gij )r ) g rk . 2 r

For the present metric g, we ﬁnd RiN N j = − But

1¨ hij + gpq ΓpiN ΓqN j − Γpij ΓqN N . 2ρ p,q

8 ΓkN N

=

1 NN 2 (gN N )N g

0

= − ρ1

for k = N , for k = N

1 NN = −2ρh˙ + 2h ΓN ij = − (gij )N g 2 and

8 ΓkiN =

0 1 2

rk ˙ −1 )ik ) = 12 (− ρ1 δik + (hh r (gri )N g

for k = N . for k = N

6.14. Holographic formula for conformally ﬂat metrics

331

These results (and some calculation) yield RiN N j

1¨ 1 1 1 −1 −1 ˙ ˙ = − hij + hpq − δip + (hh )ip − δjq + (hh )jq 2ρ 4ρ p,q ρ ρ +

1 1 1 ˙ −1 ˙ 1 1 ˙ hij − 3 hij = − ¨hij + (hh h)ij − 3 hij . 2ρ2 2ρ 2ρ 4ρ 4ρ

The combination of both formulas for RiN N j implies the assertion.

The observation that for conformally ﬂat h, (6.14.2) yields the exact Poincar´eEinstein metric, i.e., that the Feﬀerman-Graham series terminates at r4 , goes back to [220]. Remark 6.14.1. A version of the formula g = r−2 (dr2 +hr ),

hr =

2 r2 r4 1 − P(h) = 1 − r2 P(h) + P(h)2 2 4

for a conformally ﬂat metric h appears also in [195]. Let Γ be a convex-cocompact Kleinian group. It gives rise to the Kleinian manifold X = Γ\Hn+1 with boundary Γ\Ω(Γ). In Epstein’s appendix of [195] it is shown that the hyperbolic metric on an end (0, ∞) × Σn of X (Σ is a horospherically convex smooth hypersurface of X) is of the form dt2 + e−2t h− + h0 + e2t h+ , where h0 , h± are symmetric bilinear forms on Σ. Moreover, the pull-back gt of the hyperbolic metric to the parallel hypersurface Σt of (hyperbolic) distance t satisﬁes gt (X, Y ) = g0 ((cosh tI + sinh tS)(X), (cosh tI + sinh tS)(Y )) , when viewed as a metric on Σ. Here S is the shape operator on T (Σ). Hence we ﬁnd the relations 4h± (X, Y ) = g0 ((I ± S)(X), (I ± S)(Y )), 2h0 (X, Y ) = g0 (X, Y ) − g0 (SX, SY ). It follows that the linear operator T : T Σ → T Σ which is deﬁned by h0 (X, Y ) = h+ (T X, Y ) is given by

T = 2(I − S)(I + S)−1 .

In fact, (6.14.5) is equivalent to 2 g0 (X, Y ) − g0 (X, S 2 (Y )) = g0 T X, (I + S)2 (Y ) ,

(6.14.5)

332

Chapter 6. Conformally Covariant Families

i.e., g0 (T X, Y ) = 2 g0 (X, (I + S)−2 (Y )) − g0 (X, S 2 (I + S)−2 (Y )) = 2g0 (X, (I − S)(I + S)−1 (Y )). Similarly, for the linear map R : T Σ → T Σ so that h− (X, Y ) = h+ (RX, Y ),

(6.14.6)

we ﬁnd g0 ((I − S)(X), (I − S)(Y )) = g0 ((I + S)RX, (I + S)(Y )), i.e.,

g0 (RX, Y ) = g0 ((I − S)(X), (I − S)(I + S)−2 (Y )),

i.e., R = (I + S)−2 (I − S)2 =

1 2 T . 4

In other words (using r = e−t ), we can write the metric in the form r−2 dr2 + h+ + r2 h0 + r4 h− with

1 2 (h ) , 4 0 where is understood with respect to the metric h+ . Finally, note that h+ can be viewed as a metric on the boundary at inﬁnity. Similarly, h− and h0 can be viewed as symmetric bilinear forms on the boundary at inﬁnity. For a related discussion we refer to [161]. h0 = 2(I − S)(I + S)−1

and

h− =

Now combining Lemma 6.14.1 with the holographic formula (6.6.24) yields the following result. Theorem 6.14.1 (Q-curvature for conformally ﬂat metrics). For conformally ﬂat metrics on M n (n ≥ 4 even), Qn = 2

n 2 −1

n

2 −1

n ( n −j)! ∗ −1 ! P2j (0) tr ∧ 2 −j (P) . (−1)j 2−j 2 2 j! j=0

n

In particular, we can write Qn in the form n n

n n Qn = 2 2 −1 ! −1 ! tr(∧ 2 P) + δω 2 2

(6.14.7)

with a natural 1-form ω. The ﬁrst term on the right-hand side of (6.14.7) is related to the integrand in the Chern-Gauß-Bonnet formula (Euler form).

6.14. Holographic formula for conformally ﬂat metrics

333

We brieﬂy recall the formulation of that formula. It will be convenient to denote the even dimension on M by 2n. Let E2n =

1 (−1)n Pf 2n ∈ Ω2n (M ), (2π)n

where the Pfaﬃan form is deﬁned by 1 Pf 2n = n (σ)Ωσ1 σ2 ∧ · · · ∧ Ωσ2n−1 σ2n 2 n! σ∈S2n

in terms of the curvature forms Ωij = g(R(·, ·)ei , ej ) with respect to a local orthonormal basis {ei }. Then the Euler form E2n has the density 1 1 (−1)n 2n (2π)n 2 n!

(σ)(ρ)Rσ1 σ2 ρ1 ρ2 . . . Rσ2n−1 σ2n ρ2n−1 ρ2n

(6.14.8)

σ,ρ∈S2n

with respect to the corresponding Riemannian volume. In the following, we shall often identify E2n and Pf 2n with its respective density with respect to the Riemannian volume form. In particular, we have −2πE2 = R1212 and 8(2π)2 E4 =

(Rijkl Rijkl − 4Rijik Rljlk + Rijij Rklkl ) = |R|2 − 4| Ric |2 + τ 2 .

i,j,k,l

The Chern-Gauß-Bonnet theorem ([130], [107]) states that E2n = χ(M 2n ).

(6.14.9)

M 2n

Now for a conformally ﬂat metric g, n

Pf n (R) = Pf n (−P g) = (−1) 2

n

2

n

! tr(∧ 2 P)

(6.14.10)

using C(g) = 0 ([235], Proposition 8; the present formula diﬀers from the one in [235] due to a missing coeﬃcient 12 in the deﬁnition of the curvature form). (6.14.7) and (6.14.10) imply the relation n

n n Qn = 2 2 −1 (−1) 2 −1 ! Pf n +δω. (6.14.11) 2 The latter formula directly links Qn to the Pfaﬃan. It reﬁnes the result ([39]) that for a conformally ﬂat metric, the total integral of its Q-curvature is a multiple of the Euler characteristic. (6.14.11) is of interest also in connection with the following classiﬁcation result.

334

Chapter 6. Conformally Covariant Families

Theorem 6.14.2 ([81], [3]). Let Sn be a scalar Riemannian curvature invariant of weight −n on M n (n even), i.e., Sn (t2 g) = t−n Sn (g) for t > 0 and all metrics g. Assume that, for closed M , the integrals Sn vol M

are conformally invariant. Then Sn can be decomposed as the sum cn Pf n +C + δω

(6.14.12)

of a multiple of the Pfaﬃan, a local conformal invariant C and an exact divergence δω. The decomposition in Theorem 6.14.2 was proposed by Deser and Schwimmer in [81]. Alexakis’ monumental work [3] (combined with preparations in [5] and [6]) contains the ﬁrst mathematically rigorous proof of this important fact. For the formulation of a number of related conjectures we refer to [41]. In Section 6.15, we shall decompose conformal anomalies of certain functional determinants in the form (6.14.12). These examples illustrate Theorem 6.14.2. The critical Q-curvature Qn is a local scalar Riemannian curvature invariant of weight −n with a conformally invariant integral Qn vol . M

Thus Theorem 6.14.2 yields a decomposition n

Qn = (−1) 2 [(n−2)(n−4) · · · 2] Pf n +C + δω

(6.14.13)

with a generally unknown local conformal invariant C: enϕ C(e2ϕ g) = C(g) and a generally unknown ω ∈ Ω1 (M ). Concerning the nature of the decomposition (6.14.13), we recall from Section 6.6 that the holographic formula for Qn also shows that the quantity n

n

(−1) 2 −1 Δ 2 −1 (J) contributes to Qn . That contribution is the one which contains the maximal number n− 2 of covariant derivatives ∇ acting on the curvature tensor R. On the other hand, Pf n is a contribution which can be written without covariant derivatives acting on the curvature tensor R. In that sense both terms are extreme cases. Note that the coeﬃcient of Pf n in (6.14.11) and (6.14.13) can be veriﬁed by the following argument. For M = S n (n even) with the round metric gc , we have Qn = (n − 1)! (see Theorem 3.2.3). Hence n 2 (n−1)! Qn vol = vol(S n )(n−1)! = (2π) 2 (n−1)(n−3) · · · 1 Sn n

= (2π) 2 2 [(n−2)(n−4) · · · 2] .

6.14. Holographic formula for conformally ﬂat metrics

335

On the other hand, by Chern-Gauß-Bonnet, n n Pf n = (−2π) 2 χ(S n ) = 2(−2π) 2 . Sn

Thus

n

Qn (S n , gc ) = (−1) 2 [(n−2)(n−4) · · · 2] Pf(S n , gc ). There is a corresponding decomposition vn =

1 Pf n +C + δω (n(n−2) · · · 2)

(6.14.14)

of the curvature invariant vn . In the decompositions of 2Qn and vn , the respective n coeﬃcients of Pf n diﬀer by the factor (−1) 2 2n ( n2 )!( n2 − 1)!. This is the coeﬃcient of vn in (6.6.24). In dimension n = 4, the quantities Q4 and v4 are given by explicit formulas and the decompositions (6.14.13) and (6.14.14) can be made explicit. First of all, we have Q4 = 2J2 − 2|P|2 − ΔJ, −Δ = δd (see (4.1.1)) and v4 =

1 2 (J − |P|2 ) 8

(Theorem 6.9.2), i.e., 2Q4 = 32v4 − 2ΔJ. Now the quantity J − |P| satisﬁes the identity 2

2

1 (J2 − |P|2 ) + |C|2 = Pf 4 . 8

(6.14.15)

Hence

1 1 1 Q4 = 2 Pf 4 − |C|2 − ΔJ, v4 = Pf 4 − |C|2 . (6.14.16) 4 8 64 In order to prove (6.14.15), we rewrite both sides as follows. (6.14.8) implies

that

1 2 |R| − 4| Ric |2 + τ 2 . 8 On the other hand, the identities Pf 4 =

1 1 J2 − |P|2 = − | Ric |2 + τ 2 4 12 and

(6.14.17)

1 |C|2 = |R|2 − 2| Ric |2 + τ 2 (6.14.18) 3 (see the proof of Theorem 6.14.3) yield the result 18 (|R|2 − 4| Ric |2 + τ 2 ) for the left-hand side of (6.14.15).

336

Chapter 6. Conformally Covariant Families

The formula

1 16v4 = 2 Pf 4 − |C|2 4

(see (6.14.16)) plays a key role in testing the AdS/CFT correspondence (see Section 6.15). Conjecture 6.9.1 formulates an analog of the holographic formula (6.6.24) for all subcritical Q-curvatures. In the conformally ﬂat case, combining it with Lemma 6.14.1 gives (6.14.19) Q2N = 2N −1 N !(N −1)! tr(∧N P) + δω, up to terms of the form n2 − N (·). Now C = 0 implies that tr(∧N P) and tr(∧N Ω) are proportional. Thus the latter formula relates Q2N to a multiple of tr(∧N Ω). In other words, in the conformally ﬂat category, the constant term of P2N naturally splits oﬀ the quantity tr(∧N Ω). Similar observations have been discussed in the physical literature ([176]). The quantities tr(∧N Ω) also appear in the coeﬃcients in Weyl’s formula for the volume of tubes ([130]). We describe that relation. Let M n+1 ⊂ Rn+1 (n even) be an open set with compact closure and smooth boundary Σn . Let M (ε) = x ∈ Rn+1 | dist(x, M ) ≤ ε . Then the volume vol(M n+1 (ε)) − vol(M n+1 ) is given by the formula εk0 +

ε3 εn+1 k2 + · · · + kn 3 (n+1) · · · 3

(6.14.20)

(up to a linear combination of the powers ε2 , ε4 , . . . , εn ), where the coeﬃcients k2j are deﬁned by (−1)j n (6.14.21) tr(Λj Ω) vol, j = 0, . . . , . k2j = j! 2 Σn Here Ω : Λ2 (T Σ) → Λ2 (T Σ) denotes the Riemann curvature endomorphism of Σ. In particular, k2j only depends on the inner geometry of Σn . The coeﬃcients of the even powers of ε depend on the embedding. More precisely, these coeﬃcients depend linearly on the second fundamental form L; they vanish if L vanishes. For details see [130], Chapter 10. The special case ε3 3 3 2 2 vol(M (ε)) = vol(M ) + ε area (Σ ) + ε H vol + 4π 3 Σ2

6.14. Holographic formula for conformally ﬂat metrics

337

for a convex M 3 ⊂ R3 with boundary Σ2 goes back to Steiner (see [130] for more details). The numbers k0 , k2 , . . . , kn also appear as coeﬃcients in the formula for the volume of the ε-tube T (Σ, ε) = {x ∈ Rn+1 | dist(x, Σ) ≤ ε} of the closed hypersurface Σn ⊂ Rn+1 : ε3 εn+1 kn . vol(T (Σ, ε)) = 2 εk0 + k2 + · · · + 3 (n+1) · · · 3

(6.14.22)

For a closed surface Σ2 in R3 , that formula reads vol(T (Σ2 , ε)) = 2ε area (Σ2 ) +

4π 3 ε χ(Σ2 ) 3

by Gauß-Bonnet. For details see [130], Chapter 4. Note also that the coeﬃcients k2j , j = 1, . . . , n2 are naturally related to heatcoeﬃcients of Hodge-Laplacians ([86]). The relation between v4 in dimension n = 4 to heat-coeﬃcients will be the subject of Section 6.15. There are analogous tube formulas for higher codimension submanifolds of the ﬂat space Rn , but no general results for submanifolds in a curved background. However, (6.14.20) should be regarded as a relative of the following volume formula for the half-tube [0, ε] × M n with respect to the Riemannian volume of the conformal compactiﬁcation dr2+hr of the Einstein metric r−2 (dr2+hr ). Let h = h0 . Then vol({r < ε}; dr2 +hr ) ε3 εn+1 =ε v0 vol(h) + v2 vol(h) + · · · + vn vol(h) + · · · . 3 M n+1 M M

(6.14.23)

The structure of (6.14.23) resembles the structure of the half-tube formula (6.14.20). We refer to Section 6.16 for more details in the case of an Einstein metric h. In the conformally ﬂat case, the holographic coeﬃcients are given by Lemma 6.14.1, i.e., are proportional to the quantities tr(∧j Ω). For general metrics, it seems reasonable to ask for the existence of decompositions of the holographic coeﬃcients v2j , 2j ≤ n which naturally split oﬀ a multiple of tr(∧j Ω). The case v0 is trivial. The following result establishes such formulas for v2 and v4 (for general n). Lemma 6.14.3. v2 = for n ≥ 2 and v4 = for n ≥ 4.

1 tr(Ω) 2(n−1)

1 1 1 tr(∧2 Ω) − |C|2 8 (n−2)(n−3) 4

338

Chapter 6. Conformally Covariant Families

Proof. The observations v2 =

1 1 1 1 tr(h(2) ) = − J = − τ= tr(Ω) 2 2 4(n−1) 2(n−1)

prove the ﬁrst assertion. In order to prove the assertion for v4 , we recall the formula v4 =

1 (tr(P))2 − |P|2 8

(6.14.24)

for all n ≥ 4 (Theorem 6.9.1). The deﬁnition (2.5.10) yields 1 4−3n 2 2 |P|2 = + τ | Ric | . (n−2)2 4(n−1)2 Hence v4 =

1 n 2 2 τ + −| Ric | . 8(n−2)2 4(n−1)

(6.14.25)

On the other hand, for all n ≥ 4, 1 1 2 tr(∧2 Ω) = |R| − 4| Ric |2 + τ 2 2 8 (see [130], Lemma 4.2). For n = 4, that expression coincides with Pf 4 . Finally, we determine |C|2 . Using C = R + P g (see (2.5.13)) and the identities (R, P g) = −4(Ric, P),

|P g|2 = 4(n−2)|P|2 + 4J2 ,

we obtain |C|2 = |R|2 + 2(R, P g) + |P g|2 = |R|2 − 8(P, Ric) + 4(n−2)|P|2 + 4J2 4 2 | Ric |2 + τ 2. = |R|2 − n−2 (n−2)(n−1) Hence

1 n−3 n tr(∧2 (Ω)) − |C|2 = τ2 . −| Ric |2 + 4 n−2 4(n−1)

(6.14.25) and (6.14.26) imply the assertion.

(6.14.26)

On the other hand, it also seems to be reasonable to ﬁnd generalizations of Lemma 6.14.1 in the form of natural decompositions of v2j which split oﬀ a multiple of tr(∧j P). In the special cases j = 1, 2, 3, such formulas easily follow from the explicit formulas for v2j . We close the present section with a comment concerning the relation to Anselmi’s work [12]. Anselmi studied the problem of ﬁnding scalar Riemannian

6.15. v4 as a conformal index density

339

invariants of the form Pf n +δω (up to local conformal invariants) with the property that their transformation law under conformal changes e2ϕ h is governed by a linear diﬀerential operator. In dimension n = 4, this question led Riegert ([208]) to the discovery of a version of Q-curvature and the associated operator P4 (see the remarks in the introduction of Chapter 4). Anselmi called such quantities pondered Euler densities, and found that, in dimension n = 6, the quantity G6 = 26 3! Pf 6 +δω with δω =

102 24 48 δ(Ric #dτ ) + Δ(τ 2 ) − 12Δ(| Ric |2 ) − Δ2 τ 5 25 5

(6.14.27)

is a pondered Euler density for conformally ﬂat metrics. This result, in fact, is a consequence of the following observation. Lemma 6.14.4. For conformally ﬂat metrics, G6 = −48Q6 . Proof. Using P = 14 (Ric −Jh) and J = 1 48 × (6.14.27) coincides with

τ 10 ,

we ﬁnd | Ric |2 = 16|P|2 + 14J2 . Hence

1 17 Δ(J2 ) − Δ(| Ric |2 ) − Δ2 J 2 4 = 8δ(P#dJ) − 2δ(JdJ) + 5Δ(J2 ) − 4Δ(|P|2 ) − Δ2 J

2δ(Ric #dJ) +

= 8δ(P#dJ) + 4Δ(J2 ) − 4Δ(|P|2 ) − Δ2 J. Therefore, 1 G6 = 8 Pf 6 +8δ(P#dJ) + 4Δ(J2 − |P|2 ) − Δ2 J = −Q6 48 by (6.14.11) and Theorem 6.10.1.

Similarly, in dimension n = 4, Anselmi’s pondered Euler density G4 is given by 8 G4 = 24 2! Pf 4 − Δτ = 32 Pf 4 −16ΔJ. 3 Hence G4 = 16Q4 + 4|C|2 by (6.14.16).

6.15 v4 as a conformal index density In the present section, we describe a relation between the holographic anomaly v4 and certain conformal anomalies in spectral theory. This relation is considered as one of the earliest successful tests of the AdS/CFT-duality.

340

Chapter 6. Conformally Covariant Families

The setting is a closed oriented spin manifold (M, h) of dimension 4 with the spinor bundle S. Associated to (M, h) are the holographic coeﬃcients v0 = 1,

1 1 v2 = − tr(P) = − J, 2 2

v4 =

1 1 tr(∧2 P) = (J2 −|P|2 ) 4 8

(Theorem 6.9.2). We relate the conformal anomaly v4 to a combination of conformal anomalies of functional determinants of diﬀerential operators acting on certain types of ﬁelds on M . More precisely, we consider • The Yamabe operator −Y = −Δ + J = −Δ + 16 τ . / : Γ(S) → Γ(S). • The square of the Dirac operator ∇ • The form Laplacians −Δp on functions and 1-forms. The Yamabe operator and the Dirac operator are conformally covariant: ˆ ◦ e−ϕ = Y (h), e3ϕ ◦ Y (h)

5 ˆ ◦ e− 32 ϕ = ∇(h). / h) / e 2 ϕ ◦ ∇(

/ 2 and the Laplacians Δ0 and Δ1 are not conformally covariant. The square ∇ However, the operator δd : Ω1 (M 4 ) → Ω1 (M 4 ) 2 ˆ = δd by Lemma 4.2.1. In contrast to Y and ∇ / , is conformally covariant: e2ϕ ◦ δd the operator δd is not elliptic. Now if D = −Y is positive, we deﬁne its determinant by

det(D) = e−ζ where ζ(s, D) =

(0,D)

,

−s λ−s ), (s) > 2 j = tr(D

j

is the spectral zeta function of D. Then we have the Polyakov-formula − log det(−Y (h))• [ϕ] = −(d/ds)|0 log det(−Y (e2sϕ h)) ϕa4 (−Y (h)) vol(h), =2 def

(6.15.1)

M

where a4 ∈ C ∞ (M ) appears in the coeﬃcient of t0 in the heat asymptotics −tD −2+ j2 tr(ϕe )∼ t ϕaj (D) vol, t → 0 j≥0

M

(see Theorem 4.10 in [31]). Moreover, the zeta-value ζ(0, D) = a4 (D) vol M

6.15. v4 as a conformal index density

341

is a conformal invariant. a4 (D) is the conformal index density of D (see [45] and Theorem 4.8 in [31]). Now by general results of Gilkey [106] (see also [194]), A(Y ) = 180(4π)2 a4 (−Y ) = Δτ + |R|2 − | Ric |2 = 6ΔJ + 4(|P|2 − J2 ) + |C|2

(6.15.2)

for the Yamabe operator. Here we have used the identity |R|2 − | Ric |2 = |C|2 + 4(|P|2 − J2 ) (see (6.14.17) and (6.14.18)). It follows that the conformal invariance of the conformal index M a4 vol is equivalent to the conformal invariance of the integral v vol. M 4 / 2 , we have (by [106], [46], Section 5 and [48]) Next, for the square ∇ 7 5 / 2 ) = 180(4π)2 a4 (−∇ / 2 ) = −6Δτ − |R|2 − 4| Ric |2 + τ 2 A(∇ 2 2 7 = −36ΔJ − 44(|P|2 −J2 ) − |C|2 2

(6.15.3)

using (6.14.17) and (6.14.18). The latter formula shows the conformal invariance / 2 ) vol. of the conformal index M a4 (−∇ Finally, for the Laplacians −Δ0 and −Δ1 , we again use [106] and (6.14.17), (6.14.18) to ﬁnd that the respective coeﬃcients a4 are given by 5 A(Δ0 ) = 180(4π)2 a4 (−Δ0 ) = 6Δτ + |R|2 − | Ric |2 + τ 2 2 5 = 36ΔJ + 4(|P|2 −J2 ) + |C|2 + τ 2 2

(6.15.4)

and A(Δ1 ) = 180(4π)2 a4 (−Δ1 ) = −6Δτ − 11|R|2 + 86| Ric |2 − 20τ 2 = −36ΔJ + 256(|P|2 −J2 ) − 11|C|2 + 5τ 2 .

(6.15.5)

The integrals of A(−Δ0 ) and A(−Δ1 ) are not conformally invariant. However, we observe that the diﬀerence A(Δ1 ) − 2A(Δ0 ) = −108ΔJ + 248(|P|2 −J2 ) − 13|C|2

(6.15.6)

integrates to a multiple of the conformal invariant ζ(0, −Δ1 ) − 2ζ(0, −Δ0 ). Here we should emphasize that although the operator (δd)1 : Ω1 (M ) → Ω1 (M ) is conformally covariant, it is not true that the zeta-value ζ(0, (δd)1 ) = ζ(0, −Δ1 ) − ζ(0, −Δ0 )

342

Chapter 6. Conformally Covariant Families

is conformally invariant. However, the function def

ρ(s) = ζ(s, −Δ1 ) − 2ζ(s, −Δ0 ) is regular at s = 0 and the value is conformally invariant. The quantity det(−Δ0 )4 det(−Δ1 )2

(6.15.7)

is a special case of the Cheeger half-torsion (see [35]). Now the formulas (6.15.2), (6.15.3), (6.15.4) and (6.15.5) imply the miraculous identity / 2 ) + [A(Δ1 ) − 2A(Δ0 )] = 360(|P|2 −J2 ), 6A(Y ) − 2A(∇ and a consequence is the Polyakov-formula • det(−Y )6 det(−Δ1 ) 1 1 = − ϕv4 vol − log 2 π2 M / 2 )2 det(−Δ0 )2 det(−∇

(6.15.8)

(6.15.9)

(ignoring complications due to non-trivial kernels). The explicit formulas of the individual spectral anomalies on the left-hand side of (6.15.8) have been known in the physics literature since the middle of the 1970s (see [87] for a historical account). (6.15.2) can be found, for instance, in [136] and [137] (Chapter 15). We refer also to the useful review [234]. (6.15.9) can be interpreted as follows as a conﬁrmation of the AdS/CFTduality on the level of conformal anomalies. In [241], Witten proposed to formulate the duality as the identity ZCF T [h] = exp(−S[g])

(6.15.10)

of partition functions, where g is a critical point of the Einstein-Hilbert action S with [h] as conformal inﬁnity. We consider a multiplet of six (bosonic) scalar ﬁelds, two (fermionic) spinors and one (bosonic) vector ﬁeld. This is the ﬁeld content of the super Yang-Mills Lagrangian on M 4 ([82]). Henningson and Skenderis ([141]) observed that the resulting conformal anomaly, i.e., the linear combination on the left-hand side of (6.15.8), is given by the conformal anomaly v4 of the volume functional which is deﬁned by holographic renormalization. In fact, the conformal anomaly of the left-hand side of (6.15.10) is given by

−1/2 (d/ds)|0 DHS (e2sϕ h) , where DHS =

det(−Y )6 det(−Δ1 ) 2

/ )2 det(−Δ0 )2 det(−∇

.

6.16. The holographic formula for Einstein metrics

343

Note that the Dirac operator contributes to the denominator in view of the fermionic character of spinors. The contribution of det(−Δ0 ) is usually attributed to ghost ﬁelds. On the other hand, holographic renormalization of the right-hand side of (6.15.10) yields exp(−2nV (g; h)) (see 1.6.21) with the conformal anomaly exp(−2n ϕvn vol). M

It remains to choose appropriate units. Using the identity 8(J2 −|P|2 ) = 8 Pf 4 −|C|2 ,

Pf 4 =

1 tr(∧2 Ω) 2

(see Theorem 6.14.3), the formulas (6.15.2), (6.15.3) and (6.15.6) can be written in the form 3 A(Y ) = 6ΔJ + |C|2 − 4 Pf 4 , 2 / 2 ) = −36ΔJ − 9|C|2 + 44 Pf 4 , A(∇ A(Δ1 ) − 2A(Δ0 ) = −108ΔJ + 18|C|2 − 248 Pf 4 . In these terms, the results appear in [141], up to the contributions of ΔJ. The idea of the latter decompositions is to write the anomalies as sums of a multiple of Pf 4 , a local scalar invariant and a divergence term (which is possible by Theorem 6.14.2). It follows that 2

/ ) + [A(Δ1 ) − 2A(Δ0 )] = 45(|C|2 −8 Pf 4 ). 6A(Y ) − 2A(∇ The cancellation of the contributions of ΔJ in the latter sum was ignored in [141] since from the physical point these terms are considered to be of no importance. The above calculations involve only free ﬁelds. A more reﬁned version of the AdS/CFT-duality, however, would involve an interacting SU (N ) super Yang-Mills multiplet. In dimension n = 6 the relation between v6 and the conformal index densities of the free (2, 0) tensor multiplet was studied in [18].

6.16 The holographic formula for Einstein metrics Here we demonstrate that the holographic formula (6.6.20) implies Gover’s formula

τ (h) Qn (h) = (n−1)! n(n−1)

n2

(Theorem 3.2.3) for the Q-curvature of an Einstein metric h on Σn .

(6.16.1)

344

Chapter 6. Conformally Covariant Families

The ﬁrst step is Theorem 6.16.1. Let h be an Einstein metric on Σn (n even). Then 1 2 dr + (1−cr2 )2 h , 2 r

τ (h) (6.16.2) 4n(n−1) √ is the Poincar´e-Einstein metric on (0, ε) × Σn (ε < 1/ c if c > 0 and ε = ∞ if c ≤ 0) with conformal inﬁnity [h], i.e., g=

c=

Ric(g) + ng = 0

(6.16.3)

and the restriction of r2 g to r = 0 coincides with h. In particular, v2j = (−1)j 2−j σj (P(h)) = (−1)j 2−j tr(∧j (P(h))), i.e., v(r) =

j

r2 r2j v2j = det 1 − P(h) . 2

Here we use the same symbol for the linear map Ph : T (Σ) → T (Σ) and the symmetric bilinear form P(h) (Schouten tensor) inducing it. √ Note that if τ (h) > 0 the metric in Theorem 6.16.1 exists only for 0 < r < 1/ c. But the relation τλh = λ−1 τ (h) implies that the Poincar´e-Einstein metric gλh with the property (r2 gλh )|r=0 = λh ∈ [h] is given by

1 2 c dr + (1− r2 )2 λh 2 r λ for 0 < r < λ/c. In particular, for Σn = S n and λ ≥ c gλh extends to a metric on the ball Bn+1 . Proof. The construction of g is a special case of the warped product metric constructions in [24] (Section 9.J). We give an independent proof. h Einstein means Ric(h) =

τ (h) h. n

In order to calculate Ric(g), we write g using the coordinates r = es . Then g is the warped product metric 2 g = ds2 + e−s −ces h = ds2 + f 2 (s)h. Since g = f 2 (f −2 ds2 + h), where f only depends on s, it is enough to combine the transformation rule (2.5.4) with the explicit formula Ric f −2 ds2 + h (X, Y ) = Ric(h)(X, Y )

6.16. The holographic formula for Einstein metrics

345

for X, Y ∈ X (Σ); all other terms vanish. We ﬁnd ∂ ∂ , Ric(g) ∂s ∂s

2 ∂ log f ∂s ∂ ∂ , = −nf −2 (f f ) = −n = −ng , ∂s ∂s ∂ , X = 0, X ∈ X (Σ) Ric(g) ∂s

= −nf −2 Δf −2 ds2 (log f ) − (n−1)f −2|d log f |2 + (n−1)

and Ric(g)(X, Y ) = Ric(h)(X, Y ) − Δf −2 ds2 (log f )h(X, Y ) − (n−1)|d log f |2 h(X, Y ) τ (h) h(X, Y ) − f f h(X, Y ) − (n−1)(f )2 h(X, Y ) n τ (h) 2 − f f − (n−1)(f ) h(X, Y ). = n

=

For f (s) = e−s − ces , we ﬁnd for the coeﬃcient in the latter formula τ (h) −2s 2 2s − e +c e −2c − (n−1) e−2s +c2 e2s +2c n 2 τ (h) = −ne−2s − nc2 e2s + − 2c(n−2) = −n e−s −ces n using the deﬁnition of c. Hence Ric(g)(X, Y ) = −nf 2 h(X, Y ) = −ng(X, Y ). This proves (6.16.3). It follows that v(r) = (1 − cr ) = 2 n

n n j=0

j

(−1)j cj r2j ,

i.e., v2j

j τ (h) n j −2j n = (−1) c = (−1) 2 j j n(n−1) j τ (h) n = (−1)j 2−j = (−1)j 2−j σj (P(h)) j 2n(n−1) j j

since

τ (h) h 2n(n−1) for the Einstein metric h. The proof is complete. P(h) =

(6.16.4)

346

Chapter 6. Conformally Covariant Families

We have seen that g can be written as a warped product metric ds2 + f 2 (s)h. Hence its Ricci curvature can be determined by applying the following well-known general formula for the Ricci curvature of such a metric G = g + f 2 h on M m × N n , f ∈ C ∞ (M ), f > 0 (see [24], Proposition 9.106). Ric(G)(X, Y ) = Ric(g)(X, Y ) −

n Hess(f )(X, Y ), f

Ric(G)(X, V ) = 0,

Ric(G)(V, W ) = Ric(h)(V, W ) − f 2 h(V, W )

Δg f | grad f |2 + (n−1) f f2

for X, Y ∈ X (M ) and V, W ∈ X (N ). Such an argument is equivalent to the self-contained proof given above. Note that if, in addition, the boundary metric h is conformally ﬂat, then the associated Poincar´e-Einstein metric g constructed in Theorem 6.16.1 is conformally ﬂat, too. In order to prove the claim, we ﬁrst note that C(h) = 0 implies R(h) = −P(h) h = −2c(h h) using P(h) = 2ch. Now as in the above proof we ﬁnd R(g) = R(ds2 + f 2 h) = f 2 R(f −2 ds2 + h) + Ξ (f −2 ds2 + h) = f 2 R(h) + Ξ (ds2 + f 2 h) = −2cf 2 (h h) + Ξ (ds2 + f 2 h), where Ξ = Ξ(f −2 ds2 +h,log f ) , i.e., 1 Ξ = Δf −2 ds2 (log f )f −2 ds2 − d log f ⊗ d log f + f 2 |d log f |2 (f −2 ds2 + h) 2 2 1 f¨ 2 1 f˙ = ds − ds2 + f˙2 h. f 2 f 2 Hence 1 ˙2 2 1 ˙2 1 ˙2 2 ¨ R(g) = −2cf + f f (h h) + ff − f + f (ds2 h). 2 2 2 Now for f (s) = (e−s − ces ), we ﬁnd f¨f = f 2 and f˙2 − 4c = f 2 , i.e., R(g) =

1 1 4 f (h h) + f 2 (ds2 h) = (g g). 2 2

But Ric(g) = −ng implies P(g) = − 21 g, i.e., R(g) = −P(g) g. Hence C(g) = 0. This proves the conformal ﬂatness of g.

6.16. The holographic formula for Einstein metrics

347

Now we combine Theorem 6.16.1 with the holographic formula

n 2Qn (h) = (−1) 2 P˙ n (h; 0) − P˙n∗ (h; 0) (1) n

2 −1 n ( n2 −j −1)! ∗ ! P2j (h; 0)(vn−2j ). + (−1) 2 2 j=0 22j j! n 2

n

We apply the following two facts. (i) The operator P˙n (h; 0) − P˙n∗ (h; 0) annihilates constants. ∗ (h; 0), j = 1, . . . , n2 − 1 annihilate constants. (ii) The operators P2j

In fact, the operators P2j (h; λ) are polynomials in the Laplacian Δh with constant coeﬃcients. In particular, these operators are self-adjoint. But P2j (h; 0) annihilates constants since the harmonic function u = 1 has a trivial asymptotics. Hence (ii) holds true. Similarly, (i) follows from the self-adjointness of Pn (h; λ). It follows that the holographic formula simpliﬁes to n n

n ! −1 !vn 2Qn (h) = (−1) 2 2n 2 2 8 n2 ;

n n n n n τ (h) n −1 ! (−1) 2 2− 2 n ! = (−1) 2 2n 2 2 2n(n−1) 2 n2 τ (h) n! , = n n(n−1) 2 i.e.,

Qn (h) = (n−1)!

τ (h) n(n−1)

n2 .

This is Gover’s formula (6.16.1) (Theorem 3.2.3, [110]). Note that, alternatively, we can use Theorem 6.6.6 and (ii) to conclude more directly that n

n

n −1 ! !vn . Qn (h) = (−1) 2 2n−1 2 2 Remark 6.16.1. More generally, for M m = M m1 × M m2 with the product metric h = h1 + h2 of two Einstein metrics hi so that τ2 τ1 =− , τi = τ (hi ) 2m1 (m1 −1) 2m2 (m2 −1) the associated Poincar´e-Einstein metric is given by 1 2 dr + (1−cr2 )2 h1 + (1+cr2 )2 h2 , 2 r

c=

τ1 4m1 (m1 −1)

(6.16.5)

348

Chapter 6. Conformally Covariant Families

for suﬃciently small r ([114]). In that case, the holographic formula says that m m

m ! −1 !vm , 2Qm (h1 +h2 ) = (−1) 2 2m 2 2 m where vm is the coeﬃcient of r in v(r) = (1 − cr2 )m1 (1 + cr2 )m2 . Note that the formula

r2 v(r) = det 1 − P(h) 2 extends to the product case since P(h1 +h2 ) = P(h1 ) + P(h2 ). In fact, for P(h1 +h2 ) we ﬁnd τ2 τ1 + τ2 1 τ1 (h1 +h2 ) . h1 + h2 − m−2 m1 m2 2(m−1) But the relation (6.16.5) implies τi τ1 + τ2 1 τi . − = m−2 mi 2(m−1) 2mi (mi −1) Using P(hi ) =

τi hi , 2mi (mi −1)

this yields the asserted formula for P. Note that the formula for the Schouten tensor of the product metric, when combined with formula (6.9.3), shows the vanishing of the Bach tensor of the product metric. Hence Theorem 6.9.1 yields h(4) =

1 1 P(h1 +h2 )2 = (P(h1 )2 + P(h2 )2 ) 4 4 2 2 τ1 τ2 1 h1 + h2 = c2 (h1 +h2 ). = 4 2m1 (m1 −1) 2m2 (m2 −1)

Thus the above result is compatible with Theorem 6.9.1. Remark 6.16.2. In the situation of Remark 6.16.1, with (M1 , M2 ) = (S k , N n−k ) and the product of the round metric on the sphere and an Einstein metric on N so that Ric +(n−k+1) = 0, the formula for the corresponding Poincar´e-Einstein metric was also mentioned in [238]. For N = Γ\Hn−k with a cocompact discrete Γ ⊂ SO(1, n−k)◦ , the product M1 × M2 is the boundary of the Kleinian manifold Γ\Hn+1 Hk+1 × Γ\Hn−k (here Γ is viewed as a subgroup of SO(1, n+1)◦ ).

6.16. The holographic formula for Einstein metrics

349

The present context is a good case to illustrate again the relation to Weyl’s tube formula. The formula r 2 v(t) vol(Σ; h)dt vol({t < r}; dt +ht ) = 0 Σ r t2 = (6.16.6) det 1 − P(h) vol(Σ; h)dt 2 0 Σ for an Einstein metric h and the formula r vol(T (Σ, r); gc ) = det(1 ± tS) vol(Σ; gc )dt ±

0

(6.16.7)

Σ

for the volume of a tube of radius r around a closed hypersurface Σ of the Euclidean space (Rn , gc ) both have a common origin. In (6.16.7) the operator S : T (Σ) → T (Σ) is the shape operator of Σ, i.e., g(S(X), Y ) = L(X, Y ). Weyl’s volume formula (6.14.22) is a consequence of (6.16.7) ([130]). First of all, note that both formulas deal with the volumes of (geodesic) tubes. But whereas (6.16.7) deals with a ﬂat background metric, in (6.16.6) the background metric is dt2 + ht so that t−2 (dt2 + ht ) is Einstein. Now the volume vol(T (Σ, r); g) of a general tube T (Σ, r) = {x ∈ M | dist(x, Σ) ≤ r} around a closed hypersurface Σ ⊂ M with respect to a background metric g is given by the formula r v± vol(Σ; g)dt, vol(T (Σ, r); g) = 0

±

Σ

where v± satisﬁes the diﬀerential equation v˙ ± (t) = tr(S± (t)). v± (t)

(6.16.8)

Here S± (t) : T (Σt ) → T (Σt ) are the shape operators of the two components Σ± t of the parallel hypersurface with geodesic distance t from Σ. S(t) itself satisﬁes the Ricatti diﬀerential equation S˙ + S 2 + G = 0,

G(X) = R(X, N )N,

(6.16.9)

where N denotes a geodesic normal ﬁeld. Gray’s studies of tubes ([130]) rest on these two fundamental diﬀerential equations. In Weyl’s case G ≡ 0 and S± (t) = ±

S(0) , det(1 ± tS(0))

is the solution of (6.16.8), (6.16.9).

v± (t) = det(1 ± tS(0))

350

Chapter 6. Conformally Covariant Families

In the case (6.16.6), the background curvature, i.e., G, is non-trivial and the equations (6.16.8), (6.16.9) are more complicated. For the warped product metric dr2 + ϕ2 (r)h, we ﬁnd · 2 ϕ˙ ϕ˙ G=− + id ϕ ϕ and S(t) =

ϕ˙ (t) id ϕ

solves (6.16.9). (6.16.8) yields v = ϕn . For ϕ(r) = 1 − cr2 (as in Theorem 6.16.1) we get 2ct 2c id, S(t) = − id, v(t) = (1−ct2 )n . G(t) = 2 1−ct 1−ct2 Note that τ (h) id = P(h). G(0) = 2c id = 2n(n−1) The arguments in the proof of Lemma 6.8.1 can be used to prove the latter relation for all metrics h. Moreover, for Einstein h, we ﬁnd the formula −1 t2 . G(t) = P(h) id − P(h) 2 In general, the equations (6.16.8), (6.16.9), however, do not yield new insight without further information on hr . In fact, for the background metric dr2 +hr , we ﬁnd 1 (6.16.10) L(t) = h˙ t . 2 In particular, L(0) = 0. (6.16.9) can be veriﬁed by an elementary calculation. (6.16.8) and (6.16.10) imply 1 v˙ (t) = tr(S(t)) = tr(h˙ t h−1 t ). v 2 Hence using the initial condition v(0) = 1, we obtain r 1 r v˙ tr(h˙ t h−1 (t)dt = exp v(r) = exp t )dt 2 0 0 v r 1 r 1 (d/dt) tr(log(ht ))dt = exp (d/dt) log det(ht )dt = exp 2 0 2 0 = (det(hr )/ det(h0 ))1/2 = vol(hr )/ vol(h0 ), i.e., (6.16.8) is just the inﬁnitesimal form of the deﬁnition of v(r). Now we add the Einstein condition Ric(g) + ng = 0, g = r−2 (dr2 + hr ).

6.16. The holographic formula for Einstein metrics

Lemma 6.16.1.

351

1 tr(G(r)) = − tr(S(r)). r

Proof. We use the identity ¯ ¯ X gradϕ, −ng(X, Y ) = Ric(g)(X, Y ) = Ric(¯ g )(X, Y ) + (n−1)¯ g(∇ Y) + g¯(X, Y )Δg¯ (ϕ) − (n−1)|dϕ|2g¯ g¯(X, Y ) + (n−1)X(ϕ)Y (ϕ) for ϕ = log r, g¯ = e2ϕ g = dr2 + hr and X = Y = ∂/∂r. Then −

n−1 n−1 n−1 n = Ric(¯ g )(∂/∂s, ∂/∂s) − 2 + Δdr2 +hr (log r) − 2 + 2 r2 r r r n−1 1 1 ∂ = tr(G) − 2 + det(hr ) , r r det(hr ) ∂r

i.e., tr(G(r)) = −

1 1 (det(hr ))· = − tr(S(r)). 2r det(hr ) r

The proof is complete. It is obvious that for any metric h, the families −1 −1 t2 t2 def def , G(t) = P(h) id − P(h) S(t) = −tP(h) id − P(h) 2 2 (for suﬃciently small t) provide a solution of the Ricatti equation S˙ + S 2 + G = 0

(6.16.11)

with the initial condition S(0) = 0. In addition, the pair (S, G) satisﬁes the necessary relation (Lemma 6.16.1) 1 G(t) = − S(t) t for r−2 (dr2 + hr ) being Einstein. These data correspond to the volume function 2 v(r) = det(id − r2 P). For Einstein h, this is the exact solution and it is an open problem to describe how the true volume function deviates from the latter one. Note that by taking traces (6.16.11) and Lemma 6.16.1 imply ˙ tr(S(t)) + |S 2 (t)|2 −

1 tr(S(t)) = 0. t

The latter relation, in turn, yields the relations 1 tr(S1 Sj−1 + · · · + Sj−1 S1 ), j ≥ 2 j −1 for the coeﬃcients Sj in the series S(t) = j≥0 tj Sj , S0 = 0. tr(Sj ) = −

352

Chapter 6. Conformally Covariant Families

Note that the Ricatti equation also plays a central role in the so-called Hamilton-Jacobi method in holographic renormalization ([193], [174]). We close the present section with an alternative more direct proof of (6.16.1). Since n2 τ (h) 1 Qn (Σn ; ch) Qn (Σn ; h) = n 2 n(n−1) with c as in Theorem 6.16.1 and τ (ch) = c−1 τ (h) = 4n(n−1) = τ (S n ; 14 gc ), it is enough to prove Qn (Σn ; h) = 2n (n−1)! (6.16.12) for an Einstein metric h so that τ (h) = 4n(n−1). In order to prove (6.16.12), we n use the identity Qn = (−1) 2 P˙n (0)(1) (see (3.2.20)). For that purpose, we consider eigenfunctions of the Laplacian of the Poincar´e-Einstein metric gh =

1 2 dr +(1−r2 )h . 2 r

The latter metric generalizes the Poincar´e metric on the ball Bn+1 , 1 1 gE = 2 dr2 +(1−r2 )2 hS n r 4 (see (6.5.9)). The Laplacian for gh is readily obtained by replacing in (6.5.10) the Laplacian of 14 hS n by the Laplacian of h. Moreover, the asymptotics of eigenfunctions for gh can be obtained directly from the asymptotics of eigenfunctions for gE just by replacing Laplacians (see Section 6.5 for details on the asymptotics). In particular, it follows that the functions n n 1 (−1) 2 P˙n (Σ; h; 0)(1) and (−1) 2 P˙n (S n ; hS n ; 0)(1) 4

coincide. Since the latter is given by 1 Qn (S n ; hS n ) = 2n Qn (S n ; hS n ) = 2n (n−1)!, 4 this proves (6.16.12). Combining the above arguments with (3.2.18), it follows also that the critical GJMS-operator Pn (Σn , h) for (normalized) Einstein h (i.e., τ (h) = 4n(n − 1)), readily can be obtained from Pn (S n ; 14 hS n ) just by replacing Laplacians in the product formula (3.2.25) on the standard sphere. We ﬁnd Pn (Σn ; h) =

n−1 j= n 2

(Δh −4j(n−1−j)).

6.16. The holographic formula for Einstein metrics

353

Hence, for (not normalized) Einstein h, we ﬁnd n

Pn (Σn ; h) = Pn (Σn ; c−1 (ch)) = c 2 Pn (Σn ; ch) n

= c2

n−1

(Δch −4j(n−1−j)) =

j= n 2

=

j(n−1−j) τ (h) . Δh − n(n−1) n

n−1

(Δh −4j(n−1−j)c)

(6.16.13)

j= n 2

n−1 j= 2

This proves Gover’s product formula (Theorem 3.2.3). The argument extends to the non-critical GJMS-operators. In certain special cases, the product formula (6.16.13) can be proved by more elementary methods. Graham ([120]) gave such an argument proving (6.16.13) for the round sphere. Here we add a similar argument for the real hyperbolic space. nc A consequence of that formula naturally leads to the families DN (λ) for certain parameters N and λ. The following arguments are taken from [150]. Let (Hn , gc ) be the upper half-space model of hyperbolic space with the metric 2 2 gc = x−2 n (dx1 + · · · + dxn ). Theorem 6.16.2. Let n be even and N ≥ 1. Then n 2 +N

n

P2N ((H , gc )) = xn

−n 2 +N ΔN Rn xn

n 2 +N −1

=

(ΔHn + κj ),

j= n 2

where κj = j(n−1−j). The ﬁrst line is a consequence of the conformal covariance of P2N . For the proof of the factorization, we recall that ΔHn = x2n

n ∂2 ∂ − (n−2)xn , 2 ∂xi ∂xn i=1

and apply a partial Fourier transform in the variable x . It follows that Theorem 6.16.2 is a consequence of Theorem 6.16.3. On R+ , y

n 2 +N

for c ∈ C.

d2 + c2 dy 2

N y

−n 2 +N

n 2 +N −1

=

j= n 2

2 d d 2 2 + κj + c − (n−2)y y dy 2 dy

354

Chapter 6. Conformally Covariant Families

For the proof of Theorem 6.16.3, we introduce the abbreviations L=

d2 + c2 , dy 2

T = y 2 L−(n−2)y

n

n d + −1 . dy 2 2

Lemma 6.16.2. (i) L ◦ y m − y m ◦ L = m(m − 1)y m−2 + 2my m−1 ◦ (ii) Lm ◦ y − y ◦ Lm = 2mLm−1 ◦

d dy

d dy

(6.16.14)

for m ∈ N.

for m ∈ N.

Proof. (i) is obvious. We prove (ii) by induction. For m = 1, it is a special case of (i). Thus assuming d Lm ◦ y − y ◦ Lm = 2mLm−1 ◦ dy we calculate Lm+1 ◦ y − y ◦ Lm+1 = L ◦ (Lm ◦ y − y ◦ Lm ) + L ◦ y ◦ Lm − y ◦ Lm+1 d = L ◦ 2mLm−1 ◦ + (L ◦ y − y ◦ L) ◦ Lm dy d d d +2 ◦ Lm = 2(m+1)Lm ◦ . = 2mLm ◦ dy dy dy

The proof is complete. Lemma 6.16.3. y ◦ L ◦ y m

m−1

=L

m−1

◦ y ◦ L − m(m−1)L m

m−1

◦y

m−2

for m ∈ N.

Proof. We calculate y ◦ Lm ◦ y m−1 = y ◦ Lm−1 ◦ L ◦ y m−1 = Lm−1 ◦ y ◦ L ◦ y m−1 − 2(m−1)Lm−1 ◦

d ◦ y m−1 dy

(by Lemma 6.16.2/(ii))

d

= Lm−1 ◦ y ◦ y m−1 ◦ L + (m−1)(m−2)y m−3 + 2(m−1)y m−2 ◦ dy d ◦ y m−1 (by Lemma 6.16.2/(i)) − 2(m−1)Lm−1 ◦ dy d = Lm−1 ◦ y m ◦ L + (m−1)(m−2)Lm−1 ◦ y m−2 + 2(m−1)Lm−1 ◦ y m−1 ◦ dy d − 2(m−1)Lm−1 ◦ ◦ y m−1 dy = Lm−1 ◦ y m ◦ L + (m−1)(m−2)Lm−1 ◦ y m−2 − 2(m−1)(m−1)Lm−1 ◦ y m−2 = Lm−1 ◦ y m ◦ L − m(m−1)Lm−1 ◦ y m−2 .

The proof is complete. Lemma 6.16.4. T = y

n 2 +1

◦L◦y

−n 2 +1

.

6.16. The holographic formula for Einstein metrics

355

Proof. Direct calculation. Theorem 6.16.4. For N ∈ N0 , n

n

T ◦ (T −2) ◦ · · · ◦ (T −N (N +1)) = y 2 +N +1 ◦ LN +1 ◦ y − 2 +N +1 . Proof. We use induction. For N = 0, Lemma 6.16.4 proves the assertion. Now T (T −2) · · · (T −N (N +1)) = T (T −2) · · · (T −(N −1)N )(T −N (N +1)) n

n

= y 2 +N LN y − 2 +N (T − N (N +1)) (by assumption) n n n n = y 2 +N LN y − 2 +N y 2 +1 Ly − 2 +1 − N (N +1) (by Lemma 6.16.4) n

n

n

n

= y 2 +N (LN y N +1 L)y − 2 +1 − N (N +1)y 2 +N LN y − 2 +N n

n

= y 2 +N +1 LN +1 y − 2 +N +1

(by Lemma 6.16.3).

The proof is complete. As a corollary of Theorem 6.16.4, we obtain for N ≥ 1, n

n

y 2 +N LN y − 2 +N = T (T −2) · · · (T −(N −1)N ) n2 +N N −1 −1 d d 2 2 n + κ 2 −1−j = + κj , = y L−(n−2)y y L−(n−2)y dy dy n j=0 j= 2

i.e., the proof of Theorem 6.16.3 is complete. Corollary 6.16.1. Let u ∈ C ∞ (Hn ) satisfy −ΔHn u = μ(n−1−μ)u. Then

n

xnn ΔR2n u = (−μ) · · · (n−1−μ)u. Proof. Theorem 5.2.2 (for N = n 2

xnn ΔE ω

n 2)

implies

n 2 −1

=

n 2 −1

(−μ(n−1−μ) + κj ) ω =

j=0

(μ−j)(μ−(n−1−j))ω =

n−1

j=0

(μ−j)ω

j=0

by using the identity −μ(n−1−μ) + j(n−1−j) = (μ−j)(μ+j) − (μ−j)(n−1). Since n is even the proof is complete.

356

Chapter 6. Conformally Covariant Families

Corollary 6.16.2. Let n be even. Any eigenfunction u ∈ C ∞ (Rn+ ) such that −ΔHn u = λ(n − λ)u with λ ∈ [0, n−1] ∩ Z satisﬁes

n

ΔR2n u = 0. Corollary 6.16.2 has an interesting consequence. A bounded eigenfunction u as in Corollary 6.16.2 can be considered as a measure M (u) on Rn by extending it by 0 to the lower half plane, i.e., we set uϕdx, ϕ ∈ C0∞ (Rn ). M (u), ϕ = Rn +

Now we form the distribution n

ΔR2n (M (u)) ∈ C −∞ (Rn ). Corollary 6.16.2 implies that n

supp(ΔR2n (M (u))) ⊂ {xn = 0} . A closer analysis shows that for λ = N ∈ [0, . . . , n−1] ∩ Z, the latter distribution can be described in terms of the adjoint operators nc nc DN (λ)∗ , Dn−1−N (λ)∗ : C −∞ (Rn−1 ) → C −∞ (Rn )

for appropriate values of λ, acting on the boundary distributions of u.

6.17 Semi-holonomic Verma modules and their role We recall that the universal enveloping algebra U(g) of a Lie algebra g is deﬁned as the quotient of the tensor algebra T (g) by the two-sided ideal which is generated by the elements X ⊗ Y − Y ⊗ X − [X, Y ] for X, Y ∈ g. Given a representation (σ, Vσ ) of a subgroup P let def

M(Vσ ) = (U(g) ⊗ Vσ ) / X ⊗ v − 1 ⊗ dσ(X)v, X ∈ p = U(g) ⊗U (p) Vσ , where · denotes the left U(g)-ideal generated by the elements indicated. In the following, let g be the Lie algebra of G = SO(1, n)◦ , p the Lie algebra of a parabolic subgroup P ⊂ G and (σ, Vσ ) an irreducible representation of P . Then the U(g)-module M(Vσ ) is called the (generalized) Verma module induced by (σ, Vσ ). Following [19], we deﬁne the algebra A(g) as the quotient of the tensor algebra T (g) by the two-sided ideal which is generated by the elements X ⊗ Y − Y ⊗

6.17. Semi-holonomic Verma modules and their role

357

X − [X, Y ] for X ∈ p and Y ∈ g. In other words, in A(g) it is only allowed to commute two elements if at least one is in p or, equivalently, in A(g) the relations X ⊗ Y − Y ⊗ X = 0 for X, Y ∈ n− do not hold true. For a representation (σ, Vσ ) of P , we deﬁne def

N (Vσ ) = (A(g) ⊗ Vσ ) / X ⊗ v − 1 ⊗ dσ(X)v, X ∈ p , where · denotes the left A(g)-ideal generated by the elements indicated. The A(g)-module N (Vσ ) is called the semi-holonomic Verma module induced by (σ, Vσ ) ([92]). The canonical projection π : A(g) → T (g) induces a canonical projection π : N (Vσ ) → M(Vσ ). Next, we recall the correspondence between G-equivariant diﬀerential operators D : C ∞ (G, Eσ )P → C ∞ (G, Fη )P and U(g)-module homomorphisms D : M(Fη∗ ) → M(Eσ∗ ) of generalized Verma modules. D induces a map D : U(g) ⊗ Fη∗ V ⊗ f ∗ → V Ti ⊗ Li (f ∗ ) ∈ U(g) ⊗ Eσ∗ , i

where Ti ∈ U(g) and the linear maps Li ∈ Hom(Fη∗ , Eσ∗ ) are characterized by the condition Ti (u)(e), Li (f ∗ ) (6.17.1) D(u)(e), f ∗ = i

for u ∈ C ∞ (G, Eσ )P . Here U(g) acts from the right on C ∞ (G). In order to prove that D induces a homomorphism of generalized Verma modules, we verify that D induces a map 3 4 X ⊗ f ∗ − 1 ⊗ dη ∗ (X)f ∗ | X ∈ p, f ∗ ∈ Fη∗ −→ X ⊗ e∗ − 1 ⊗ dσ ∗ (X)e∗ | X ∈ p, e∗ ∈ Eσ∗ of left U(g)-ideals. In fact, (6.17.1) implies (ad(X)Ti ⊗ e∗i + Ti ⊗ dσ ∗ (X)e∗i ) , X ∈ p D(1 ⊗ dη ∗ (X)f ∗ ) =

(6.17.2)

i

with e∗i = Li (f ∗ ). Hence D(X ⊗ f ∗ ) − D(1 ⊗ dη ∗ (X)f ∗ ) = XTi ⊗ e∗i − (ad(X)Ti ⊗ e∗i + Ti ⊗ dσ ∗ (X)e∗i ) i

i

=

i

Ti (X ⊗ e∗i − 1 ⊗ dσ ∗ (X)e∗i ) , X ∈ p.

358

Chapter 6. Conformally Covariant Families

This proves the claim. It remains to verify the relation (6.17.2). But for p ∈ P , we have 3 4 D(u)(e), η ∗ (p)f ∗ = η(p−1 )D(u)(e), f ∗ 4 3 = Lp−1 (D(u))(e), f ∗ 4 3 = D(Lp−1 (u))(e), f ∗ 3 4 Ti (Lp−1 (u))(e), e∗i = i

= (Ad(p)Ti )(u)(e), σ ∗ (p)e∗i by left G-equivariance of D. Here Lg denotes left translation by g. (6.17.2) follows by diﬀerentiation. In the reverse direction, we reconstruct D from D by g → Fη∗ f ∗ → D(1 ⊗ f ∗ )(u)(g) ∈ Fη . Here T ⊗ e∗ ∈ U(g) ⊗ Eσ∗ acts on u ∈ C ∞ (G, Eσ )P by T (u)(g), e∗ . The following special case is the one which is of most signiﬁcance here. Let C(λ) denote the 1-dimensional P -module with the action at → aλt = eλ t. Then C(λ)∗ = C(−λ). We use the notation Mλ (g) = U(g) ⊗U (p) C(λ) for the corresponding Verma module. Note that C ∞ (G, C(λ)∗ )P IndG P (ξλ ) (Section 5.2). In order to construct conformally covariant curved analogs of G-equivariant diﬀerential operators on sections of homogeneous vector bundles on spheres G/P , Baston ([19]) introduced an induction mechanism which combines Cartan’s conformal connection with homomorphism of semi-holonomic Verma modules. Here it is necessary to go to the semi-holonomic category since the curvature of Cartan’s conformal connection does not vanish, in general. We brieﬂy explain the idea. For a conformal manifold (M, [g]), there is a replacement of the principal P -ﬁbre bundle G → G/P together with the Maurer-Cartan form ω. More precisely, there exists a principal P -ﬁbre bundle G → M (P is a parabolic subgroup in G = SO(1, n + 1) if M has dimension n) and a one-form ω ∈ Ω1 (G, g) so that (i) ωm : Tm G → g is an isomorphism for all m ∈ G. (ii) ω −1 (X) = ζX for X ∈ p, where ζX denotes the corresponding fundamental vector ﬁeld of the P -action. (iii) p∗ (ω) = Ad(p)ω for p ∈ P . By (i), the horizontal vector ﬁelds ω −1 (X) ∈ X (G), X ∈ g are well deﬁned. For X ∈ p, these are the fundamental vector ﬁelds of the P -action. For details on Cartan connections and its applications to constructions of invariant diﬀerential operators we refer to [57], [58], [59], [22].

6.17. Semi-holonomic Verma modules and their role

359

Now an A(g)-module homomorphism D : N (Fη∗ ) → N (Eσ∗ ) induces a diﬀerential operator D : C ∞ (G, Eσ )P → C ∞ (G, Fη )P as follows. For u ∈ C ∞ (G, Eσ )P = u ∈ C ∞ (G, Eσ ) | u(gp) = σ −1 (p)u(g), p ∈ P , we deﬁne Du(m) by ∗ Fη f ∗ → D(1 ⊗ f ∗ )(u)(g) ∈ Fη . Here an element T ⊗ e∗ ∈ A(g) ⊗ Eσ∗ acts on u by ω −1 (T )u, e∗ . The fact that Du ∈ C ∞ (G, Fη )P is a consequence of (iii). The inﬁnitesimal form of (iii) states that ω −1 ([X, Y ]) = ω −1 (X), ω −1 (Y ) for X ∈ p, Y ∈ g. But other commutators are not preserved by ω −1 . That is the reason to go from U(g) to A(g). Now the problem of constructing a conformally covariant curved analog of an invariant diﬀerential operator on homogeneous vector bundles on G/P S n can be treated in the following three steps. 1. Identify the invariant diﬀerential operator with a homomorphism of (generalized) Verma modules. 2. Lift the homomorphism of Verma modules to a homomorphism of semiholonomic Verma modules. 3. Use the lift to induce a diﬀerential operator on associated vector bundles on G/P M . An unfortunate aspect of that procedure is that the lifts required by the second step do not always exist. In particular, Eastwood and Slovak ([92]) proved that the Verma module homomorphisms which correspond to the critical GJMS-operators on the spheres (viewed as conformally invariant operators on densities) do not lift to the semi-holonomic category. Thus the above method cannot be used to induce the critical GJMS-operators. c Now the fact that the equivariant families DN (λ) : C ∞ (S n ) → C ∞ (S n−1 ) are induced by the families DN (λ) of homomorphisms of Verma modules (Theorem 5.1.5) suggests that we 1. lift the family DN (λ) to the semi-holonomic category and c 2. use the lifts to induce curved analogs of DN (λ). These problems will be discussed in the following sections. Although we shall be able to construct certain rational lifts of all polynomial families DN (λ) (Secc tion 6.18), the curved analogs of DN (λ) will not arise by an induction procedure in terms of conformal connections. Instead, the speciﬁc form of the constructed semi-holonomic lifts will allow us to apply tractor calculus for that purpose. The consequences of the non-uniqueness of semi-holonomic lifts remain to be understood.

360

Chapter 6. Conformally Covariant Families

6.18 Zuckerman translation and DN (λ) In the present section, we lift the families DN (λ) to rational families of homomorphisms of semi-holonomic Verma modules using Zuckerman translation. We ﬁrst recall the construction of GJMS-operators by Zuckerman translation of the Yamabe operator. The idea is as follows. The Yamabe operator P2 (·) : C ∞ (M n ) → C ∞ (M n ) satisﬁes the relation n

n

P2 (e2ϕ g) = e−( 2 +1)ϕ ◦ P2 (g) ◦ e( 2 −1)ϕ . Using Cartan’s conformal connection, it is induced by a homomorphism N− n2 −1 (gn+1 ) → N− n2 +1 (gn+1 )

(6.18.1)

of semi-holonomic Verma modules Nλ (gn+1 ) = A(gn+1 ) ⊗ C(λ)/X ⊗ 1 − 1 ⊗ dξλ (X)1, X ∈ p, where ξλ denotes the irreducible P -representation on C(λ) deﬁned by ξλ (mat )z = eλt z. In explicit terms that homomorphism is induced by the map

n n A(gn+1 ) ⊗ C − −1 T ⊗ 1 → T Δ− +1 . ⊗ 1 ∈ A(g ) ⊗ C − n+1 n 2 2 The fact that (6.18.1) is well deﬁned is a consequence of the commutator relations − − (6.18.3), [H0 , Δ− n ] = −2Δn and [Mij , Δn ] = 0. The homomorphism (6.18.1) covers the corresponding homomorphism n n Δ− n : M− 2 −1 (gn+1 ) → M− 2 +1 (gn+1 )

of generalized Verma modules for gn+1 . Zuckerman translation constructs a new homomorphism using a given one. In the present context, let (π, F ) be the fundamental representation of gn+1 and consider the composition of the homomorphism n n Δ− n ⊗ I : M− 2 −1 (gn+1 ) ⊗ F → M− 2 +1 (gn+1 ) ⊗ F

with an embedding i : M− n2 −2 → M− n2 −1 ⊗ F and a projection p : M− n2 +1 ⊗ F → M− n2 +2 of modules. Here X ∈ g acts on Mλ ⊗ F by T ⊗ 1 ⊗ v → XT ⊗ 1 ⊗ v + T ⊗ 1 ⊗ π(X)v.

6.18. Zuckerman translation and DN (λ)

361

For (π, F ) we use the model given by the deﬁning representation of SO(1, n+1) on R1,n+1 . The intertwining map n n p ◦ (Δ− n ⊗ I) ◦ i : M− 2 −2 → M− 2 +2

is the Zuckerman translate of Δ− n . The procedure can be iterated, and yields nontrivial intertwining maps M− n2 −j (gn+1 ) → M− n2 +j (gn+1 ) for j = 1, . . . , n2 − 1. However, it fails to construct a non-trivial intertwining map M−n (gn+1 ) → M0 (gn+1 ). The method can be used in the semi-holonomic category as well since the corresponding embedding i and projection p can be lifted. For later use, it will be important to have explicit formulas for these maps. We introduce some more notation. We choose a basis {v+ , vi , v− , i = 1, . . . , n} of F R1,n+1 by v+ = (1, 1, 0, . . . , 0)t ,

v− = (1, −1, 0, . . . , 0)t ,

vi = (0, 0, ei )t , i = 1, . . . , n.

Then Yi+ v− = 2vi ,

Yi+ vj = δij v+ ,

Yi+ v+ = 0,

Yi− v+ = 2vi ,

Yi− vj = δij v− ,

Yi− v− = 0.

(6.18.2)

Moreover, the decomposition R1,n+1 Rv− ⊕ vi ⊕ Rv+ = V−1 ⊕ V0 ⊕ V1 is M A-equivariant. Here M acts trivially on V±1 , by rotation on the n-dimensional space V0 , and A acts by at : v± → e±t v± , vi → vi . Let J(λ) be the left ideal generated by the elements X ⊗ 1 − 1 ⊗ ξλ (X)1, X ∈ p. Lemma 6.18.1. The homomorphism i : M− n2 −2 (g) → M− n2 −1 (g) ⊗ F of U(g)modules is induced by the U(g)-module map

n

n U(g) ⊗ C − −2 → U(g) ⊗ C − −1 ⊗ F 2 2 which is determined by n

1 i : 1 ⊗ 1 → +1 (1 ⊗ 1 ⊗ v− ) + Yi− ⊗ 1 ⊗ vi + Δ− ⊗ 1 ⊗ v+ . 2 8 n i Similarly, the homomorphism p : M− n2 +1 (g) ⊗ F → M− n2 +2 (g) of U(g)-modules is induced by the U(g)-module map

n

n p : U(g) ⊗ C − +1 ⊗ F → U(g) ⊗ C − +2 2 2

362

Chapter 6. Conformally Covariant Families

which is determined by 1 − (Δ ⊗ 1), 1 ⊗ 1 ⊗ vi → Yi− ⊗ 1, 1 ⊗ 1 ⊗ v+ → (4−n)(1 ⊗ 1). 2 n Proof. In order to prove the assertion for i, it is enough to verify that n

n

i : J − −2 → J − −1 ⊗ F. 2 2 1 ⊗ 1 ⊗ v− →

But (6.18.2) says that modulo terms in J(− n2 −1) ⊗ F

n i(Yj+ ⊗ 1) = +1 Yj+ ⊗ 1 ⊗ v− + 1 ⊗ 1 ⊗ Yj+ v− 2 Yj+ Yi− ⊗ 1 ⊗ vi + Yi− ⊗ 1 ⊗ Yj+ vi + i

1 + − + + Yj Δn ⊗ 1 ⊗ v+ + Δ− n ⊗ 1 ⊗ Yj v+ 8

n +1 (1 ⊗ 1 ⊗ 2vj ) ≡ 2 Mji ⊗ 1 ⊗ vi + Yi− ⊗ 1 ⊗ δij v+ + 2H0 ⊗ 1 ⊗ vj + 2 i=j

i

1 + 2(n−2)Yj− + 4Yj− H0 ⊗ 1 ⊗ v+ 8 ≡ 0. Here we have used the commutator relation + − Yr− Mjr Yj , Δn = 2(n−2)Yj− + 4Yj− H0 + 4

(6.18.3)

r=j

which follows from Lemma 5.1.3 for j = 1, and by a rotation for all j. Similarly, we calculate

n

n i(H0 ⊗ 1 + +2 ⊗ 1) = +1 (H0 ⊗ 1 ⊗ v− − 1 ⊗ 1 ⊗ v− ) 2 2

n n n +2 +1 ⊗ 1 ⊗ v− + +2 Yi− ⊗ 1 ⊗ vi H0 Yi− ⊗ 1 ⊗ vi + + 2 2 2 i

1 n − H0 Δ− +2 Δ− + n ⊗ 1 ⊗ v+ + Δn ⊗ 1 ⊗ v+ + n ⊗ 1 ⊗ v+ , 8 2 i.e.,

n

n n i(H0 ⊗ 1 + +2 ⊗ 1) = +1 H0 ⊗ 1 ⊗ v− + +1 ⊗ 1 ⊗ v− 2 2 2 n

+ Yi− H0 ⊗ 1 ⊗ vi + +1 Yi− ⊗ 1 ⊗ vi 2 i

n

1 +1 Δ− + Δ− n H0 ⊗ 1 ⊗ v+ + n ⊗ 1 ⊗ v+ ≡ 0 8 2

6.18. Zuckerman translation and DN (λ)

363

− using [H0 , Δ− n ] = −2Δn . Finally, i(Mij ⊗ 1) ≡ 0 since M acts trivially on v± and − − Δn and n V0 as a M -module. We continue with the proof for p. In order to verify that p deﬁnes a map J(− n2 +1) ⊗ F → J(− n2 +2), we ﬁrst calculate using (6.18.3)

p(Yj+ ⊗ 1 ⊗ v− ) = p(Yj+ (1 ⊗ 1 ⊗ v− ) − 2 ⊗ 1 ⊗ vj ) 1 − − − − ⊗1≡0 = Yj+ Δ− n ⊗ 1 − 2Yj ⊗ 1 ≡ (n−2)Yj + 2Yj H0 − 2Yj 2 modulo terms in J(− n2 +2). Similarly, we ﬁnd p(Yj+ ⊗ 1 ⊗ vi ) = p(Yj+ (1 ⊗ 1 ⊗ vi ) − 1 ⊗ 1 ⊗ δij v+ ) = Yj+ Yi− ⊗ 1 + (n−4)δij (1 ⊗ 1) ≡ 2H0 δij ⊗ 1 + (n−4)δij (1 ⊗ 1) ≡ 0 modulo J(− n2 +2) and p(Yj+ ⊗ 1 ⊗ v+ ) = p(Yj+ (1 ⊗ 1 ⊗ v+ )) = (4−n)Yj+ ⊗ 1 ≡ 0.

The proof is complete.

The proof of Theorem 6.18.1 shows that the maps i and p also induce homomorphisms of semi-holonomic Verma modules. − Next, we determine the translation p ◦ (Δ− n ⊗ I) ◦ i of Δn . n n Theorem 6.18.1. The composition p ◦ (Δ− n ⊗ I) ◦ i : N− 2 −2 (g) → N− 2 +2 (g) is given by right multiplication with

1 2 , 4L4,n − n(Δ− n) 8 where L4,n =

− Yi− Δ− n Yi .

i 2 (Δ− n)

−

Since π(L4,n ) = ∈ U(n ), the homomorphism p ◦ (Δ− n ⊗ I) ◦ i : M− n2 −2 (g) → M− n2 +2 (g) of Verma modules is induced by right multiplication 2 with 18 (4 − n)(Δ− n ) . In particular, for n = 4, the latter operator vanishes. Notice, however, that it is still non-trivial in the semi-holonomic category since L4,n = 2 (Δ− n) . Proof. The proof is a straightforward calculation. We have to calculate the image of

n 1 − 2 +1 (Δ− Yi− Δ− n ⊗ 1 ⊗ v− ) + n ⊗ 1 ⊗ vi + ((Δn ) ⊗ 1 ⊗ v+ ) 2 8 i under p. For that purpose, we use the formulas − Yi− Δ− Yi− Δ− n (1 ⊗ 1 ⊗ vi ) = n ⊗ 1 ⊗ vi + (n+2)Δn (1 ⊗ 1 ⊗ v− ) (6.18.4) i

i

364

Chapter 6. Conformally Covariant Families

and 2 (Δ− n ) (1 ⊗ 1 ⊗ v+ ) 2 = (Δ− n ) ⊗ 1 ⊗ v+ + 4

Yi− Δ− n ⊗ 1 ⊗ vi + 4

i

=

2 (Δ− n)

⊗ 1 ⊗ v+ + 4

− Δ− n Yi (1 ⊗ 1 ⊗ vi )

i

Yi− Δ− n (1

⊗ 1 ⊗ vi )

i

− 4(n+2)Δ− n (1 ⊗ 1 ⊗ v− ) + 4

− Δ− n Yi (1 ⊗ 1 ⊗ vi ) (by (6.18.4)),

i

the proofs of which are left to the reader. Now for the composition we obtain n+2 n+2 − 2 − 2 ((Δn ) ⊗ 1) + L4,n ⊗ 1 − (Δn ) ⊗ 1 4 2 1 2 − 2 − 2 + ⊗1 (4−n)(Δ− n ) − 4L4,n + 2(n+2)(Δn ) − 4(Δn ) 8 1 n − 2 L4,n − (Δn ) ⊗ 1. = 2 8

The proof is complete. Lemma 6.18.1 admits the following family version. Lemma 6.18.2. The family of U(g)-module maps i(λ) : U(g) ⊗ C(λ−1) → U(g) ⊗ C(λ) ⊗ F which is determined by 1 ⊗ 1 → −λ(1 ⊗ 1 ⊗ v− ) +

i

Yi− ⊗ 1 ⊗ vi −

1 (Δ− ⊗ 1 ⊗ v+ ) 2(n−2+2λ) n (6.18.5)

induces a family of homomorphisms Mλ−1 (g) → Mλ (g) ⊗ F of U(g)-modules. Similarly, the family of U(g)-module maps p(λ) : U(g) ⊗ C(λ−1) ⊗ F → U(g) ⊗ C(λ) which is determined by 1 ⊗ 1 ⊗ v− →

1 (Δ− ⊗ 1), 1 ⊗ 1 ⊗ vi → Yi− ⊗ 1, 1 ⊗ 1 ⊗ v+ → 2λ(1 ⊗ 1) n−2+2λ n

induces a family of homomorphisms Mλ−1 (g) ⊗ F → Mλ (g) of U(g)-modules. i(λ) and p(λ) induce analogous homomorphisms in the semi-holonomic category.

6.18. Zuckerman translation and DN (λ)

365

The proof is analogous to that of Lemma 6.18.1 and we omit the details. We will also need the following generalization of Lemma 6.18.2. Lemma 6.18.3. For any g-module W , the family i(λ) : U(g) ⊗ C(λ−1) ⊗ W → U(g) ⊗ C(λ) ⊗ W ⊗ F of U(g)-module maps deﬁned by 1⊗1⊗w → −λ(1⊗1⊗w⊗v−)+

Yi− ⊗1⊗w⊗vi −

i

1 (Δ− ⊗1⊗w⊗v+) 2(n−2+2λ) n

descends to a family Mλ−1 (g) ⊗ W → Mλ (g) ⊗ W ⊗ F of homomorphisms of U(g)-modules. Similarly, the family of U(g)-module maps p(λ) : U(g) ⊗ C(λ−1) ⊗ W ⊗ F → U(g) ⊗ C(λ) ⊗ W deﬁned by 1 (Δ− ⊗ 1 ⊗ w), n−2+2λ n 1 ⊗ 1 ⊗ w ⊗ vi → Yi− ⊗ 1 ⊗ w, 1 ⊗ 1 ⊗ w ⊗ v+ → 2λ(1 ⊗ 1 ⊗ w)

1 ⊗ 1 ⊗ w ⊗ v− →

induces a family Mλ−1 (g) ⊗ W ⊗ F → Mλ (g) ⊗ W of homomorphisms of U(g)-modules. i(λ) and p(λ) induce analogous homomorphisms in the semi-holonomic category. Proof. The proof is analogous to that of Lemma 6.18.2. In particular, we have to prove that i(λ)(Yj+ ⊗1⊗w) ∈ J(λ)⊗W , where J(λ) = X ⊗1−1⊗ξλ(X)1, X ∈ p. Now Yj+ ⊗ 1 ⊗ w = Yj+ (1 ⊗ 1 ⊗ w) − 1 ⊗ 1 ⊗ Yj+ w implies i(λ)(Yj+ ⊗ 1 ⊗ w) = −λ Yj+ ⊗ 1 ⊗ w ⊗ v− + 1 ⊗ 1 ⊗ Yj+ (w ⊗ v− ) Yj+ Yi− ⊗ 1 ⊗ w ⊗ vi + Yi− ⊗ 1 ⊗ Yj+ (w ⊗ vi ) + i

i

+ − 1 + Yj Δn ⊗ 1 ⊗ w ⊗ v+ + Δ− − n ⊗ 1 ⊗ Yj (w ⊗ v+ ) 2(n−2+2λ) + λ(1 ⊗ 1 ⊗ Yj+ (w) ⊗ v− ) − Yj− ⊗ 1 ⊗ Yj+ (w) ⊗ vi i

+

1 (Δ− ⊗ 1 ⊗ Yj+ (w) ⊗ v+ ) 2(n−2+2λ) n

366

Chapter 6. Conformally Covariant Families

= −λ Yj+ ⊗ 1 ⊗ w ⊗ v− + 1 ⊗ 1 ⊗ w ⊗ Yj+ (v− ) + Yj+ Yi− ⊗ 1 ⊗ w ⊗ vi + Yi− ⊗ 1 ⊗ w ⊗ Yj+ (vi ) i

i

+ − 1 + − Yj Δn ⊗ 1 ⊗ w ⊗ v+ + Δ− n ⊗ 1 ⊗ w ⊗ Yj (v+ ) . 2(n−2+2λ) Modulo terms in (J(λ) ⊗ W ) ⊗ F , the latter sum equals − λ(1 ⊗ 1 ⊗ w ⊗ Yj+ (v− )) + 2λ(1 ⊗ 1 ⊗ w ⊗ vj ) + Yj− ⊗ 1 ⊗ w ⊗ v+ + − 1 Y , Δn ⊗ 1 ⊗ w ⊗ v+ − 2(n−2+2λ) j 1 2(n−2)Yj− + 4λYj− ⊗ 1 ⊗ w ⊗ v+ ≡ Yj− ⊗ 1 ⊗ w ⊗ v+ − 2(n−2+2λ) ≡0 using (6.18.3). The remaining parts of the proof are left to the reader.

Lemma 6.18.3 allows us to relate i(λ) and p(λ) as follows. Corollary 6.18.1. p(λ) = trF ◦i(λ), where i(λ) is viewed as a homomorphism U(g) ⊗ C(λ−1) ⊗ F → U(g) ⊗ C(λ) ⊗ F ⊗ F, and trF (v ⊗ w) = (v, w) is the contraction deﬁned by the scalar product on F R1,n+1 . Proof. It suﬃces to notice that for v ∈ F , trF i(λ)(1 ⊗ 1 ⊗ v) = −λ ⊗ (v, v− ) +

i

Yi− ⊗ (v, vi ) −

1 Δ− ⊗ (v, v+ ). 2(n−2+2λ) n

Thus using (v± , v± ) = 0, (v± , v∓ ) = −2, (v± , vi ) = 0 and (vi , vj ) = δij , we get the assertion. Corollary 6.18.1 shows that the mapping properties of p(λ) are consequences of those of i(λ). In fact, trF ◦i(λ) deﬁnes a g-module map U(g) ⊗ C(λ − 1) ⊗ F → U(g) ⊗ C(λ) since (Xv, w) + (v, Xw) = 0 for X ∈ g and v, w ∈ F . Moreover, the composition descends to a homomorphism Mλ−1 (g) ⊗ F → Mλ (g). Note that i(λ) and p(λ) are rational in λ. A calculation shows that the composition i(λ−1)

p(λ)

Mλ−2 (gn+1 ) −−−−→ Mλ−1 (gn+1 ) ⊗ F −−−→ Mλ (gn+1 ) vanishes.

6.18. Zuckerman translation and DN (λ)

367

Deﬁnition 6.18.1. The formulas Θ2N (λ) = p(λ) ◦ (Θ2N −2 (λ−1) ⊗ In ) ◦ i(λ−(2N −1)),

Θ0 (λ) = D0 (λ) = 1,

and Θ2N +1 (λ) = p(λ) ◦ (Θ2N −1 (λ−1) ⊗ In ) ◦ i(λ−2N ),

Θ1 (λ) = D1 (λ)

deﬁne rational families ΘN (λ) : Mλ−N (gn ) → Mλ (gn+1 ) of homomorphisms of Verma modules and rational families ˜ N (λ) : Nλ−N (gn ) → Nλ (gn+1 ) Θ ˜ N (λ) lifts ΘN (λ). of homomorphisms of semi-holonomic Verma modules. Θ Here we use the Gn -equivariant map In : Fn → Fn+1 which corresponds to the embedding SO(1, n) → SO(1, n + 1), i.e., SO(1, n) is the subgroup which preserves the subspace R1,n → R1,n+1 deﬁned by xn+1 = 0. Then v− , vi (i = 1, . . . , n − 1) and v+ ∈ Fn will be viewed as elements of Fn+1 . The deﬁnition of ΘN (λ) is correct since D0 and D1 (λ) = Yn− induce homomorphisms of semi-holonomic Verma modules. In fact, it suﬃces to note that [Yi+ , Yn− ] = 2Min for i = 1, . . . , n − 1. The poles of the families ΘN (λ) are caused only by the poles of the coeﬃcients of i(λ) and p(λ). An obvious question is to relate ΘN (λ) and DN (λ). The answer is given by Theorem 6.18.7 and Theorem 6.18.8. In addition, it is instructive and useful to ﬁnd explicit formulas for low order translates. The ﬁrst non-trivial translate is Θ2 (λ). Its discussion will be followed by establishing formulas for Θ3 (λ) and Θ4 (λ). Theorem 6.18.2. The homomorphism Θ2 (λ) : Nλ−2 (gn ) → Nλ (gn+1 ) is induced by right multiplication with (n−3+λ) D2 (λ), (n−5+2λ)(n−2+2λ) where

(6.18.6)

− D2 (λ) = −(n−3+2λ)Δ− n + (n−2+2λ)Δn−1 .

Proof. We recall that Θ2 (λ) = p(λ) ◦ (1 ⊗ In ) ⊗ i(λ−1). Thus we have to calculate the image of i(λ−1)(1 ⊗ 1) ∈ A(gn ) ⊗ Fn+1 , i.e., of −(λ−1) ⊗ 1 ⊗ v− +

n−1 i=1

Yi− ⊗ 1 ⊗ vi −

1 (Δ− ⊗ 1 ⊗ v+ ) 2(n−5+2λ) n−1

368

Chapter 6. Conformally Covariant Families

under p(λ). For that purpose, we use that (n−1)(1 ⊗ v− ) and − Δ− n−1 ⊗ v+ = Δn−1 (1 ⊗ v+ ) − 4

n−1

n−1 1

Yi− ⊗ vi =

n−1 1

Yi− (1 ⊗ vi ) −

Yi− (1 ⊗ vi ) + 2(n−1)(1 ⊗ v− ).

1

We apply p(λ) and obtain (for β −1 = n − 2 + 2λ) − − − (λ−1)βΔ− n ⊗ 1 + Δn−1 ⊗ 1 − (n−1)βΔn ⊗ 1 1 − − 2λΔ− − n−1 − 4Δn−1 + 2(n−1)βΔn ⊗ 1 2(n−5+2λ) 1 =− [(n−2+λ)(n−5+2λ) + (n−1)] Δ− n ⊗1 (n−2+2λ)(n−5+2λ) n−3+λ − Δ ⊗1 + n−5+2λ n−1 (n−3+λ) − −(n−3+2λ)Δ− = n + (n−2+2λ)Δn−1 . (n−2+2λ)(n−5+2λ)

The proof is complete.

We continue with a discussion of Θ3 (λ). Theorem 6.18.3. The homomorphism Θ3 (λ) : Nλ−3 (gn ) → Nλ (gn+1 ) is induced by right multiplication with 3 (n−4+λ)D3T (λ), (n−7+2λ)(n−2+2λ)

(6.18.7)

where 1 D3T (λ) = − (n−5+2λ) (Yn− )3 + Yn− Δ− n−1 3 1 (n−2+2λ) − − − − − (n−5+2λ) + Yi Yn Yi − (λ−1)Δn−1 Yn . (6.18.8) 3 (n−4+λ) i Note that πD3T (λ) = − 31 (n−5+2λ)(Yn− )3 + Yn− Δ− n−1 = D3 (λ). In other words, D3T (λ) is a lift of D3 (λ) which deﬁnes a homomorphism of semi-holonomic Verma modules. It arises by translation of Θ1 (λ) = D1 (λ) = Yn+ . Note that the factorization identities n

n−5 − D3 − +1 = Yn− Δ− and D − = Δ− 3 n n−1 Yn 2 2 continue to hold true for the lift D3T (λ).

6.18. Zuckerman translation and DN (λ)

369

Although D3 (λ) is polynomial, its lift has a simple pole at λ = 4 − n with residue 1 (n−3)(n−6) Yi− Yn− Yi− − Yi− Yn− ∈ ker π. 3 1 n−1

Res4−n (D3T (λ)) =

(6.18.9)

In particular, the residue deﬁnes a homomorphism N1−n (gn ) → N4−n (gn+1 ). It is of interest to determine the conformally covariant diﬀerential operator C ∞ (M n ) → C ∞ (Σn−1 ) which is induced by that homomorphism. The critical case n = 4, λ = 0 is of particular interest. We see that, for n = 4, the family D3T (λ) has a simple pole at λ = 0 with residue 2 − − − Y (Y4 Yi − Yi− Y4− ) ∈ ker π 3 1 i 3

− and constant term CT0 (D3T )

1 = 3

(Y4− )3

+

Y4− Δ− 3

+2

3

Yi− Y4− Yi−

1

so that πCT0 (D3T ) = 13 (Y4− )3 + Y4− Δ− 3 = D3 (0). Note that the coeﬃcients (n − 7 + 2λ)−1 and (n − 2 + 2λ)−1 are caused by the poles of i and p. On the other hand, the pole of D3T (λ) at λ = 4 − n is due to the overall coeﬃcient n − 4 + λ in (6.18.7); Θ3 (λ) is regular at λ = 4 − n. More precisely, n−1 n−3 − − − Θ3 (4 − n) = Y (Yn Yi − Yi− Yn− ) ∈ ker π. n−1 1 i Proof. We have to calculate the image of n−1

1 − (Δ− n−1 Yn ⊗ 1 ⊗ v+ ) 2(n−7+2λ) 1 (6.18.10) under p(λ). For that purpose, we use the formulas −(λ−2)(Yn− ⊗ 1 ⊗ v− ) +

n−1 1

Yi− Yn− ⊗ 1 ⊗ vi −

Yi− Yn− (1 ⊗ 1 ⊗ vi ) =

n−1

Yi− Yn− ⊗ 1 ⊗ vi + (n−1)Yn− (1 ⊗ 1 ⊗ v− )

1

and − − − Δ− n−1 Yn (1 ⊗ 1 ⊗ v+ ) = Δn−1 Yn ⊗ 1 ⊗ v+ − + 2Δ− n−1 (1 ⊗ 1 ⊗ vn ) − 2(n−1)Yn (1 ⊗ 1 ⊗ v− ) + 4

n−1 1

Yi− Yn− (1 ⊗ 1 ⊗ vi )

370

Chapter 6. Conformally Covariant Families

the proofs of which are left to the reader. We apply p(λ) to (6.18.10) and obtain 2 ⊗ 1 + 1 + Yi− Yn− Yi− ⊗ 1 − [(λ−2)+(n−1)] βYn− Δ− n n−7+2λ i −

1 − − − − − 2λΔ− n−1 Yn − 2Δn−1 Yn + 2(n−1)βYn Δn ⊗ 1, 2(n−7+2λ)

where β −1 = (n−2+2λ). A direct calculation yields the assertion.

It is interesting to notice that D3T (λ) is an element in a one-parameter family of lifts of D3 (λ). Theorem 6.18.4. Let ¯ 3 (λ) = − 1 (2λ+n−5)(Yn− )3 + D 3

− aΔ− n−1 Yn

+b

n−1

Yj− Yn− Yj−

+

cYn− Δ− n−1

1

¯ 3 (λ) = D3 (λ). Moreover, assume that the map be a lift of D3 (λ), i.e., π D ¯ 3 (λ) ⊗ 1 ∈ A(gn+1 ) ⊗ C(λ) A(gn ) ⊗ C(λ−3) T ⊗ 1 → i(T )D induces a homomorphism Nλ−3 (gn ) → Nλ (gn+1 ) of semi-holonomic Verma modules. Then a + b + c = 1, 2 (n−4+λ)b + (n−4+2λ)c = (n−5+2λ). 3 ¯ 3 (λ) deﬁnes a rational famConversely, if (a, b, c) satisﬁes these conditions, then D ily of homomorphisms of semi-holonomic Verma modules. The family D3T (λ) appears for c = − 31 (n−5+2λ). ¯3 (λ) = D3 (λ) is equivalent to a + b + c = 1 since n− is Proof. The condition π D abelian. It is enough to analyze the conditions ¯ 3 (λ)] ∈ J(λ) = A(gn+1 )(mn+1 ⊕ C(H0 −λ)) [Yi+ , D for i = 1, . . . , n − 1. The arguments are similar to those in the proof of Theorem 5.1.1. However, it is now forbidden to use the relations X ⊗ Y − Y ⊗ X = [X, Y ] for X, Y ∈ n− . It suﬃces to consider the case i = 1. The general case follows by a rotation as in the proof of Theorem 5.1.1. We determine the commutators ⎡ ⎤ + − ⎣Y1+ , Yj− Yn− Yj− ⎦ , Y1+ , Yn− Δ− Y1 , (Yn− )3 , Y1+ , Δ− n−1 Yn , n−1 . j

6.18. Zuckerman translation and DN (λ)

371

First of all, we have in A(gn+1 ), + Y1 , (Yn− )3 = 2Y1− Yn− + 4Yn− Y1− + 6(Yn− )2 M1n .

(6.18.11)

Next, by Lemma 5.1.3, n−1 + − Y1 , Δn−1 = −2Y1− + 4Y1− H0 + (2Y1− +4Yr− M1r ) r=2

= 2(n−3)Y1− + 4Y1− H0 + 4

n−1

Yr− M1r .

(6.18.12)

r=2

Hence − + − + − − Y1 , Δn−1 Yn− = Y1+ , Δ− n−1 Yn + Δn−1 Y1 , Yn = 2(n−3)Y1− Yn− + 4Y1− H0 Yn− + 4

n−1

Yr− M1r Yn− + 2Δ− n−1 M1n

r=2

= 2(n−5)Y1− Yn− + 4Y1− Yn− H0 + 4

n−1

Yr− Yn− M1r + 2Δ− n−1 M1n

r=2

≡ 2(n−5+2λ)Y1− Yn−

(6.18.13)

mod J(λ). Similarly, using (5.2.7) and [M1n , (Y1− )2 ] = Y1− [M1n , Y1− ] + [M1n , Y1− ]Y1− = −(Yn− Y1− + Y1− Yn− ), we get + − − + − − Y1 , Yn Δn−1 = Y1 , Yn Δn−1 + Yn− Y1+ , Δ− n−1 n−1 − − − = 2M1n Δ− Yr− M1r n−1 + Yn 2(n−3)Y1 + 4Y1 H0 + 4 r=2

− )2 M1n = 2M1n (Y1− )2 + 2 (Y2− )2 + · · · + (Yn−1

n−1 + Yn− 2(n−3)Y1− + 4Y1− H0 + 4 Yr− M1r , r=2

i.e.,

Y1+ , Yn− Δ− n−1

= 2(n−4)Yn− Y1− − 2Y1− Yn− + 4Yn− Y1− H0 + 4Yn− ≡

n−1

Yr− M1r + 2Δ− n−1 M1n

r=2 2(n−4+2λ)Yn− Y1−

− 2Y1− Yn−

(6.18.14)

372

Chapter 6. Conformally Covariant Families

mod J(λ). Finally, we ﬁnd + − − − Y1 , Yj Yn Yj = [Y1+ , Yj− ]Yn− Yj− + Yj− [Y1+ , Yn− Yj− ] = 2M1j Yn− Yj− + 2Yj− (M1n Yj− + Y >− n M1j ) = 2Yn− Y1− + 2(Yn− Yj− + Yj− Yn− )M1j + 2(Yj− )2 M1n for j > 1 and + − − − + − − − Y1 , Y1 Yn Y1 [Y1 , Y1 ]Yn Y1 + Y1− [Y1+ , Yn− Y1− ] = 2H0 Yn− Y1− + 2Y1− (M1n Y1− + Yn− H0 ) = −4Yn− Y1− − 2Y1− Yn− + 2(Yn− Y1− + Y1− Yn− )H0 . Hence

< Y1+ ,

n−1

= Yj− Yn− Yj− ≡ 2(n−4+λ)Yn− Y1− + 2(λ − 1)Y1− Yn−

1

¯ 3 (λ)] is of the form mod J(λ). These results imply that [Y1+ , D 1 − (n−5+2λ) 2Y1− Yn− + 4Yn− Y1− 3 + 2[a(n−5+2λ) + b(n−4+λ)Yn− Y1− + b(λ−1)Y1− Yn− + c(n−4+2λ)Yn− Y1− − cY1− Yn− ], i.e., 1 2 − (n−5+2λ) + a(n−5+2λ) + b(λ−1) − c Y1− Yn− 3 2 + 2 − (n−5+2λ) + b(n−4+λ) + c(n−4+2λ) Yn− Y1− . 3 Now the condition 13 (n−5+2λ) = a(n−5 +2λ) + b(λ−1) − c is a consequence of 2 (n−5+2λ) = b(n−4+λ) + c(n−4+2λ) 3 and a + b + c = 1. The proof is complete.

Next, we discuss the fourth-order family Θ4 (λ) : Nλ−4 (gn ) → Nλ (gn+1 ) constructed by translation of Θ2 (λ). Theorem 6.18.5. The homomorphism Θ4 (λ) : Nλ−3 (gn ) → Nλ (gn+1 ) is induced by right multiplication with 3

1 (n−5+λ)(n−4+λ)D4T (λ), (n−9+2λ)(n−7+2λ)(n−4+2λ)(n−2+2λ)

6.18. Zuckerman translation and DN (λ)

373

where 1 − − 2 (n−5+2λ)(n−7+2λ) (Yn− )4 + (Yn− )2 Δ− n−1 − 2(n−5+2λ)Δn−1(Yn ) 3 λ−2 1 n−7+2λ 2 −(n−5+2λ)(Δ− (Δ− )2 + n−1 ) + (n−2+2λ)L4,n−1 − 3 n−5+λ n−5+λ n−1 n−1 1 (n−7+2λ)(n−5+2λ) − − −2 − 2 − (λ+3)Δn−1 Yn − (n−2+2λ) + Yj (Yn ) Yj . 3 n−5+λ 1 D4T (λ) =

Note that πD4T (λ) = D4 (λ). In other words, D4T (λ) is a lift of D4 (λ) which deﬁnes a homomorphism of semi-holonomic Verma modules. Although D4 (λ) is polynomial in λ, its lift has a simple pole at λ = 5 − n with residue (n−3)(n−8) Res5−n (D4T ) = − 3 = < n−1 − 2 − − − 2 − 2 − (Δn−1 ) − L4,n−1 + (n−5) Δn−1 (Yn ) − Yj (Yn ) Yj 1

and value D4T

n−5 1 2 (n−1)(Δ− − = n−1 ) − 4L4,n−1 2 n−5

(6.18.15)

− 2 for n = 5. We recall that D4 (− n−5 2 ) = (Δn−1 ) . The proof is a straightforward calculation and we omit the details. − 2 (6.18.15) shows why the lift D4T (− n−5 2 ) of (Δn−1 ) does not exist for n = 5. This is the case which is not covered by the translation in the category of Verma − 2 module: (Δ− 4 ) can not be constructed from Δ4 by translation. Instead, we have the rational family

2 − 2 (Δ4 ) − L4 λ < = 1 − 2 2 − − 2 − − 2 − + (Δ ) + L4 − 4 Δ4 (Y5 ) − Yi (Y5 ) Yi + ··· . 3 4 3 i

(6.18.16)

We can write D4T (λ) in the form 1 (n−5+2λ)(n−7+2λ)(Yn− )4 − 2(n−5+2λ) 3 8 ×

a(Yn− )2 Δ− n−1

+

− bYn− Δ− n−1 Yn

+

− 2 cΔ− n−1 (Yn )

+d

n−1

; Yj− (Yn− )2 Yj−

1

+

2 (1−α)(Δ− n−1 )

+ αL4,n−1 ,

(6.18.17)

374

Chapter 6. Conformally Covariant Families

where the coeﬃcients are given by 1 a = − (n−7+2λ), 6 b = 0, 1 n−7+2λ (λ+3), c=1− 6 n−5+λ 1 n−7+2λ d= (n−2+2λ), 6 n−5+λ 1 n−7+2λ 1 α= (n−2+2λ) (= d). 2 n−5+λ 2 In analogy to Theorem 6.18.4, D4T (λ) is an element in a one-parameter family of lifts of D4 (λ). ¯ 4 (λ) is a lift of ¯ 4 (λ) be of the form (6.18.17). Assume that D Theorem 6.18.6. Let D D4 (λ) =

1 − 2 (2λ+n−5)(2λ+n−7)(Yn− )4 − 2(2λ+n−5)(Yn− )2 Δ− n−1 + (Δn−1 ) . 3

Moreover, assume that the map ˜ 4 (λ) ⊗ 1 ∈ A(gn+1 ) ⊗ C(λ) A(gn ) ⊗ C(λ−4) T ⊗ 1 → i(T )D induces a homomorphism Nλ−4 (gn ) → Nλ (gn+1 ) of A(gn )-modules. Then a + b + c + d = 1, α = 2d, (n−5+λ)α + 2(n−5+2λ)a = n−7+2λ, 1 (n−6+2λ)b − 2a = (n−7+2λ). 3 ¯ 4 (λ) deﬁnes a raConversely, if (a, b, c, d) and α satisfy these conditions, then D tional family of homomorphisms of semi-holonomic Verma modules. The family D4T (λ) appears for b = 0. ¯ 4 (λ) = D4 (λ) is equivalent to a + b + c + d = 1 since n− is Proof. The condition π D abelian. As in the proof of Theorem 6.18.4, it is enough to analyze the condition ¯4 (λ)] ∈ J(λ) = A(gn+1 )(mn+1 ⊕ C(H0 −λ)). [Y1+ , D We determine the structure of the commutators Y1+ , (Yn− )4 ,

⎡

⎣Y1+ , Y1+ , (Yn− )2 Δ− n−1 ,

j

⎤ Yj− (Yn− )2 Yj− ⎦ , + − − − 2 Y1 , Yn Δn−1 Yn− , Y1+ , Δ− n−1 (Yn )

6.18. Zuckerman translation and DN (λ)

375

+ 2 and Y1+ , (Δ− n−1 ) , Y1 , L4,n−1 . Since

Y1+ , (Yn− )2 = Y1+ , Yn− Yn− + Yn− Y1+ , Yn− = 2M1n Yn− + 2Yn− M1n = 2Y1− + 4Yn− M1n

and M1n , (Yn− )2 = M1n , Yn− Yn− + Yn− M1n , Yn− = Y1− Yn− + Yn− Y1− we get + Y1 , (Yn− )4 = Y1+ , (Yn− )2 (Yn− )2 + (Yn− )2 Y1+ , (Yn− )2 = 2Y1− + 4Yn− M1n (Yn− )2 + (Yn− )2 2Y1− + 4Yn− M1n = 2Y1− (Yn− )2 + 4Yn− M1n , (Yn− )2 + 2(Yn− )2 Y1− + 8(Yn− )3 M1n ≡ 2Y1− (Yn− )2 + 4Yn− Y1− Yn− + 6(Yn− )2 Y1−

(6.18.18)

, we use the mod J(λ). In order to determine the commutator Y1+ , (Yn− )2 Δ− n−1 − − − − − identity M1n , Δn−1 = −(Yn Y1 + Y1 Yn ) which follows from M1n , (Yr− )2 = M1n , Yr− Yr− +Yr− M1n , Yr− =

8

0 r = 1 . − − − − −(Yn Y1 + Y1 Yn ) r = 1

Hence (6.18.12) yields + + − + − − 2 − 2 Y1 , (Yn− )2 Δ− n−1 = Y1 , (Yn ) Δn−1 + (Yn ) Y1 , Δn−1 = 2Y1− + 4Yn− M1n Δ− n−1 + (Yn− )2

2(n−3)Y1− + 4Y1− H0 + 4

n−1

Yr− M1r

r=2 − − − − 2 − − 2 − ≡ 2Y1− Δ− n−1 − 4Yn Y1 Yn + 2(n−5)(Yn ) Y1 + 4λ(Yn ) Y1 (6.18.19)

mod J(λ). Again, using (6.18.12), we see that + − − 2 + − − 2 Y1 , Δn−1 (Yn− )2 = Y1+ , Δ− n−1 (Yn ) + Δn−1 Y1 , (Yn ) n−1 − − − = 2(n−3)Y1 + 4Y1 H0 + 4 Yr M1r (Yn− )2 r=2 − − + Δ− n−1 (2Y1 + 4Yn M1n ) − ≡ 2(n−7)Y1− (Yn− )2 + 4λY1− (Yn− )2 + 2Δ− n−1 Y1

(6.18.20)

376

Chapter 6. Conformally Covariant Families

− mod J(λ). Next, (6.18.12) and [H0 , Δ− n−1 ] = −2Δn−1 yield + − − + − + − 2 Y1 ,(Δ− n−1 ) = Y1 ,Δn−1 Δn−1 + Δn−1 Y1 ,Δn−1 − − = 2(n−3)Y1− Δ− n−1 + 4Y1 H0 Δn−1 + 4

n−1

Yr− M1r Δ− n−1

r=2 − − − − + 2(n−3)Δ− n−1 Y1 + 4Δn−1 Y1 H0 + 4Δn−1

n−1

Yr− M1r

r=2

− − − − − − ≡ 2(n−7)Y1− Δ− n−1 + 2(n−3)Δn−1 Y1 + 4λ Y1 Δn−1 + Δn−1 Y1 (6.18.21) mod J(λ). (6.18.14) and [M1n , Yn− ] = Y1− imply + − − − + − − − Y1 , Yn Δn−1 Yn− = Y1+ , Yn− Δ− n−1 Yn + Yn Δn−1 Y1 , Yn = 2(n−4)Yn− Y1− − 2Y1− Yn− n−1

− + 4Yn− Y1− H0 + 4Yn− Yr− M1r + 2Δ− n−1 M1n Yn r=2

+ 2Yn− Δ− n−1 M1n − ≡ 2(n−6)Yn− Y1− Yn− − 2Y1− (Yn− )2 + 4λYn− Y1− Yn− + 2Δ− n−1 Y1 (6.18.22)

mod J(λ). It remains to determine the commutators

− Y1+ , Y1+ , Yj− (Yn− )2 Yj− , Yj− Δ− n−1 Yj . j

j

We ﬁnd [Y1+ , L4,n−1 ] − − − ≡ (2n−6+2λ)Δ− n−1 Y1 + (2λ−4)Y1 Δn−1 + 2(n−5+2λ)

Yj− Y1− Yj−

j

mod J(λ) and

Y1+ , Yj− (Yn− )2 Yj− j

≡ 2(n−5+λ)(Yn− )2 Y1− + (2λ−4)Y1− (Yn −)2 + 2

Yj− Y1− Yj−

j

¯ 4 (λ)] is a linear combinamod J(λ). Now it follows that the commutator [Y1+ , D tion of Y1− (Yn− )2 ,

Yn− Y1− Yn− ,

(Yn− )2 Y1− ,

Y1− Δ− n−1 ,

− Δ− n−1 Y1

6.18. Zuckerman translation and DN (λ)

377

− − − and j Yj Y1 Yj modulo J(λ). The vanishing of the resulting six coeﬃcients implies the six conditions listed in Table 6.5. Here we use the notation a0 (λ) =

1 (n−5+2λ)(n−7+2λ), a1 (λ) = −2(n−5+2λ). 3

term

condition

1

Y1− (Yn− )2

2a0 + a1 (−2b + 2c(n−7+2λ) + 2d(λ−2)) = 0

2

Yn− Y1− Yn− (Yn− )2 Y1− Y1− Δ− n−1 Δ− Y1− n−1 − − − j Yj Y1 Yj

4a0 + a1 (−4a + 2b(n−6+2λ)) = 0

3 4 5 6

6a0 + a1 (2a(n−5+2λ) + 2d(n−5+λ)) = 0 2a1 a + 2(1−α)(n−7+2λ) + 2α(λ−2) = 0 2a1 (b + c) + 2(1−α)(n−3+2λ) = 0 2a1 d + 2α(n−5+2λ) = 0.

Table 6.5: The vanishing conditions We analyze the conditions in Table 6.5 together with a + b + c + d = 1. The condition 6 means 2d = α. The ﬁrst three and the last three conditions are linearly dependent if a + b + c + d = 1; that can be seen immediately by adding these sets of equations. Now condition 4 is equivalent to 2a(n−5+2λ) + α(n−5+λ) = (n−7+2λ),

(6.18.23)

which, in turn, is equivalent to condition 3 using 2d = α. Thus all conditions are satisﬁed if we choose a + b + c + d = 1, α = 2d, (6.18.23) and condition 2. It is worth emphasizing a special case. The relations − − − [Yj+ , L4,n ] ≡ (2n−4+2λ)Δ− n Yj + (2λ−4)Yj Δn + 2(n−4+2λ)

Yj− Y1− Yj−

j

and

+ 2 − − ≡ 2(n−6+2λ)Yj− Δ− Yj , (Δ− n) n + 2(n−2+2λ)Δn Yj

mod J(λ) for j = 1, . . . , n show that for λ = − n2 + 2 the linear combination 2 B4,n = n(Δ− n ) − 4L4,n 2 satisﬁes [Yj+ , B4,n ] ≡ 0 mod J(− n2 + 2). But πB4,n = (n − 4)(Δ− n ) , i.e., B4,n deﬁnes a homomorphism N− n2 −2 → N− n2 +2 which lifts the intertwining operator 2 M− n2 −2 → M− n2 +2 of Verma modules deﬁned by right multiplication with (Δ− n) . − It is this operator which is the translation of Δn in the semi-holonomic category. If n = 4, then B4,n induces the Paneitz operator (n − 4)P4,n on C ∞ (M n ). For n = 4, the product (n−4)P4,n vanishes although B4,n is non-trivial.

378

Chapter 6. Conformally Covariant Families

Theorem 6.18.7. The family Θ2N (λ) : Nλ−2N (gn ) → Nλ (gn+1 ) (N ≥ 0) is induced by right multiplication with ⎧ ⎫ 2N N ⎨ ⎬ 1 1 c2N ⎩ n−1−2j +2λ j=1 n−2j +2λ ⎭ j=N +1

×

N

T (n−1−(j +N )+λ) D2N (λ),

(6.18.24)

j=1 T where D2N (λ) is a rational family with the property T π ◦ D2N (λ) = D2N (λ), T T (λ) lifts D2N (λ). Here c2N = (2N −1)(2N −3) · · · 1. The poles of D2N (λ) i.e., D2N are at the zeros of the product N

(n−1−(j +N )+λ),

j=2

i.e., at λ ∈ {−n+N +3, . . . , −n+1+2N }. These N −1 poles are (at most ) simple. T (λ) has (at most ) a simple pole In particular, for odd n, the critical family Dn−1 at λ = 0. Note that in (6.18.24) the zeros of the denominator are disjoint from the zeros of the nominator. There is no pole at the zero of (n−2−N +λ) since D2T (λ) has no pole. Since we do not give formulas for the residues, it is not excluded that some of these vanish. For an explicit example we refer to Theorem 6.18.5 and the discussion following it. Notice that for λ = − n−1 2 + N , the identity (6.18.24) reads N (−1)N n−1 n−1 n−1 T +N = c2N −j D2N − +N . Θ2N − 2 (2N )! j=1 2 2 − n−1 T N Hence using πD2N (− n−1 2 +N ) = D2N (− 2 +N ) = (Δn−1 ) , we ﬁnd N (−1)N n−1 n−1 N +N = (2N −1) · · · 3 · 1 −j (Δ− πΘ2N − n−1 ) 2 (2N )! j=1 2

=

N (−1)N 1 N (j − )(n−1−2j)(Δ− n−1) (2N )! j=1 2

(6.18.25)

for the N th translation of i : gn → gn+1 in the holonomic category. In particular, for odd n πΘn−1 (0) = 0. (6.18.25) should be compared with the formula N j=2

(j −1)(n−1−2j)ΔN n−1

6.18. Zuckerman translation and DN (λ)

379

for an analogous composition of operators in [116]. Since the latter product vanishes for 2N = n − 1, it does not yield the critical GJMS-operator in dimension n − 1 even in the ﬂat case. Proof. We use induction on N . The assertion is valid for N = 0. In the following, we identify operators with multiplicators. We assume that πΘ2N (λ)(1 ⊗ 1) ⎧ ⎫ 2N N N ⎨ ⎬ 1 1 (n−1−(j +N )+λ)D2N (λ). = c2N ⎩ n−1−2j +2λ j=1 n−2j +2λ ⎭ j=1 j=N +1

We claim that πΘ2N +2 (λ)(1 ⊗ 1) is of the form − N +1 . αN +1 (λ)(Yn− )2N +2 + αN (λ)(Yn− )2N Δ− n−1 + · · · + α0 (λ)(Δn−1 )

Since πΘ2N +2 (λ) is an intertwining operator of Verma modules, we can apply Theorem 5.1.1. It follows that πΘ2N +2 (λ)(1 ⊗ 1) is the product of D2N +2 (λ) with a rational function in λ. It only remains to identify the latter rational multiplier. In order to prove the claim, we ﬁrst notice that πΘ2N +2 (λ)(1 ⊗ 1) = p(λ)πΘ2N (λ−1)i(λ−(2N +1))(1 ⊗ 1) (by deﬁnition) ⎫ ⎧ 2N +1 N +1 N ⎬ ⎨ 1 1 = c2N (n−1−(j +N + 1)+λ) ⎩ n−1−2j +2λ n−2j +2λ ⎭ j=N +2

j=2

j=1

× p(λ) (D2N (λ−1) ⊗ In ) i(λ−(2N +1))(1 ⊗ 1). Hence it is enough to prove that p(λ) (D2N (λ−1) ⊗ In ) i(λ−(2N +1))(1 ⊗ 1) is of the form 1 1 N +1 αN +1 (λ)(Yn− )2N +2 + · · · + α0 (λ)(Δ− , n−1 ) n−1−2(2N +2)+2λ n−2+2λ (6.18.26) where i(λ) and p(λ) are deﬁned by i(λ)(1 ⊗ 1) = −λ(1 ⊗ 1 ⊗ v− ) +

n−1

Yi− ⊗ 1 ⊗ vi −

1

1 (Δ− ⊗ 1 ⊗ v+ ) 2(n−3+2λ) n−1

and − 1 Δn ⊗ 1 , n−2+2λ p(λ)(1 ⊗ 1 ⊗ vi ) = Yi− ⊗ 1 (i = 1, . . . , n),

p(λ)(1 ⊗ 1 ⊗ v− ) =

p(λ)(1 ⊗ 1 ⊗ v+ ) = 2λ ⊗ 1

380

Chapter 6. Conformally Covariant Families

(notice the diﬀerent dimensions). Now (D2N (λ−1) ⊗ In ) i(λ−(2N +1))(1 ⊗ 1) is a linear combination of elements of the form a (Yn− )2N −2a (Δ− n−1 ) ⊗ 1 ⊗ v− , n−1

a Yi− (Yn− )2N −2a (Δ− n−1 ) ⊗ 1 ⊗ vi ,

a+1 (Yn− )2N −2a (Δ− ⊗ 1 ⊗ v+ . n−1 )

1

These elements can be written as linear combinations of elements of the form n−1

a (Yn− )2N −2a (Δ− n−1 ) (1 ⊗ 1 ⊗ v− ),

a Yi− (Yn− )2N −2a (Δ− n−1 ) (1 ⊗ 1 ⊗ vi ),

1 a+1 (1 ⊗ 1 ⊗ v+ ). (Yn− )2N −2a (Δ− n−1 )

(6.18.27)

Now applying p(λ), yields an element of the form (6.18.26). Next, we prove that the proportionality coeﬃcient is as asserted. We use induction on N . Assuming the validity of (6.18.24)2N for Θ2N , it suﬃces to prove that the coeﬃcient of (Yn− )2N +2 in Θ2N +2 coincides with the corresponding coefﬁcient given by (6.18.24)N +1 . Now contributions of the form (Yn− )2N +2 can arise only from contributions of (Yn− )2N (1 ⊗ v− ) in − (λ−2N −1)D2N (λ−1) ⊗ 1 ⊗ v− +

n−1

Yi− D2N (λ−1) ⊗ 1 ⊗ vi −

i=1

1 Δ− D2N (λ−1) ⊗ 1 ⊗ v+ . 2(n−3 −2(2N +1)+2λ) n−1 (6.18.28)

In order to determine these contributions, we write each term as a linear combination of terms as in (6.18.27). The ﬁrst term equals −(λ−2N−1)D2N (λ−1)(1⊗1⊗v−), i.e., it contributes −(λ−2N −1)A2N (λ−1)(Yn− )2N (1 ⊗ 1 ⊗ v− ), where A2N (λ) is the coeﬃcient of (Yn− )2N in D2N (λ). The sum in (6.18.28) contributes −(n−1)A2N (λ−1)(Yn− )2N (1 ⊗ 1 ⊗ v− ). Finally, the third term in (6.18.28) yields the contribution −

2(n−1) A2N (λ−1)(Yn− )2N (1 ⊗ 1 ⊗ v− ). 2(n−3 −2(2N +1)+2λ)

Thus we ﬁnd − (n−2−2N +λ) + =−

n−1 A2N (λ−1)(Yn− )2N (1 ⊗ 1 ⊗ v− ) n−5−4N +2λ

1 (n−3−2N+λ)(n−3+2λ−4N )A2N (λ−1)(Yn− )2N (1 ⊗ 1 ⊗ v−). n−5−4N +2λ

6.19. From Verma modules to tractors

381

An easy calculation shows that (5.1.3) implies the relation −

1 (n−3+2λ−4N )A2N (λ−1) = A2N +2 (λ). 2N +1

Hence the coeﬃcient of (Yn− )2N +2 (1 ⊗ 1 ⊗ v− ) in (6.18.28) is 1 (2N +1)(n−3−2N +λ)A2N +2(λ). n−5−4N +2λ Now we apply p(λ) and ﬁnd that the coeﬃcient of (Yn− )2N +2 in π◦Θ2N +2 (λ)(1⊗1) is given by ⎫ ⎧ +2 N +1 ⎬ ⎨ 2N 1 1 (2N +1)c2N ⎩ n−1−2j +2λ n−2j +2λ ⎭ j=1

j=N +2

×

N +1

(n−1−(j +N +1)+λ)A2N +2(λ).

j=1

Since c2N +2 = (2N+1)c2N , this product coincides with the coeﬃcient of (Yn− )2N +2 in (6.18.24)N +1 . The proof is complete. There is an analogous result for odd order families. Theorem 6.18.8. The family Θ2N +1 (λ) : Nλ−2N −1 (gn ) → Nλ (gn+1 ) (N ≥ 0) is induced by right multiplication with ⎧ ⎫ 2N N N ⎨ ⎬ 1 1 T c2N +1 (n−2−(j +N )+λ)D2N +1(λ), ⎩ n−3−2j +2λ j=1 n−2j +2λ ⎭ j=1 j=N +1

(6.18.29) T where D2N +1 (λ) is a rational family with the property T π ◦ D2N +1 (λ) = D2N +1 (λ), T T i.e., D2N (λ) lifts D2N +1 (λ). The poles of D2N +1 (λ) are at the zeros of the >+1 N product j=1 (n−2−(j+N )+λ), i.e., at λ ∈ {−n+N+3, . . . , −n+2+2N }. These N T poles are (at most ) simple. In particular, for even n, the critical family Dn−1 (λ) has (at most ) a simple pole at λ = 0.

Theorem 6.18.3 and the discussion following it illustrate the general result.

6.19 From Verma modules to tractors The projection p(λ) : Mλ−1 ⊗ F → Mλ , which is given by 1 ⊗ 1 ⊗ v− →

1 (Δ− ⊗ 1), n−2+2λ n

1 ⊗ 1 ⊗ vi → Yi− ⊗ 1,

1 ⊗ 1 ⊗ v+ → 2λ ⊗ 1,

382

Chapter 6. Conformally Covariant Families

induces a left G-equivariant diﬀerential operator D(λ) : C ∞ (G, C(λ))P → (C ∞ (G, C(λ−1)) ⊗ F ∗ )P by sending u to

(6.19.1)

F f → p(λ)(1 ⊗ 1 ⊗ f )(u) ∈ C ∞ (G),

where U(g) acts on C ∞ (G) by the right regular representation R. In explicit terms, we ﬁnd 1 ∗ ∗ D(λ) : u → R(Δ− R(Yi− )u ⊗ vi∗ + 2λu ⊗ v+ (6.19.2) n )u ⊗ v− + n−2+2λ i ∗ using a dual basis {v± , vj∗ } of F ∗ . It is instructive to verify the mapping properties of D(λ) directly. The relation − ∗ ∗ π ∗ (m)D(λ)u(gm) = u(g) for m ∈ M follows from Ad(m)Δ− n = Δn , π (m)v± = − ∗ − v± and the fact that Yj → vj is an M -equivariant isomorphism n → V0 . It remains to prove that

R(H0 )D(λ)u + π ∗ (H0 )D(λ)u = (λ−1)D(λ)u,

R(Yj+ )D(λ)u + π ∗ (Yj+ )D(λ)u = 0

for j = 1, . . . , n. In order to prove the second set of conditions, we calculate the sum 1 ∗ ∗ R(Yj+ Δ− R(Yj+ Yi− )u ⊗ vi∗ + 2λR(Yj+ )u ⊗ v+ n )u ⊗ v− + n−2+2λ i 1 + ∗ ∗ R(Δ− + R(Yi− )u ⊗ Yj+ (vi∗ ) + 2λu ⊗ Yj+ (v+ ) n )u ⊗ Yj (v− ) + n−2+2λ i for u ∈ C ∞ (G, C(λ))P , i.e., R(Yj+ )u = 0, R(H0 )u = λu, R(m)u = u. The identities ∗ ∗ ∗ ) = 0, Yj+ (v+ ) = −vj∗ , Yj+ (vi∗ ) = −2δij v− Yj+ (v− show that the sum equals 1 ∗ ∗ R([Yj+ , Δ− R([Yj+ , Yi− ])u ⊗ vi∗ − 2R(Yj− )u ⊗ v− − 2λu ⊗ vj∗. n ])u ⊗ v− + n−2+2λ i Now the commutator relations + − Yj , Δn = 2(n−2)Yj− + 4Yj− H0 + 4 Yr− Mjr r=j

(see (6.18.3)) and [Yj+ , Yi− ] = 2δij H0 + 2Mji simplify that sum to 1 ∗ 2(n−2)R(Yj− )u + 4λR(Yj− )u ⊗ v− n−2+2λ ∗ + 2λu ⊗ vj∗ − 2R(Yj− )u ⊗ v− − 2λu ⊗ vj∗ = 0.

6.19. From Verma modules to tractors

383

∗ ∗ In order to prove the ﬁrst set of conditions, we use H0 (v± ) = ∓v± , H0 (vi∗ ) = 0 and − − − − [H0 , Δn ] = −2Δn , [H0 , Yi ] = −Yi . It follows that R(H0 )D(λ)u + π ∗ (H0 )D(λ)u equals

1 − ∗ ∗ (R(H0 Δ− R(H0 Yi− )u⊗vi∗ +2λ(R(H0 )u−u)⊗v+ , n )u+R(Δn )u)⊗v− + n−2+2λ i i.e., λ−1 ∗ ∗ R(Δ− R(Yi− )u ⊗ vi∗ + 2λ(λ−1)u ⊗ v+ n )u ⊗ v− + (λ−1) n−2+2λ i = (λ−1)D(λ)u. Similarly, the embedding i(λ) : Mλ−1 → Mλ ⊗ F deﬁned by 1 ⊗ 1 → −λ ⊗ 1 ⊗ v− +

Yi− ⊗ 1 ⊗ vi −

i

1 (Δ− ⊗ 1 ⊗ v+ ) 2(n−2+2λ) n

induces a left G-equivariant diﬀerential operator C(λ) : (C ∞ (G, C(λ)) ⊗ F ∗ )P → C ∞ (G, C(λ−1))P

(6.19.3)

by u → i(λ)(1 ⊗ 1)(u). Here U(g) ⊗ F acts on the tensor product module C ∞ (G) ⊗ F ∗ by T ⊗ f : u ⊗ f ∗ → T (u ⊗ f ∗ ), f , where C ∞ (G) is viewed as a right G-module. In order to determine an explicit formula, we ﬁrst calculate how Yj− ⊗ vj and Δ− n ⊗ v+ act on ∗ + u = u− ⊗ v−

∗ ui ⊗ vi∗ + u+ ⊗ v+ ∈ C ∞ (G) ⊗ F ∗ .

i

We ﬁnd 4 3 4 3 ∗ ∗ ∗ (Yj− ⊗vj )(u− ⊗v− ) = Yj− (u− ⊗ v− ), vj = Yj− (u− ) ⊗ v− − u− ⊗ vj∗ , vj = −u− , (Yj− ⊗ vj )(ui ⊗ vi∗ ) 4 3 4 3 ∗ , vj = δij Yj− (ui ) = Yj− (ui ⊗ vi∗ ), vj = Yj− (ui ) ⊗ vi∗ − 2ui ⊗ δij v+ and 4 3 4 3 ∗ ∗ ∗ (Yj− ⊗ vj )(u+ ⊗ v+ ) = Yj− (u+ ⊗ v+ ), vj = Yj− (u+ ) ⊗ v+ , vj = 0. Here we have used ∗ Yj− (v− ) = −vj− ,

∗ Yj− (vi∗ ) = −2δij v+ ,

∗ Yj− (v+ ) = 0.

384

Chapter 6. Conformally Covariant Families

Hence

(Yj− ⊗ vj )(u) = −nu− +

j

Yi− (ui ).

i

By deﬁnition, − ∗ (Δ− n ⊗ v+ )(u) = Δn (u− ⊗ v− ), v+ +

∗ − ∗ Δ− n (ui ⊗ vi ), v+ + Δn (u+ ⊗ v+ ), v+ .

i

Thus using ∗ ∗ (Yj− )2 (u− ⊗ v− ) = (Yj− )2 (ui ) ⊗ v− − 2Yj− (u− ) ⊗ vj∗ + 2u− ⊗ V+∗ , ∗ , (Yj− )2 (ui ⊗ vi∗ ) = (Yj− )2 (ui ) ⊗ vi∗ − 4Yj− (ui ) ⊗ δij v+ ∗ ∗ (Yj− )2 (ui ⊗ v+ ) = (Yj− )2 (ui ) ⊗ v+ ,

we ﬁnd

(Δ− n ⊗ v+ )(u) = 2nu− − 4

Yi− (ui ) + Δ− n (u+ ).

i

Finally, we have (λ ⊗ v− )(u) = λu− . Therefore, C(λ)u = −λu− − nu− +

Yi− (ui )

i

=

1 − 2(n−2+2λ)

2nu− − 4

Yi− (ui )

+

Δ− n (u+ )

i

" # 1 − 2(n+2λ)(n+λ−1)u− + 2(n+2λ) Yi− (ui ) − Δ− n (u+ ) . 2(n−2+2λ) i (6.19.4)

It is instructive to verify the mapping properties of C(λ) directly. The P invariance of u ∈ C ∞ (G, C(λ)) ⊗ F ∗ means that ∗ ∗ ∗ Yj+ (u− ) ⊗ v− + Yj+ (ui ) ⊗ vi∗ + Yj+ (u+ ) ⊗ v+ − 2uj ⊗ v− − u+ ⊗ vj∗ = 0, i

i.e., Yj+ (u+ ) = 0, Yj+ (ui ) = δij u+ , Yj+ (u− ) = 2uj . Moreover, ∗ + H0 (u− ) ⊗ v−

∗ ∗ ∗ H0 (ui ) ⊗ vi∗ + H0 (u+ ) ⊗ v+ + u− ⊗ v− − u+ ⊗ v+

i ∗ = λ(u− ⊗ v− +

i

i.e., H0 (u± ) = (λ ± 1)u± , H0 (ui ) = λui and m(u± ) = u± , Mji (ui ) = uj .

∗ ui ⊗ vi∗ + u+ ⊗ v+ )

6.19. From Verma modules to tractors

385

Now using these results, we ﬁnd that 2(n−2+2λ)Yj+ C(λ)u equals − 2(n+2λ)(n+λ−1)Yj+ (u− ) + 2(n+2λ)

Yj+ Yi− (ui ) − Yj+ Δ− n (u+ )

i

= −4(n+2λ)(n+λ−1)uj + 2(n+2λ)

[Yj+ , Yi− ](ui )

+

Yi− Yj+ (ui )

i

− [Yj+ , Δ− n ](u+ ) i.e., − 4(n+2λ)(n+λ−1)uj + 2(n+2λ) (n−1)uj + 2λuj + Yj− (u+ ) − 2(n−2)Yj− (u+ ) − 4(λ+1)Yj− (u+ ) = 0 using (6.18.3). Similarly, it follows that H0 C(λ)u = (λ−1)C(λ)u and mC(λ)u = C(λ)u. In other words, we have proved C(λ)u ∈ C ∞ (G, C(λ−1))P . p(λ) and i(λ) also induce twisted versions of D(λ) and C(λ) which involve an additional G-module (W, τ ). Let ∞ ∞ ∇W X : C (G) ⊗ W → C (G) ⊗ W, u → (R ⊗ τ )(X)(u)

and ΔW =

∇W ◦ ∇W , Y− Y− j

j

j

i.e., ΔW (u ⊗ w) = R(Δ− n )u ⊗ w + 2

R(Yj− )u ⊗ τ (Yj− )w + u ⊗ τ (Δ− n )w.

j

Lemma 6.19.1. p(λ) induces a left G-equivariant operator DW (λ) : (C ∞ (G, C(λ)) ⊗ W )P → (C ∞ (G, C(λ−1)) ⊗ W ⊗ F ∗ )P by u → (F ∈ f → p(λ)(1 ⊗ 1 ⊗ f )u). In explicit terms it is given by u →

1 ∗ ∗ ΔW u ⊗ v− + ∇W u ⊗ vi∗ + 2λu ⊗ v+ Yi− n−2+2λ i

In order to verify the mapping properties of DW (λ), the same arguments as in the untwisted case apply by replacing the G-module C ∞ (G) by C ∞ (G) ⊗ W . In fact, the arguments only rest on the commutator relations. In order to work out the case W = F ∗ , we write an element of C ∞ (G) ⊗ F ∗ in the form ∗ ∗ u = u− ⊗ v− + ui ⊗ vi∗ + u+ ⊗ v+ . i

386

Chapter 6. Conformally Covariant Families

We ﬁnd ∗

∗ ∇F (u) = Yj− (u− ) ⊗ v− + Y−

j

∗ ∗ Yj− (ui ) ⊗ vi∗ + Yj− (u+ ) ⊗ v+ − u− ⊗ vj∗ − 2uj ⊗ v+

i

and ∗

∗ ΔF (u) = Δ− n (u− ) ⊗ v− +

−2

∗ − ∗ Δ− n (ui ) ⊗ vi + Δn (u+ ) ⊗ v+

i

Yj− (u− ) ⊗ vj∗ − 4

j

∗ ∗ Yj− (ui ) ⊗ δij v+ + 2nu− ⊗ v+ .

i,j

It is convenient to write the latter formulas in terms of matrices as ⎞⎛ ⎞ ⎛ ⎞ ⎛ − 0 0 Yj u− u − ∗ ⎠⎝ ω ⎠, ⎝ ω ⎠ → ⎝−ej Yj− 0 ∇F − : Yj u+ u+ 0 −2etj Yj− and

(6.19.5)

⎞ ⎞⎛ ⎞ ⎛ − 0 0 u− u− Δn ⎠ ⎝ Δ− 0 ω ⎠, ΔF : ⎝ ω ⎠ → ⎝−2Y − (6.19.6) n u+ 2n −4(Y − )t Δ− u + n ⎛ ⎞ ⎛ −⎞ u1 Y1 ⎜ ⎟ ⎜ ⎟ where ω = ⎝ ... ⎠ and Y − = ⎝ ... ⎠. In the following sections, these constructions ⎛

∗

un Yn− will be recognized as the ﬂat case of the tractor connection and its associated tractor Laplacian (see Deﬁnition 6.20.1 and formula (6.22.2)). Corollary 6.18.1 has the following counterpart in terms of C(λ) and D(λ). ∗

Lemma 6.19.2. C(λ) = trF ∗ ◦DF (λ), where trF ∗ : F ∗ ⊗ F ∗ → C denotes the contraction deﬁned by the scalar product ⎛ ⎞ ⎛ ⎞ τ− μ− 1 (⎝ τ ⎠ , ⎝ μ ⎠) = − (τ− μ+ + τ+ μ− ) + (τ, μ) 2 τ+ μ+ ∗ ∗ ∗ , vi , v+ . in terms of the basis v− Proof. A routine calculation shows that the scalar product is induced by the scalar product of F R1,n+1 . Now for u ∈ (C ∞ (G, C(λ)) ⊗ F ∗ )P , we have ∗

DF (λ)u =

∗ ∗ 1 ∗ ∗ ΔF u ⊗ v− + ∇F u ⊗ vi∗ + 2λu ⊗ v+ Yi− n−2+2λ i

6.19. From Verma modules to tractors

387

by Lemma 6.19.1. Now (6.19.5) and (6.19.6) imply ⎛ ⎞ ∗ 1 1 ⎝2nu− − 4 ⎠ Yj− (ωj ) + Δ− tr(DF (λ)u) = − n u+ 2 n−2+2λ j (−u− + Yj− (ωj )) − λu− = C(λ)u + j

using (6.19.4). The proof is complete. The relation n

n n−4 − 2 p − −1 ◦ (Δ− +2 (Δn ) , =− ⊗ I) ◦ i − n 2 2 8

(6.19.7)

2 − i.e., the fact that (Δ− n ) is the Zuckerman translate of Δn (see Theorem 6.18.1), induces analogous relations for diﬀerential operators. In the ﬂat case (Rn , gc ), let ⎞ ⎛ 1

n 2 Δn u D − +2 u = ⎝ du ⎠ 2 (4−n)u

and

⎛ ⎞ n

u−

1 1 n ⎝ ⎠ ω =− C − −1 −2 u− + 2δω − Δn u+ . 2 2 2 4 u+

In order to write the analog of (6.19.7), we use the operator ⎛ ⎞ 0 0 Δn ΔTn = ⎝−2d Δn 0 ⎠ 2n −4δ Δn

(6.19.8)

on the space C ∞ (Rn ) ⊕ Ω1 (Rn ) ⊕ C ∞ (Rn ) (this is the ﬂat version of (6.22.2)). Lemma 6.19.3.

n n n−4 2 Δn . C − −1 ◦ ΔTn ◦ D − +2 = − 2 2 8

Proof. We notice that

⎛1 2⎞ n

2 Δn ΔTn ◦ D − +2 = ⎝ 0 ⎠ . 2 0

Now the result follows by a direct calculation.

Lemma 6.19.3 is the ﬂat special case of a formula which yields the Paneitz operator P4 as a composition of operators on tractors (Theorem 6.20.7, Corollary 6.20.3).

388

Chapter 6. Conformally Covariant Families

6.20 Some elements of tractor calculus In the present section, we discuss some basic ingredients of tractor calculus. The results will be used in Section 6.21 for the construction of the tractor families, which are conformally covariant curved versions of the constructions in Section 6.19. The following presentation is self-contained. For more information on tractor calculus we refer to [17], [88], [109] and [54]. The main objects will be the conformally invariant tractor connection ∇T and the conformally invariant tractor D-operator D(λ). The latter is conformally invariant as an operator on sections of certain vector bundles. Both the operator and the bundles depend on the parameter λ. By choosing appropriate trivializations of the bundles, the operator corresponds to a conformally covariant family of operators which acts on functional spaces which are independent of λ. We identify the pair consisting of ⎛ ⎞ u− ⎝ ω ⎠ ∈ C ∞ (M ) ⊕ Ω1 (M ) ⊕ C ∞ (M ) u+ and the metric g with the pair consisting of ⎛ ⎞ u− T (g, ϕ) ⎝ ω ⎠ ∈ C ∞ (M ) ⊕ Ω1 (M ) ⊕ C ∞ (M ) u+ and the metric e2ϕ g. Here ⎛ −ϕ e def T (g, ϕ) = ⎝

⎞⎛

eϕ

1 (dϕ, ·) ⎠ ⎝0 1 eϕ 0 0

⎞

1 2 4 |dϕ| 1 ⎠, 2 dϕ

(6.20.1)

1

where (·, ·) and | · | refer to g, of course. In view of T (e2ψ g, ϕ) ◦ T (g, ψ) = T (g, ϕ + ψ), this deﬁnes an equivalence relation. The equivalence classes are called dual tractors. Alternatively, dual tractors can be deﬁned as sections of the dual standard tractor bundle T ∗ M on M . T ∗ M is an invariant of the conformal class c containing g. It can be regarded as an associated vector bundle for the P -principal ﬁbre bundle G → M (associated to c). The deﬁning representation is the restriction to P of the standard representation of G on F ∗ . The choice of a metric g ∈ c deﬁnes a trivialization of T ∗ M and an isomorphism Γ(T ∗ M ) C ∞ (M ) ⊕ Ω1 (M ) ⊕ C ∞ (M ). Such identiﬁcations will be used throughout without further notice and, abusing notation, Γ(T ∗ M ) will also denote the direct sum.

6.20. Some elements of tractor calculus

389

The following result is a consequence of the transformation rule (2.5.7) for scalar curvature. Theorem 6.20.1 (Tractor D-operator). The rational family ⎛ 1 ⎞ n−2+2λ (Δg +λJ(g))u ⎠ ∈ Γ(T ∗ M ) DM (g; λ) : C ∞ (M ) u → ⎝ du 2λu

(6.20.2)

is conformally covariant in the sense that g ; λ) ◦ eλϕ = e(λ−1)ϕ ◦ TM (g, ϕ) ◦ DM (g; λ), DM (ˆ

gˆ = e2ϕ g

(6.20.3)

∞

for all ϕ ∈ C (M ). Proof. We start by proving the important identity ˆ + λˆJ)(eλϕ u) e−(λ−2)ϕ (Δ = (Δ + λJ)u + (n−2+2λ)(du, dϕ) + λ

n−2+2λ |dϕ|2 u. (6.20.4) 2

In fact, by Lemma 4.2.1 and (2.5.7), the left-hand side equals Δu + λδ(udϕ) + (n−2+λ)(dϕ, du) + λ(n−2+λ)|dϕ|2 u + λ(J−Δϕ−

n−2 |dϕ|2 )u 2

n−2+2λ |dϕ|2 u + (n−2+λ)(dϕ, du) 2 n−2+2λ |dϕ|2 u. = (Δ+λJ)u + (n−2+2λ)(dϕ, du) + λ 2

= (Δ+λJ)u + λ(δ(udϕ) − uΔϕ) + λ

Notice that, in contrast to Chapter 4, we use here the convention that −Δ is the non-negative Laplacian and −δ is adjoint to d. It follows that e−(λ−2)ϕ

1 1 1 ˆJ)(ueλϕ ) = ˆ (Δ+λ (Δ+λJ)u + (dϕ, du) + λ|dϕ|2 u. n−2+2λ n−2+2λ 2

In other words, the ﬁrst components of both sides of (6.20.3) coincide. For the second components, (6.20.3) asserts the obvious identity d(eλϕ u) = eλϕ (du + λudϕ). The proof is complete. It is natural to interpret (6.20.3) as stating that DM (λ) deﬁnes a conformally invariant operator from densities to dual tractors coupled with densities. A disadvantage of such a formulation, however, is that the functional spaces depend on the parameter λ. The above choice of trivializations of all bundles avoids such complications. (6.20.1) gives rise to a representation of the conformal group of (M, g). Let κ be a conformal diﬀeomorphism of g, i.e., κ∗ (g) = e2Φ g for some Φ ∈ C ∞ (M ). We deﬁne the operator ηλ (κ) = T −1 (g, Φ) ◦ e−λΦ ◦ κ∗ , λ ∈ C

(6.20.5)

390

Chapter 6. Conformally Covariant Families

on C ∞ (M ) ⊕ Ω1 (M ) ⊕ C ∞ (M ). Then ηλ (κ1 )ηλ (κ2 ) = ηλ (κ1 κ2 ). In fact, let Φ1 and Φ2 correspond to κ1 and κ2 . Then ηλ (κ1 )ηλ (κ2 ) = T −1 (g, Φ1 )e−λΦ1 (κ1 )∗ T −1 (g, Φ2 )e−λΦ2 (κ2 )∗ = e−λ(Φ1 +(κ1 )∗ Φ2 ) T −1 (g, Φ1 )(κ1 )∗ T −1 (g, Φ2 )(κ2 )∗ . But using

T (κ∗ (g), κ∗ (ϕ)) = κ∗ ◦ T (g, ϕ) ◦ κ∗ ,

we simplify the right-hand side to e−λ(Φ1 +(κ1 )∗ Φ2 ) T −1 (g, Φ1 )T −1 ((κ1 )∗ (g), (κ1 )∗ (Φ2 )) (κ1 κ2 )∗ = e−λ(Φ1 +(κ1 )∗ Φ2 ) T −1 (g, Φ1 + (κ1 )∗ Φ2 ) (κ1 κ2 )∗ = ηλ (κ1 κ2 ). In particular, for (M, g) = (S n , gc ), (6.20.5) deﬁnes a representation ηλc of Gn+1 on C ∞ (S n ) ⊕ Ω1 (S n ) ⊕ C ∞ (S n ). It corresponds to the left regular representation on the space (C ∞ (G, C(λ)) ⊗ F ∗ )P . Therefore, Theorem 6.20.1 shows that on the sphere S n the tractor D-operator deﬁnes an equivariant operator: c (g) ◦ DS n (gc ; λ), g ∈ Gn+1 . DS n (gc ; λ) ◦ πλc (g) = ηλ−1

It corresponds to the D-operator (6.19.1). Similar comments apply to the following construction. Theorem 6.20.2 (Tractor C-operator). The rational family ⎛ ⎞ u− 1 CM (g; λ) : Γ(T ∗ M ) ⎝ ω ⎠ → 2(n−2+2λ) u+ × {−2(n+2λ)(n+λ−1)u− + 2(n+2λ)δg (ω) − Δg u+ +(n+λ−1)J(g)u+} (6.20.6) is conformally covariant in the sense that g ; λ) ◦ eλϕ ◦ TM (g, ϕ) = e(λ−1)ϕ ◦ CM (g; λ), CM (ˆ

gˆ = e2ϕ g

(6.20.7)

for all ϕ ∈ C ∞ (M ). Proof. Note that ⎛ ⎞ u− CM (λ) ⎝ ω ⎠ u+ =−

1 (2nu− −4δω +Δu+ −(n+λ−1)Ju+) − (λ+n)u− + δω 2(n−2+2λ)

6.20. Some elements of tractor calculus

391

is equivalent to (6.20.6). In these terms, (6.20.7) asserts that −

1 1 2n u− +(dϕ, ω)+ |dϕ|2 u+ e(λ−1)ϕ 2(n−2+2λ) 4 1 ˆ (λ+1)ϕ u+ ) − 4δˆ (ω + u+ dϕ)e(λ+1)ϕ + Δ(e 2 1 − (λ+n) (u− +(dϕ, ω)+ |dϕ|2 u+ )e(λ−1)ϕ 4 1 + δˆ (ω + u+ dϕ)e(λ+1)ϕ + ηˆJu+ e(λ+1)ϕ 2 1 [2nu− −4δω +Δu+] e(λ−1)ϕ =− 2(n−2+2λ) − (λ+n)u− e(λ−1)ϕ + δωe(λ−1)ϕ + ηJu+ e(λ−1)ϕ

for def

η =

n+λ−1 . 2(n−2+2λ)

Lemma 4.2.1 shows that this identity is equivalent to η

n−2 |dϕ|2 + ηΔϕ + ηˆJe2ϕ = ηJ. 2

But the latter identity is a consequence of (2.5.7).

The following Lemma extends the identity p(λ) ◦ i(λ−1) = 0 to the curved case. Lemma 6.20.1. The composition CM (g; λ−1)◦DM (g; λ) on C ∞ (M ) vanishes identically.

Proof. Direct calculation. Next, we deﬁne a connection on dual standard tractors. Deﬁnition 6.20.1 (Tractor connection). Let ⎞ ⎛ 0 ∇X (P(X), ·) 1 ⎠, ∇TX = ⎝−X ∇X 2 P(X) 0 −2X, · ∇X

X ∈ X (M ).

(6.20.8)

Here the 1-form P(X) ∈ Ω1 (M ) is deﬁned by P via the relation P(X, Y ) = P(X), Y . In Deﬁnition 6.20.1, all constructions (∇X , P etc.) are to be understood with respect to a chosen metric g. The basic property of ∇T is its conformal invariance. For a detailed discussion of the relation to the normal conformal Cartan connection we refer to [22].

392

Chapter 6. Conformally Covariant Families

Theorem 6.20.3. The connection ∇T is conformally invariant, i.e., ˆ T ◦ T (g, ϕ) T (g, ϕ) ◦ ∇TX = ∇ X

(6.20.9)

ˆ T denote the respective tractor for all vector ﬁelds X ∈ X (M ). Here ∇T and ∇ 2ϕ connections for g and gˆ = e g. Proof. The assertion follows from the conformal transformation rules for ∇X and P(X). Since Xω, Y = ∇X (ω), Y + ω, ∇X (Y ), the rule (2.5.1) implies ˆ X (ω) = ∇X (ω) − ωdϕ, X − dϕω, X + X (dϕ, ω). ∇ ˆ = P − Ξ (see (2.5.9)) yields Moreover, the identity P 1 ˆ P(X) = P(X) − ∇X (dϕ) − dϕ, Xdϕ + |dϕ|2 X . 2

(6.20.10)

(6.20.11)

Now we prove that the matrix entries ·23 of both sides of (6.20.9) coincide. For the other entries, the arguments are analogous and we omit the details. The assertion is that 1 1 1 ϕ ˆ ˆ |dϕ|2 · + 1 ∇ ˆ X (eϕ dϕ·) + 1 eϕ P(X) e P(X) · + eϕ dϕ∇X (·) = − e−ϕ X 2 2 4 2 2 as operators C ∞ (M ) → Ω1 (M ). (6.20.10) and (6.20.11) show that the right-hand side equals

1 1 ∇X · −dϕ, X · −dϕX, · + X (dϕ, ·) (eϕ udϕ) − eϕ X |dϕ|2 u + 4 2 1 1 + eϕ P(X) − ∇X (dϕ) + dϕ, Xdϕ − |dϕ|2 X u, 2 2 i.e., 1 1 1 1 − eϕ X |dϕ|2 u + eϕ ∇X (dϕ)u + eϕ dϕdϕ, Xu + eϕ dϕ∇X u 4 2 2 2 1 ϕ 1 ϕ 1 ϕ − e dϕ, Xudϕ − e dϕX, dϕu + e |dϕ|2 X u 2 2 2 1 ϕ 1 ϕ 1 ϕ 1 + e P(X)u − e ∇X (dϕ)u + e dϕ, Xudϕ − eϕ |dϕ|2 X u. 2 2 2 4 But the latter sum simpliﬁes to 1 1 ϕ e P(X)u + eϕ dϕ∇X u. 2 2 The proof is complete.

6.20. Some elements of tractor calculus

393

Next, the connection ∇TX gives rise to the tractor curvature endomorphisms RT (X, Y ) = ∇TX ∇TY − ∇TY ∇TX − ∇T[X,Y ] . A calculation yields the explicit formula ⎛ 0 (C(X, Y ), ·) RT (X, Y ) = ⎝0 C(X, Y ) 0 0

⎞ 0 1 ⎠ 2 C(X, Y ) , 0

(6.20.12)

(6.20.13)

where the endomorphism C(X, Y ) : Ω1 (M ) → Ω1 (M ) is deﬁned by the Weyl tensor C via C(X, Y )ω, Z = C(X, Y, ω , Z), and C(X, Y ) = ∇X (P)(Y, ·) − ∇Y (P)(X, ·) ∈ Ω1 (M )

(6.20.14)

is the Cotton tensor (see Lemma 4.2.7). In fact, we verify the central term in formula (6.20.13). We ﬁnd ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] − P(X)Y, · + P(Y )X, · − X (P(Y ), ·) + Y (P(X), ·) = R(X, Y ) + (P g)(X, Y ) = C(X, Y ). We omit the details for the remaining entries. The conformal invariance of RT is equivalent to the conformal transformation laws ˆ ˆ C(X, Y ) = C(X, Y ) and C(X, Y ) = C(X, Y ) − C(X, Y )(dϕ).

(6.20.15)

(6.20.13) and the relation (4.2.14) between C and C imply that the curvature RT vanishes iﬀ C = 0. Moreover, the tractor connection preserves a scalar product on dual tractors. Deﬁnition 6.20.2 (Tractor scalar product). The tractor scalar product is deﬁned by ⎛⎛ ⎞ ⎛ ⎞⎞T u− v− ⎝⎝ ω ⎠ , ⎝ η ⎠⎠ = − 1 (u− v+ + u+ v− ) + (ω, η)g . 2 u+ v+ g

(6.20.16)

Lemma 6.20.2. The tractor scalar product is conformally invariant and preserved by the connection ∇TX , i.e., T

(T (g, ϕ)u, T (g, ϕ)v)gˆ = (u, v)Tg and

T T ∇X (u, v)Tg = ∇TX (u), v g + u, ∇TX (v) g .

Proof. The assertions follow by direct calculations. We omit the details.

394

Chapter 6. Conformally Covariant Families

Remark 6.20.1. We use trivializations of tractor bundles and a deﬁnition of the tractor connection which slightly diﬀer from those used in the literature. The choice here is dictated by the choice of conventions in connection with the discussion of homomorphisms of Verma modules in Section 6.18. For the convenience of the reader, we relate the conventions explicitly. We use the diagonal matrix diag(1, 1, 2) to conjugate ⎞⎛ ⎞ ⎛ −ϕ ⎞ ⎛ 0 ∇X (P(X), ·) 1 (dϕ, ·) 14 |dϕ|2 e 1 1 ⎠ ⎝0 ⎠ ⎝ ⎠ , ⎝−X eϕ 1 ∇X 2 dϕ 2 P(X) ϕ e 0 0 1 0 −2X, · ∇X into

⎛ ⎝

⎞⎛

e−ϕ eϕ

1 ⎠ ⎝0 0 eϕ

(dϕ, ·) 1 0

1 2 2 |dϕ|

⎞

⎛

∇X ⎝−X 0

dϕ ⎠ , 1

⎞ (P(X), ·) 0 ∇X P(X)⎠ . −X, · ∇X

For standard tractors, Bailey, Eastwood and Gover ([17]) use the convention ⎛ −ϕ ⎞⎛ ⎞ ⎞ ⎛ 0 ∇X −P(X), · e 1 −(dϕ, ·) − 21 |dϕ|2 ⎝ ⎠ ⎝0 e−ϕ 1 grad(ϕ) ⎠ , ⎝ X P(X) ⎠ ∇X ϕ e 0 −(X, ·) ∇X 0 0 1 for trivializations and tractor connection. The diagonal matrix diag(1, −1, −1) conjugates these into ⎞ ⎞⎛ ⎞ ⎛ −ϕ ⎛ 0 ∇X P(X), · e 1 (dϕ, ·) 12 |dϕ|2 ⎠ ⎝0 ⎝ e−ϕ 1 grad(ϕ)⎠ , ⎝−X ∇X P(X) ⎠ . ϕ 0 0 1 e 0 −(X, ·) ∇X The tractor connection ∇T : Γ(T ∗ ) → Γ(T ∗ ⊗ T ∗ ) gives rise to the tractor Bochner-Laplacian ΔT = tr(∇T ◦ ∇T ) : Γ(T ∗ ) → Γ(T ∗ ). Here

(6.20.17)

∇T : Γ(T ∗ ⊗ T ∗ ) → Γ(T ∗ ⊗ T ∗ ⊗ T ∗ )

couples ∇T with the Levi-Civita connection ∇LC on Ω1 , i.e., ∇T (u ⊗ ω) = ∇T (u) ⊗ ω + u ⊗ ∇LC (ω), and tr denotes the contraction T ∗ ⊗ T ∗ → C deﬁned by g. A standard calculation yields Lemma 6.20.3.

ΔT =

∇Tei ∇Tei − ∇T∇LC = δ T ∇T e ei

i

in terms of a local orthonormal frame ei .

i

6.20. Some elements of tractor calculus

395

The following result extends the conformal covariance of the Yamabe operator P2 = Δ − ( n2 −1)J. Theorem 6.20.4. The operator T = ΔT − def

n 2

−1 J : Γ(T ∗ ) → Γ(T ∗ )

is conformally covariant in the sense that n

n

Tgˆ ◦ e(− 2 +1)ϕ ◦ T (g, ϕ) = e−( 2 +1)ϕ ◦ T (g, ϕ) ◦ Tg . Proof. We use Lemma 6.20.3 and Theorem 6.20.3. Since eˆi = e−ϕ ei , we get −ϕ ∇LC ˆi = e−ϕ ∇LC ei ) = e−2ϕ ∇LC eˆi e ei (e ei ei − dϕ, ei ei and the transformation rule (2.5.1) yields ˆ LC eˆi = ∇LC eˆi + 2e−2ϕ dϕ, ei ei − e−2ϕ grad ϕ ∇ eˆi eˆi = e−2ϕ ∇LC ei ei + dϕ, ei ei − grad ϕ . Hence

ˆ T LC + dϕ, ei ∇ ˆ Tˆ LC = e−2ϕ ∇ ˆT − ∇ ˆT ∇ ei grad ϕ . ∇ eˆi eˆ ∇ e ˆi

Moreover, we ﬁnd

i

e ˆi

ˆ Teˆ ∇ ˆ Teˆ = e−2ϕ ∇ ˆ Te − dϕ, ei ∇ ˆ Te ∇ ˆ Te ∇ i i i i i T T −2ϕ T ∇ei ∇ei − dϕ, ei ∇Tei T −1 , =e

where T = T (g, ϕ). It follows that T −1 e2ϕ ΔTgˆ T = ΔTg − 2

dϕ, ei ∇Tei + n∇Tgrad ϕ i

=

ΔTg

+ (n−2)∇Tgrad ϕ .

On the other hand, a calculation shows that ΔT (eλϕ u) = eλϕ ΔT u + 2λ∇Tgrad ϕ u + λuΔϕ + λ2 |dϕ|2 u . Therefore, ΔTgˆ (eλϕ u) = e−2ϕ T ΔTg + (n−2)∇Tgrad ϕ (T −1 (eλϕ u)) + e(λ−2)ϕ T (n−2+2λ)∇Tgrad ϕ + λΔg (ϕ) + λ(n−2+λ)|dϕ|2 (T −1 u), i.e., ΔTgˆ (eλϕ u) − e(λ−2)ϕ T ΔTg (T −1 u) = e(λ−2)ϕ T (n−2+2λ)∇Tgrad ϕ + λΔg (ϕ) + λ(n−2+λ)|dϕ|2 (T −1 u).

396

Chapter 6. Conformally Covariant Families

Together with n−2 |dϕ|2 eλϕ ˆJ − e(λ−2)ϕ J = e(λ−2)ϕ −Δϕ − 2 (see (2.5.7)) we obtain (ΔTgˆ + λˆJ)(eλϕ u) − e(λ−2)ϕ T (ΔTg + λJ)(T −1 u) n−2 = e(λ−2)ϕ T (n−2+2λ)∇Tgrad ϕ + (λ+ )λ|dϕ|2 (T −1 u) = 0 2

for λ = − n2 + 1. The proof is complete.

The latter proof also shows that the tractor D-operator can be iterated. More precisely, we have Theorem 6.20.5. The tractor D-operator DM (λ) : C ∞ (M ) → Γ(T ∗ M ) extends to a conformally covariant family DM (λ) : Γ(T ∗ M ) → Γ(T ∗ M ⊗ T ∗ M ) by

⎛ DM (g; λ) : u →

1 T n−2+2λ (Δ T ⎝

∇ u 2λu

+λJ)u

⎞ ⎠.

(6.20.18)

Here all constructions are to be understood with respect to g and we use the identiﬁcation Γ(T ∗ M ) ⊕ Γ(T ∗ M ⊗ T ∗ M ) ⊕ Γ(T ∗ M ) Γ(T ∗ M ⊗ T ∗ M ). The conformal covariance means g ; λ) ◦ eλϕ ◦ TM (g, ϕ) = e(λ−1)ϕ ◦ TM (g, ϕ) ◦ DM (g; λ), DM (ˆ

(6.20.19)

where TM denotes the respective trivializations of T ∗ M and T ∗ M ⊗ T ∗ M . Proof. The proof is analogous to the proof of Theorem 6.20.1. It rests on the identity ˆ T +λˆJ)(eλϕ u) e−(λ−2)ϕ (Δ = T (ΔT +λJ)(T −1 u) + (n−2+2λ)T ∇Tgrad ϕ T −1 u + λ(λ+

n−2 )|dϕ|2 u 2

6.20. Some elements of tractor calculus

397

which was established at the end of the proof of Theorem 6.20.4; compare also with (6.20.4). We divide by (n−2+2λ). It follows that e−(λ−1)ϕ

1 ˆ T +λˆJ)(eλϕ u) (Δ n−2+2λ ⎞ ⎛ 1 T −1 u n−2+2λ T (Δ +λJ)T 1 ⎠. = e−ϕ 1, (dϕ, ·), |dϕ|2 ⎝ T ∇T T −1 u 4 2λu

In other words, the ﬁrst components of both sides of (6.20.19) coincide. For the second components, the assertion is the obvious identity ˆ T (eλϕ u) = eλϕ λudϕ + T ∇T T −1 u . ∇

The proof is complete.

The extension of DM (λ) from C ∞ (M ) to Γ(T ∗ M ) described in Theorem 6.20.5 can be continued, and yields conformally covariant tractor D-operators DM (λ) : Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ) → Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ). @A B @A B ? ? r

r+1

These are deﬁned by (6.20.18) using the connection ∇T acting on the space ∗ Γ(T · · ⊗ T ∗ MB) and the tractor Bochner-Laplacian ? M ⊗ ·@A r

ΔT : Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ) → Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ). @A B @A B ? ? r

r def

The corresponding operator T = ΔT − n−2 2 J is conformally covariant. We omit the details. The following result is the curved analog of Lemma 6.19.2. It can be used to give an alternative proof of Theorem 6.20.2. Lemma 6.20.4. CM (λ) = tr ◦DM (λ), where DM (λ) : Γ(T ∗ M ) → Γ(T ∗ M ⊗ T ∗ M ) and the trace denotes the contraction induced by the tractor scalar product. Proof. By deﬁnition, ⎛ D(λ)u =

1 T n−2+2λ (Δ T ⎝

∇ u 2λu

+λJ)u

⎞ ⎠

398

Chapter 6. Conformally Covariant Families

⎞ u− (see (6.20.18)). Let u = ⎝ ω ⎠. The explicit formulas (6.20.8) and (6.22.2) imply u+ ⎛

1 tr(D(λ)u) = −λu− + (−nu− + δω + Ju+ ) 2 1 1 1 1 {2nu− − 4δω + (Δ−J)u+ } − λJu+ . − 2 n−2+2λ 2 n−2+2λ A calculation shows that the latter sum equals C(λ)u. The proof is complete.

Similarly, the family CM (λ) : Γ(T ∗ M ) → C ∞ (M ) extends to conformally covariant families Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ) → Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ) @A B @A B ? ? r

r+1 def

by CM (λ) = tr DM (λ) using a contraction. In explicit terms, 1 CM (λ)u = 2(n−2−2λ) × −2(n+2λ)(n+λ−1)u− + 2(n+2λ)δ T ω − ΔT u+ + (n+λ−1)Ju+

(6.20.20)

⎛ ⎞ u− for u = ⎝ ω ⎠ with u+ ∗ · · ⊗ T ∗ MB), u± ∈ Γ(T ? M ⊗ ·@A

∗ ω ∈ Γ(T · · ⊗ T ∗ MB ⊗T ∗ M ). ? M ⊗ ·@A

r

r

Here δ T denotes the divergence which is deﬁned by the connections. Now for an oriented codimension one submanifold (i.e., a hypersurface) Σ in M , the family D1 (λ) = ∇N − λH : C ∞ (M ) → C ∞ (Σ) is conformally covariant (Theorem 6.2.1). An analogous family D1T (λ) : Γ(T ∗ M ) → Γ(T ∗ Σ) can be deﬁned by the composition of two conformally covariant families. The following result deﬁnes one factor. D1T (λ) will be deﬁned in Lemma 6.20.8. Lemma 6.20.5. The family DT (λ) = ∇TN − λH def

is conformally covariant in the sense that e(λ−1)ϕ ◦ T (g, ϕ) ◦ DT (g; λ) = DT (ˆ g; λ) ◦ eλϕ ◦ T (g, ϕ).

6.20. Some elements of tractor calculus

399

Proof. We observe that the composition g ; λ) ◦ eλϕ ◦ T (g, ϕ) T (g, ϕ)−1 ◦ e−(λ−1)ϕ ◦ DT (ˆ ˆ = e−ϕ N , Theorem 6.20.3 implies that the absolute coeﬃis linear in λ. Using N cient is ˆ T ◦ T (g, ϕ) = ∇T . T (g, ϕ)−1 ◦ ∇ N N ˆ = e−ϕ (H + ∇N (ϕ)), we ﬁnd that the linear coeﬃcient is Moreover, using H

ˆ TN − eϕ H(ˆ ˆ TN ◦ ϕ ◦ T (g, ϕ) g) + ∇ T (g, ϕ)−1 ◦ −ϕ ◦ ∇ = T (g, ϕ)−1 ◦ [∇N (ϕ) − (H(g) + ∇N (ϕ))] ◦ T (g, ϕ) = −H(g).

The proof is complete.

Next, we deﬁne a tractor analog of the unit normal vector ﬁeld of a hypersurface. Deﬁnition 6.20.3 (Normal tractor). For a hypersurface Σ of M and a metric g, let ⎛ ⎞ H def N T = ⎝N ⎠ , 0 where all quantities are to be understood with respect to g. N T is called the normal tractor of Σ for the metric g. Note that N T has length 1. The following result says that the normal tractor is a section of the dual tractor bundle on M , deﬁned on the hypersurface Σ. ˆ T = TM (g, ϕ)N T , where N T and N ˆ T are the respective normal Lemma 6.20.6. N 2ϕ tractors for the metrics g and gˆ = e g. ˆ = eϕ N and H ˆ = e−ϕ (H + dϕ, N ). Proof. The assertion is equivalent to N

As a consequence, we ﬁnd Corollary 6.20.1. D1 (M, Σ; g; λ) = (DM (g; λ), NgT )Tg . Corollary 6.20.1 re-proves the conformal covariance of D1 (M, Σ; g; λ) (Theorem 6.2.1). In fact, Lemma 6.20.2, Theorem 6.20.1 and Lemma 6.20.6 imply e(λ−1)ϕ D1 (g; λ)u = e(λ−1)ϕ (D(g; λ)u, NgT )Tg T = e(λ−1)ϕ T (g, ϕ)D(g; λ)u, T (g, ϕ)NgT gˆ = (D(ˆ g ; λ)eλϕ u, NgˆT )Tgˆ = D1 (ˆ g ; λ)eλϕ u.

400

Chapter 6. Conformally Covariant Families

Similar arguments can be used to re-prove the conformal covariance of DT (λ) (Lemma 6.20.5). It will be important to have an invariant linear map from tractors on M to tractors on a hypersurface Σ. The following result provides such a map. Theorem 6.20.6 (Projection operator). The linear operator ⎛∗ ⎛ ⎞ ⎞ i (u− )−Hω, N + 14 H 2 i∗ (u+ ) u− def ⎠ i∗ (ω) ΠΣ ⎝ ω ⎠ = ⎝ u+ i∗ (u+ ) is conformally covariant in the sense that ∗

g ) ◦ eλϕ ◦ TM (g, ϕ) = eλi ΠΣ (ˆ

(ϕ)

◦ TΣ (i∗ (g), i∗ (ϕ)) ◦ ΠΣ (g)

(6.20.21)

for all ϕ ∈ C ∞ (M ). Proof. It is enough to prove the assertion for λ = 0. Then for the ﬁrst component, (6.20.21) claims that 5 6 1 1 1 ϕ ˆ2 −ϕ 2 ϕ ˆ ˆ e u− +(dϕ, ω)+ |dϕ| u+ + e H u+ − e H ω + u+ dϕ, N 4 4 2 1 2 1 ∗ −ϕ ∗ ∗ 2 =e u− + H u+ −Hω, N + (i (dϕ), i (ω)) + |i (dϕ)| u+ 4 4 on Σ. The assertions for the remaining two components are trivial. The above identity is equivalent to ˆ ˆ = e−ϕ (i∗ (dϕ), i∗ (ω)) − e−ϕ Hω, N N e−ϕ (dϕ, ω) − eϕ Hω, and

ˆ 2 − 2eϕ Hdϕ, ˆ ˆ = e−ϕ H 2 + e−ϕ |i∗ (dϕ)|2 . N e−ϕ |dϕ|2 + eϕ H ˆ = H + ∇N (ϕ). The proof is complete. Both identities follow from eϕ H

We extend the map ΠΣ to a conformally covariant operator Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ) → Γ(T ∗ Σ ⊗ · · · ⊗ T ∗ Σ) @A B @A B ? ? r

r

by ΠΣ (u ⊗ · · · ⊗ v) = ΠΣ (u) ⊗ · · · ⊗ ΠΣ (v). Remark 6.20.2. The projection operator ΠΣ does not ⎛ coincide ⎞ with the orthogonal u− projection πΣ along N T . In fact, for the tractor u = ⎝ ω ⎠, we ﬁnd u+ ⎞ ⎛ u− − Hω, N + 12 H 2 u+ πΣ (u) = u − (u, N T )T N T = ⎝ω − ω, N N + 12 u+ HN ⎠ . u+

6.20. Some elements of tractor calculus

401

πΣ and ΠΣ actually satisfy diﬀerent transformation laws. In fact, it follows from the deﬁnitions and Lemmas 6.20.2 and 6.20.6 that g ) ◦ TM (g, ϕ) = TM (g, ϕ) ◦ πΣ (g). πΣ (ˆ In order to relate πΣ and ΠΣ , we introduce the linear operator ⎛ ⎞ ⎞ ⎛ v− + 14 H 2 v+ v− I(g) : ⎝ η ⎠ → ⎝η + 12 HN v+ ⎠ ∈ N T (g)⊥ . v+ v+ It deﬁnes an isomorphism between dual standard tractors on Σ and dual standard tractors on M (supported on Σ) which are orthogonal to N T . It is easy to verify that I(ˆ g ) ◦ TΣ (i∗ (g), i∗ (ϕ)) = TM (g, ϕ) ◦ I(g), i.e., I is conformally covariant. Moreover, we ﬁnd I(g) ◦ ΠΣ (g) = πΣ (g). Finally, note that I(g) reduces to the identity if H = 0. The following simple result illustrates these constructions by identifying the trace-free second fundamental form L0 as a composition of operators on tractors. Lemma 6.20.7. In the situation of Deﬁnition 6.20.3, let ⎛ ⎞ u− 1 σ : ⎝ ω ⎠ → ω − du+ . 2 u+ Then σ ◦ T (g, ϕ) = eϕ ◦ σ and the bilinear form C D T (X, Y ) → σΠΣ ∇M,T (N ), Y ∈ C ∞ (Σ), X, Y ∈ X (Σ) X coincides with L0 . Proof. (6.20.22) is equivalent to the obvious identity 1 1 ϕ 1 ϕ ϕ e ω + dϕu+ − d(e u+ ) = e ω − du+ . 2 2 2 Now since

⎛

⎞ dH + P(N ), X ⎠ ∇M,T (N T ) = ⎝−HX + ∇M X (N ) , X 0

(6.20.22)

402

Chapter 6. Conformally Covariant Families

⎞ ∗ ⎠ ΠΣ ∇M,T (N T ) = ⎝−HX + i∗ ∇M X (N ) . X 0 ⎛

we ﬁnd

Hence the composition is given by −HX + L(X) = L0 (X).

Lemma 6.20.7 implies that L0 = 0 if N T is parallel, i.e., ∇TX N T = 0 for all X ∈ X (Σ). Conversely, if L0 = 0, then the proof of Lemma 6.20.7 shows that ⎛ ⎞ dH + P(N ), X ⎠ 0 ∇TX (N T ) = ⎝ 0 for all X ∈ X (Σ). But L0 = 0 and Lemma 6.25.2 imply that P(N ) + dH = 0, i.e., ∇TX (N T ) = 0. Hence we have proved Corollary 6.20.2. Σ is totally umbilic iﬀ N T is parallel. This is Proposition 2.9 in [17]. These considerations also show that ⎛ 1 ⎞ n−2 δ(L0 )(X) ⎠ ∈ Γ(T ∗ Σ). ΠΣ ∇TX (N T ) = ⎝ L0 (X) 0 Now Lemma 6.20.5 and Theorem 6.20.6 yield Lemma 6.20.8. The family D1T (M, Σ; g; λ) = ΠΣ DT (M, Σ; g; λ) : Γ(T ∗ M ) → Γ(T ∗ Σ) def

is conformally covariant, i.e., e(λ−1)ϕ ◦ TΣ (g, ϕ) ◦ D1T (M, Σ; g; λ) = D1T (M, Σ; gˆ; λ) ◦ eλϕ ◦ TM (g, ϕ). The latter extension of D1 (λ) from C ∞ (M ) to Γ(T ∗ M ) can be continued, and yields conformally covariant operators ∗ ∗ D1T (λ) : Γ(T · · ⊗ T ∗ MB) → Γ(T · · ⊗ T ∗ ΣB). ? M ⊗ ·@A ? Σ ⊗ ·@A r

r

Finally, we brieﬂy consider the Paneitz operator from the point of view of tractor calculus. The following result is due to Eastwood and Gover ([109]). Theorem 6.20.7. On M n ,

⎞ ⎛

n

1 P4,n −1 J ◦ D − +2 = ⎝ 0 ⎠ . ΔT − 2 2 2 0 n

T 6.21. The tractor families DN (M, Σ; g; λ)

403

Proof. The explicit formula (6.22.2) implies that the left-hand side is given by ⎛ ⎞ ⎞⎛1 1 n 2 Δ− n2 J −(dJ, ·) + 2δ(P#·) 2 |P| 2 (Δ−( 2 −2)J)u 1 ⎝ −2d ⎠. ⎠⎝ du Δ + n−4 2 (2P# − J) 2 dJ + P#d −(n−4)u 2n −4δ Δ− n2 J Here we made use of Ric −2P −

n−4 −1 Jg = (2P − Jg). 2 2

n

Hence the upper entry of the resulting tractor is given by

n n 1 1 Δ− J Δ−( −2)J u + 2δ(P#du) − (dJ, du) − (n−4)|P|2 u. 2 2 2 2 The latter sum equals 1" 2 Δ u − (n−2)JΔu − (n−4)(dJ, du) 2 + 4δ(P#du) − 2(dJ, du) +

# 1 n−4 n 2 J − ΔJ − 2|P|2 u = P4,n u 2 2 2

(see (4.1.7) and (4.2.11)); note that the Laplacian here has the opposite sign. The vanishing of the remaining two components is easy to check. We omit the details. The following consequence is the curved version of Theorem 6.18.1. Corollary 6.20.3. On M n ,

n

n n n−4 −1 J ◦ D − +2 = − P4,n . C − −1 ◦ ΔT − 2 2 2 8 Corollary 6.20.3 implies that on M 4 the composition C(−3) ◦ (ΔT −J) ◦ D(0) vanishes, i.e., does not yields a construction of P4 . In this respect, Theorem 6.20.7 is more useful, and motivated analogous constructions of the higher order critical GJMS-operators ([116]). T 6.21 The tractor families DN (M, Σ; g; λ) In the present section, we use tractor D-operators for the deﬁnition of a series of conformally covariant families of diﬀerential operators T DN (M n , Σn−1 ; g; λ) : C ∞ (M n ) → C ∞ (Σn−1 )

404

Chapter 6. Conformally Covariant Families

of any order N (even and odd) for a hypersurface Σ of M and an arbitrary background metric g. The superscript T stands for (curved) translation. These families, c in particular, generalize the families DN (λ) = DN (S n , S n−1 ; gc ; λ). The idea is as c ∞ n follows. We know that DN (λ) : C (S ) → C ∞ (S n−1 ) is induced by the families DN (λ) of homomorphisms of Verma modules. By Theorem 6.18.7, DN (λ) is an iterated Zuckerman translate of either D0 (λ) = i or D1 (λ) = Yn− (depending on the parity of N ); here we simplify notation by writing DN (λ) for the normalized 0 version DN (λ). The building blocks of the curved version of Zuckerman translation are the families D(λ) and C(λ). Theorem 6.18.7 and Theorem 6.18.8 immediately suggest the deﬁnitions. Now two basic questions concerning these families are: 1. Which parts of the construction only depend on the metric on the submanifold Σ? 2. What can be said about the relation to GJMS-operators and Q-curvature on the submanifold? A closely related question concerns the relation between the tractor families and the residue families. In order to address these questions, we formulate a system of conjectures and prove the holographic duality for a conformally ﬂat metric h. The tractor families may also shed some light on the non-existence of certain conformally covariant powers of the Laplacian. For a manifold of even dimension n ≥ 4, there exists no conformally covariant power of the Laplacian of order ≥ n + 2 ([118], [112]). That non-existence is reﬂected in the theory of the families T (·, Σn ; ·; λ) as follows. Although the tractor construction is holomorphic in the D2N spectral parameter (apart from trivial rational coeﬃcients), the operator families of interest, i.e., those which generalize the families D2N (S n , S n−1 ; λ), in general, will have simple poles. These poles arise precisely under those conditions on the parameters (dimension, order and conformal weight) for which the existence of a conformally covariant power of the Laplacian (for a general metric) is forbidden. Therefore, we regard the corresponding residues as obstructions to the existence of such operators. For certain metrics, the obstructions may vanish, however. In such cases, the value of the family at the corresponding parameter is well deﬁned, and is a conformally covariant power of the Laplacian on the target space. In particular, on the conformally ﬂat round sphere S n−1 there are conformally covariant powers of the Laplacian for all orders. That corresponds to the vanishing of all obstructions in that case, and is reﬂected by the existence of the holomorphic families D2N (S n+1 , S n ; λ) (for all N ) with the property

n D2N S n+1 , S n ; gc ; − +N = P2N (S n , gc )i∗ . 2 We continue with the details. First, observe that for even order 2N , the deﬁnition of Θ2N (λ) : Mλ−2N (gn ) → Mλ (gn+1 )

T 6.21. The tractor families DN (M, Σ; g; λ)

405

(see Deﬁnition 6.18.1) is equivalent to Θ2N (λ) = p(λ) ◦ p(λ−1) ◦ · · · ◦ p(λ−N +1) ◦ (i ⊗ In ) ◦ i(λ−N ) ◦ · · · ◦ i(λ−2N +1), which realizes the composition Mλ−2N (gn ) → Mλ−2N +1 (gn ) ⊗ Fn → · · · → Mλ−N (gn ) ⊗ Fn ⊗ · · · ⊗ Fn ? @A B N factors i⊗In

−−−→ Mλ−N (gn+1 ) ⊗ Fn+1 ⊗ · · · ⊗ Fn+1 → · · · → Mλ (gn+1 ). ? @A B N factors

Here i(·) and p(·) denote the usual embedding and projection as well as all twisted versions. Note also that both maps are used for the Lie algebras gn and gn+1 . i : gn → gn+1 embeds the Lie algebras, and In : Fn → Fn+1 denotes the Gn equivariant embedding of standard representations. For the induced embeddings of tensor products we use the same symbol. Deﬁnition 6.21.1. Let Σ ⊂ M be a hypersurface and g a metric on M . For N ≥ 0, set T (M, Σ; g; λ) = CΣ (g; λ−2N +1) ◦ · · · ◦ CΣ (g; λ−N ) D2N

◦ ΠΣ (g) ◦ DM (g; λ−N +1) ◦ · · · ◦ DM (g; λ).

(6.21.1)

The composition (6.21.1) is conformally covariant since all factors are. More precisely, we have T T (M, Σ; e2ϕ g; λ) ◦ eλϕ = D2N (M, Σ, g; λ) e−(λ−2N )ϕ ◦ D2N

for all λ ∈ C and all ϕ ∈ C ∞ (M ). Similarly, the odd order versions are deﬁned in Deﬁnition 6.21.2. Let Σ ⊂ M be a hypersurface and g a metric on M . We ﬁx a unit normal ﬁeld N = N (g). For N ≥ 0, set T D2N +1 (M, Σ; g; λ) = CΣ (g; λ−2N ) ◦ · · · ◦ CΣ (g; λ−N +1)

◦ D1T (M, Σ; g; λ−N ) ◦ DM (g; λ−N +1) ◦ · · · ◦ DM (g; λ). We recall that D1T (λ) = ΠΣ ◦DT (λ) (Lemma 6.20.8). Again, the composition is conformally covariant since all factors are. More precisely, we have T 2ϕ λϕ T e−(λ−2N −1)ϕ ◦ D2N = D2N +1 (M, Σ; e g; λ) ◦ e +1 (M, Σ, g; λ)

for all λ ∈ C and all ϕ ∈ C ∞ (M ).

406

N

th

Chapter 6. Conformally Covariant Families T T We rephrase these deﬁnitions by saying that D2N (λ) and D2N +1 (λ) are the ∗ curved translations of i and ∇N , respectively.

T (λ) (for speciﬁc In [40], Branson and Gover used the tractor families DN values of the parameter λ), for the construction of conformally covariant elliptic self-adjoint boundary value problems on manifolds M with boundary Σ. Such boundary value problems can be used to construct conformally covariant pseudodiﬀerential operators on Σ (see also [111]). The convention in [40] diﬀer from those used here in two respects: trivializations of tractor bundles (see Remark 6.20.1) and normalizations of tractor D-operators. T (λ), the projection ΠΣ operates on In the composition which deﬁnes D2N sections in ∗ · · ⊗ T ∗ MB). Γ(T ? M ⊗ ·@A N factors T (λ) with powIt follows that the mean curvature H appears in the family D2N T (λ) the operator ΠΣ ers up to 2N . Similarly, in the composition deﬁning D2N +1 operates on

Γ(T ∗ M ⊗ · · · ⊗ T ∗ M ). @A B ? N factors

This yields powers of H up to 2N . The additional factor DT contributes one more T power of H so that in the family D2N +1 (λ) the mean curvature H appears with powers up to 2N + 1. In particular, the top power of the mean curvature in the T (λ) always coincides with the order of the family. family DN Theorem 6.18.7 and Theorem 6.18.8 say which coeﬃcients have to be removed in order to get the families of interest. There are actually two types of coeﬃcients. The ﬁrst two products trivially arise by the deﬁnition of the rational C and D families. However, the product N

(n−1−(j +N )+λ)

j=1

is of a diﬀerent nature. In the context of (holonomic) Verma modules, it appears as an overall coeﬃcient in the formula for the N th Zuckerman translation. We have seen that in the semi-holonomic setting that coeﬃcient has the eﬀect that the families of interest are no longer polynomial but rational with possible poles at the zeros of the above product. In the curved case, the situation is analogous with the only diﬀerence that some residues possibly vanish. Now in the following deﬁnition, we remove the irrelevant coeﬃcients from T (λ). DN

T 6.21. The tractor families DN (M, Σ; g; λ)

407

Deﬁnition 6.21.3. For the pair (M n , Σn−1 ), the factorizations T (M, Σ; g; λ) = c2N D2N

⎧ 2N ⎨ ⎩

⎫ ⎬

N

j=N +1

1 1 n−1−2j +2λ j=1 n−2j +2λ ⎭ ×

N

T (n−1−(j +N )+λ)D2N (M, Σ; g; λ)

j=1

and T D2N +1 (M, Σ; g; λ) = c2N +1

⎧ 2N ⎨ ⎩

N

j=N +1

×

⎫ ⎬

1 1 n−3−2j +2λ j=1 n−2j +2λ ⎭ N

T (n−2−(j +N )+λ)D2N +1(M, Σ; g; λ)

j=1 T deﬁne rational families DN (M, Σ; g; λ) of conformally covariant diﬀerential operT ators. The poles of the even order family D2N (λ) are at the zeros of the product >N j=1 (n−1−(j +N )+λ), i.e., at

λ ∈ {−n+N +2, . . . , −n+1+2N }.

(6.21.2)

T These N poles are (at most) simple. The poles of the odd order family D2N +1 (λ) >N are at the zeros of the product j=1 (n−2−(j +N )+λ), i.e., at

λ ∈ {−n+N +3, . . . , −n+2+2N }. These N poles are (at most) simple. Theorem 6.18.7 says that, for even n, the critical family Θn (λ) : Nλ−n (gn+1 ) → Nλ (gn+2 ) (notice the choice of Lie algebras) factorizes as n

2 n

+λ DnT (λ). n− j + (· · ·) 2 j=1

Similarly, we factorize DnT (M, Σ; g; λ) : C ∞ (M n+1 ) → C ∞ (Σn ) as n

(· · ·)

2

j=1

n

n− j + +λ DnT (M, Σ; g; λ). 2

(6.21.3)

408

Chapter 6. Conformally Covariant Families

Therefore, it is natural to expect poles of DnT (·, Σn ; g; λ) (at most) at the points # " n (6.21.4) λ ∈ − +1, . . . , 0 . 2 DnT (λ) has no pole at λ = − n2 +1. We expect that the same is true for DnT (λ). Much more important, however, is the behaviour at λ = 0. This is the critical case. DnT (λ) (n ≥ 4) has a simple pole at λ = 0 with a non-vanishing residue. However, in the curved case we conjecture that the residue actually vanishes. Conjecture 6.21.1. For even n, the critical family DnT (·, Σn ; g; λ) is regular at λ = 0, i.e., DnT (·, Σn ; g; 0) = 0. Note that the operator Bn (·, Σ; g) = DnT (·, Σn ; g; 0) is a conformally covariant operator enϕ Bn (ˆ g ) = Bn (g). The regularity of DnT (·, Σn ; g; λ) at λ = 0 has the consequence that the value PnT (M, Σ; g) = DnT (M, Σn ; g; 0) : C ∞ (M ) → C ∞ (Σ)

(6.21.5)

and the quantity QTn (M, Σ; g) = D˙ nT (M, Σn ; g; 0)(1) ∈ C ∞ (Σ)

(6.21.6)

are well deﬁned. Moreover, Lemma 6.21.1. If DnT (·, Σn ; g; 0) = 0, let def P˜nT (g) = PnT (g) − PnT (g)(1).

Then the natural pair (P˜nT (g), QTn (g)) satisﬁes the fundamental identity enϕ QTn (ˆ g) = QTn (g) − P˜nT (g)(ϕ). Proof. We diﬀerentiate the identity T e−(λ+n)ϕ ◦ DN (ˆ g ; λ) ◦ eλϕ = DnT (g; λ)

at λ = 0 and apply the result to u = 1. Then −ϕenϕ DnT (ˆ g ; 0)(1) + enϕ D˙ nT (ˆ g ; 0)(1) + enϕ DnT (ˆ g ; 0)(ϕ) = D˙ nT (g; 0)(1), i.e.,

enϕ D˙ nT (ˆ g ; 0)(1) + DnT (ˆ g ; 0)(ϕ) − ϕDnT (ˆ g ; 0)(1) = D˙ nT (g; 0)(1)

using enϕ DnT (ˆ g; 0) = DnT (g; 0). In other words, enϕ QTn (ˆ g) + P˜nT (g)(ϕ) = QTn (g). The proof is complete.

T 6.21. The tractor families DN (M, Σ; g; λ)

409

It follows that, if the operator PnT (g) annihilates constants, then the pair (PnT (g), QTn (g)) satisﬁes the fundamental identity enϕ QTn (ˆ g ) = QTn (g) − PnT (g)(ϕ),

gˆ = e2ϕ g.

T For even n, the subcritical family D2N (·, Σn ; g; λ) (2N < n) is regular at n λ = − 2 + N . Hence the value

n T T ·; Σ; g; − +N (6.21.7) (·, Σ; g) = D2N P2N 2 T is well deﬁned. Similarly, for odd n and all N , the family D2N (·, Σ; g; λ) is regular at λ = − n2 + N . Hence the value

n T T ·; Σ; g; − +N (6.21.8) (·; Σ; g) = D2N P2N 2 N is well deﬁned. Now the relation D2N (− n2 + N ) = (Δ− suggests n)

Conjecture 6.21.2. For n ≥ 3, let 2N < n if n is even. Then

n T ·, Σn ; g; − +N = P2N (Σn , i∗ (g))i∗ D2N 2 for a conformally covariant diﬀerential operator P2N on Σ with leading part ΔN g . In particular, the operator on the left-hand side does not depend on the embedding. The critical case for even n behaves diﬀerently. Conjecture 6.21.3 (Decomposition). Assuming Conjecture 6.21.1, the operator PnT (·, Σn ; g) (see (6.21.5)) admits a decomposition PnT (·, Σ; g) = Pn (Σ, i∗ (g))i∗ + Pne (·, Σ; g) into conformally covariant diﬀerential operators Pn : C ∞ (Σ) → C ∞ (Σ),

Pne : C ∞ (M ) → C ∞ (Σ),

n

where Pn has leading term Δ 2 . The operator Pn only depends on the metric i∗ (g). The operator Pne depends on the metric in a neighborhood of Σ but lives on Σ. Similarly, QTn (·, Σ; g) ∈ C ∞ (Σ) (see (6.21.6)) admits a decomposition n

−(−1) 2 QTn (·, Σ; g) = Qn (Σ, i∗ (g)) + Qen (·, Σ; g). Moreover, the pairs (Pn , Qn ) and (Pne , Qen ) satisfy the respective fundamental identities g ) = Qn (g) − Pn (g)(ϕ) enϕ Qn (ˆ on Σ and g ) = Qen (g) − Pne (g)(ϕ). enϕ Qen (ˆ

410

Chapter 6. Conformally Covariant Families

The point of these decompositions is that the intrinsic pair (Pn , Qn ) only depends on the metric i∗ (g) on Σ. For odd n, the situation for the critical families DnT (g; 0) is the following. In that case, the value DnT (·, Σn ; g; 0) is not well deﬁned, in general, since the family DnT (·, Σn ; g; λ) has a simple pole at λ = 0. In fact, Theorem 6.18.8 says that, for odd n, the critical family Θn (λ) : Nλ−n (gn+1 ) → Nλ (gn+2 ) factorizes as

n−1 n−1− j + (· · ·) +λ DnT (λ). 2 j=1 n−1 2

Similarly, we factorize DnT (M, Σ; g; λ) : C ∞ (M ) → C ∞ (Σ) as n−1 n−1− j + +λ DnT (M, Σ; g; λ). 2 j=1 n−1 2

(· · ·)

Therefore, it is natural to expect poles of DnT (·, Σ; g; λ) (at most) at the points n−1 +1, . . . , 0 . λ∈ − 2 In particular, there is a possible pole at λ = 0. Let Bn (M, Σ; g) = Res0 (DnT (M, Σ; g; λ)) : C ∞ (M ) → C ∞ (Σ) be the residue. Bn is a conformally covariant diﬀerential operator, i.e., g ) = Bn (g). enϕ Bn (ˆ Problem 6.21.1. Let n be odd. Determine the residue Bn . Characterize in geometric terms the metrics with Bn = 0. Does Bn vanish if L0 = 0? Is Bn induced by the residue Res0 (ΘTn (λ)) of the semi-holonomic family? Note that it is not true that the semi-holonomic Res0 (ΘTn (λ)) ∈ T (n− n+2 ) is − tangential, i.e., does not contain components Yn+1 . As an example, compare for n = 3 the algebraic structure given in Theorem 6.18.3 with the geometric structure given in Theorem 6.25.1. Deﬁnition 6.21.4. Let n be odd and assume that Bn (M, Σ; g) = 0. Then the critical family DnT (M, Σn ; g; λ) is regular at λ = 0. We set PnT (M, Σ; g) = DnT (M, Σ; g; 0), P˜nT (g) = PnT (g) − PnT (g)(1), QTn (M, Σ; g) = D˙ nT (M, Σ; g; 0)(1).

T 6.21. The tractor families DN (M, Σ; g; λ)

411

Lemma 6.21.2. For odd n, the natural pair (P˜nT (g), QTn (g)) satisﬁes the fundamental identity g) = QTn (g) − P˜nT (g)(ϕ). enϕ QTn (ˆ The proof is the same as for Lemma 6.21.1. In particular, the behaviour of odd order Q-curvature QTn (M, Σ; g) under g → e2ϕ g is described by a linear diﬀerential operator. If for odd n the obstruction operator Res0 (DnT (·, Σn ; g; λ)) : C ∞ (M ) → C ∞ (Σ) is non-trivial, then it comes together with a notion of Q-curvature. In fact, the usual family proof shows that the pair

DnT (·, Σn ; g; 0), D˙ nT (·, Σn ; g; 0)(1) satisﬁes a fundamental identity. Note that DnT (·,Σn ;g;0)(1) = 0 since DM (0)(1) = 0. Another notion of an odd order Q-curvature was introduced in [97]. We brieﬂy recall the deﬁnition. Assume that Σn is the boundary of the manifold M , and that g is a Poincar´e-Einstein metric with conformal inﬁnity [h]. Let S(h; λ) : C ∞ (Σ) → C ∞ (Σ) be the scattering operator (as in Chapter 3). Then n

S(h; n)(1) = (−1) 2 c n2 Qn (h) for even n and S(h; n)(1) = 0 for odd n. Set G ˙ QF n (g; h) = S(h; n)(1). G G Here the notation indicates that QF does not depend only on h. QF satisﬁes n n the fundamental identity G 2ϕ FG enϕ QF n (g; e h) = Qn (g; h) − S(h; n)(ϕ).

The operator S(h; n) is a non-local pseudo-diﬀerential operator of order n on Σ. In particular, it follows that both notions of odd order Q-curvature diﬀer. G Note that in [97], QF n is deﬁned by a slightly diﬀerent normalization (similar to that for the even order Qn ). G The integral of QF n (g; h) only depends on [h] and we have Theorem 6.21.1 ([97]). Let n be odd. Then G − QF n (g; h) vol(h) = V, Σn

where V is the renormalized volume in vol(g) = c0 ε−n + c2 ε−n+2 + · · · + cn−1 ε−1 + V + o(1), ε → 0 r≥ε

(compare with (6.6.25)).

412

Chapter 6. Conformally Covariant Families

G An interesting problem is to uncover the relation between QF (as deﬁned n T in [97]) and Qn (as deﬁned here). In the case n = 3, we have G 4 3 T 4 3 2v Lemma 6.21.3. QF 3 (M , Σ ; g; h) coincides with Q3 (M , Σ ; e g), up to a con∞ stant multiple. Here v ∈ C (M ) is a certain solution of the equation Δg v = −3 ([97]) which induces a conformal compactiﬁcation e2v g of the Poincar´e-Einstein metric g.

Proof. In the following formulas, we suppress constant multiples. By [70] (Lemma 2.2), G 4 3 2v QF 3 (M , Σ ; g; h) = T (e g), where T = ∇N τ is the order 3 Chang-Qing Q-curvature for the metric e2v g (see Theorem 6.26.1). Here we use that L = 0. On the other hand, by Theorem 6.26.2, QT3 (M 4 , Σ3 ; e2v g) = T (e2v g).

This proves the assertion.

Does this relation generalize? Note also that the conformal compactiﬁcation e2v g has the property ([70]) Qn (M n , e2v g) = 0

(6.21.9)

for even n. In order to prove the vanishing result (6.21.9), we apply Theorem 3.2.3. Since τg = −n(n−1), we ﬁnd Pn (g)(v) =

n−2 j(n−1−j) τg Δg (v) = j(n−1−j)Δg (v) = −(n−1)! Δg − n(n−1) n n

n−2 j= 2

j= 2

n

and Qn (g) = (−1) 2 (n−1)!. Hence n

env Qn (M n , e2v g) = Qn (M n , g) + (−1) 2 Pn (g)(v) n

n

= (−1) 2 (n−1)! − (−1) 2 (n−1)! = 0. Problem 6.21.2. Determine the relation between (Pn , Qn ) and (Pn , Qn ). There is one important special class of background metrics for which we can prove a result in the direction of Problem 6.21.2. We recall that for conformally ﬂat h, the Feﬀerman-Graham expansion terminates at the third term (see Lemma 6.14.1 and its proof). Theorem 6.21.2 (Holographic duality). Let n ≥ 3. On M n+1 = (0, ε) × Σn we consider a Poincar´e-Einstein metric r−2 (dr2 +hr ) with conformal inﬁnity h. Let h be conformally ﬂat. Then T res (dr2 +hr ; λ) = D2N (h; λ). D2N

T 6.21. The tractor families DN (M, Σ; g; λ)

413

Proof. We work in local coordinates so that h = e2ϕ hc with hc = proof rests on the identity

i

dx2i . The

res T (hc ; λ) = D2N (dr2 +hc ; λ). D2N T (dr2 +hc ; λ) is induced by D2N (λ), i.e., In fact, Theorem 6.18.7 implies that D2N T nc D2N (dr2 +hc ; λ) = D2N (λ).

On the other hand, Theorem 5.2.5 and Deﬁnition 6.6.2 yield res nc (hc ; λ) = D2N (λ). D2N

As in Section 6.11, let κ be the local diﬀeomorphism with the properties

κ∗ r−2 (dr2 + ˆ hr ) = r−2 (dr2 +hr ) and i∗

κ∗ (r) r

= eϕ ,

ˆ = e2ϕ h. h

Then for h = hc we get ∗ −λ ∗ −λ κ (r) κ (r) res ∗ T 2 ◦ κ = D2N (dr +hc ; λ) ◦ ◦ κ∗ . D2N (hc ; λ) ◦ r r T (g; λ) and the relation Now we apply (6.11.4), the conformal covariance of D2N ∗ ∗ ϕ i (κ (r)/r) = e . We ﬁnd ∗ res T e−(λ−2N )ϕ ◦ D2N (h; λ) = e−(λ−2N )ϕ ◦ D2N (κ (r)/r)2 (dr2 +hc ); λ ◦ κ∗ T = e−(λ−2N )ϕ ◦ D2N (dr2 +hr ; λ)

using i∗ κ∗ = id.

There is an analogous result for odd order families. We omit the details. As consequences, we obtain the following tractor formulas for GJMS-operators and critical Q-curvature. Theorem 6.21.3. Let n ≥ 3 be even and assume that h is conformally ﬂat. Then T (dr2 +hr ) = P2N (h)i∗ P2N

and

n D˙ nT (dr2 +hr ; 0)(1) = −(−1) 2 Qn (h).

Proof. We ﬁnd

n T T P2N dr2 +hr ; − +N (dr2 +hr ) = D2N

2 n res = D2N h; − +N 2 = P2N (h)i∗

(by (6.21.7)) (by Theorem 6.21.2) (by (6.6.15))

414

Chapter 6. Conformally Covariant Families

and D˙ nT (dr2 +hr ; 0)(1) = D˙ nres (0)(1)

(by Theorem 6.21.2)

n 2

= −(−1) Qn (h)

(by (6.6.2)).

The proof is complete.

Theorem 6.21.3 shows that the conformal ﬂatness of h implies that DnT (dr2+ hr ; λ) = Pn (h)i∗ , i.e., the extrinsic part Pne (dr2 +hr ) vanishes. Similarly, we have Qen (dr2 +hr ) = 0. We expect that the same is true for any h. More precisely, Conjecture 6.21.4. Pne (dr2 +hr ) = 0 and Qen (dr2 +hr ) = 0. The holographic duality very likely holds true for more general metrics h. In Section 6.22 we prove its validity for any h if the order of the family is ≤ 4 (Theorem 6.22.6). The proof rests on factorization identities. For a conformally ﬂat h, factorization identities for residue families (Theorem 6.11.1) immediately imply factorization identities for tractor families using Theorem 6.21.2. More precisely, Theorem 6.21.4. For a conformally ﬂat metric h, the tractor family T (dr2 +hr ; λ) : C ∞ ([0, ε) × Σ) → C ∞ (Σ) D2N

factorizes for " # n+1 n+1 n n λ∈ − ,...,− +N ∪ − +N, . . . , − +2N 2 2 2 2 into products of lower order tractor families and GJMS-operators. More precisely, for j = 1, . . . , N , the identities

n n T T 2 dr2 +hr ; − +2N −j = P2j (h) ◦ D2N D2N −2j dr +hr ; − +2N −j 2 2 and T D2N

n+1 n+1 2 T 2 +j = D2N −2j dr +hr ; − −j ◦ P2j (dr2 +hr ) dr +hr ; − 2 2

hold true. Above we used an indirect argument to prove Theorem 6.21.4. It rests on Theorem 6.21.2, i.e., on the relation to residue families and their factorizations. For the convenience of the reader, we next present the argument in a way which is independent of residue families. It only rests on the conformal covariance of tractor families and their factorizations for the ﬂat metric hc = i dx2i . For that metric, the factorization identities follow from T nc D2N (dr2 +hc; λ) = D2N (λ)

T 6.21. The tractor families DN (M, Σ; g; λ)

415

and the factorization identities for D2N (λ) (see the arguments in the proof of Theorem 6.21.2). In fact, we compose the identity

n n T T 2 dr2 +hc ; − +2N −j = P2j (hc ) ◦ D2N D2N −2j dr +hr ; − +2N −j 2 2 with

κ∗ (r) r

n2 −2N +j

◦ κ∗ ,

where κ is the local diﬀeomorphism in the proof of Theorem 6.21.2, i.e., κ∗ (κ∗ (r)/r) (dr2 +hc) = dr2 +hr . 2

Using conformal covariance, we ﬁnd

n n T (κ∗ (r)/r)2 (dr2 +hc ); − +2N −j ◦ κ∗ e( 2 +j)ϕ ◦ D2N 2

n n T ∗ 2 2 ∗ = P2j (hc ) ◦ e( 2 −j)ϕ ◦ D2N −2j (κ (r)/r) (dr +hc ); − +2N −j ◦ κ , 2 i.e.,

n T D2N dr2 +hr ; − +2N −j

n2 n n T 2 = e( 2 +j)ϕ ◦ P2j (hc ) ◦ e( 2 −j)ϕ ◦ D2N −2j dr +hr ; − +2N −j 2 by naturality and i∗ κ∗ = id. Now the ﬁrst system of factorization identities follows from n n P2j (h) = e−( 2 +j)ϕ ◦ P2j (hc ) ◦ e( 2 −j)ϕ , h = e2ϕ hc . Similarly, in order to prove the second system we compose n+1 n+1 T 2 T 2 +j = D2N −2j dr +hc ; − −j ◦ P2j (dr2 +hc ) D2N dr +hc; − 2 2 with

κ∗ (r) r

n+1 2 −j

and ﬁnd T D2N

n+1 +j (κ∗ (r)/r)2 (dr2 +hc ); − 2 n+1 T ∗ 2 2 −j ◦ P2j (κ∗ (r)/r)2 (dr2 +hc) = D2N (r)/r) (dr +h ); − (κ c −2j 2

416

Chapter 6. Conformally Covariant Families

using conformal covariance. Finally, we compose with κ∗ . Naturality and i∗ κ∗ = id imply n+1 n+1 T 2 T 2 +j = D2N −2j dr +hr ; − −j ◦ P2j dr2 +hr . D2N dr +hr ; − 2 2

The proof is complete.

Of course, it would be interesting to extend Theorem 6.21.4 beyond the conformally ﬂat category. In the situation of Theorem 6.21.2, the trace-free part L0 of L vanishes. In fact, it is a well-known observation of LeBrun ([164], Proposition 2.8) that the Einstein condition Ric(g) + ng = 0 implies L0 = 0, i.e., Σ is totally umbilic; see also (6.16.10). In the case n = 3, we will see later (Theorem 6.25.1) that L0 actually determines the obstruction B3 , i.e., the vanishing of L0 implies the vanishing of the obstruction B3 . For even n, the poles of the families T D2N (λ) : Nλ−2N (gn+1 ) → Nλ (gn+2 )

and

T (M, Σn ; g; λ) : C ∞ (M n+1 ) → C ∞ (Σn ) D2N

at the zeros of the product N

(n−(j +N )+λ)

j=2

seem to be related to the non-existence of certain conformally covariant powers of the Laplacian. More precisely, we recall (Theorem 5.2.2) that for all n, n

N (6.21.10) D2N − +N = (Δ− n ) , N ≥ 1. 2 N Thus in order to induce a curved version of (Δ− n ) , we would use the semiT T holonomic lift D2N (λ) or the tractor construction D2N (M, Σn ; g; λ) at λ = − n2 +N . T But the family D2N (·; Σn ; g; λ) has a pole at λ = − n2 + N if

n − +N ∈ {−n+N +2, . . . , −n+2N } , 2 i.e., for n = 4, 6, . . . , 2N. For even n, obstructions arise as follows. In view of the above mentioned T (·, Σn ; g; − n2 + N ) is possibly not well deﬁned if the order 2N poles, the value D2N exceeds the dimension n of the target manifold Σ, i.e., 2N ≥ n. In the extreme case 2N = n, however, Conjecture 6.21.1 states that the value DnT (·; Σn ; g; 0) is still well

T 6.21. The tractor families DN (M, Σ; g; λ)

417

deﬁned. Thus we are left with the cases 2N > n (n even). These are precisely the cases for which the GJMS-construction does not work for general metrics (because of the obstructed Feﬀerman-Graham expansion). Moreover, Graham [118] proved that on a four-manifold Σ there exists no natural conformally covariant operator of the form Δ3Σ + LOT . For more details we refer to the discussion in Section 6.12. More generally, Gover and Hirachi ([112]) proved that the condition 2N ≤ n in the construction of [124], i.e., in Theorem 3.1.1, is sharp, i.e., Graham’s non-existence extends to all cases 2N > n (n even). From the above perspective, an obstruction N is the residue to the construction of a curved analog of D2N (− n2 +N ) = (Δ− n) T (M, Σn ; g; λ)) : C ∞ (M ) → C ∞ (Σ). B2N (g) = Res− n2 +N (D2N def

The conformal covariance of the tractor family implies that the operator B2N is conformally covariant, i.e., n

n

g ) ◦ e(− 2 +N )ϕ = B2N (g). e( 2 +N )ϕ ◦ B2N (ˆ In Lemma 6.12.1, we have seen that on manifolds of dimension n = 4, the secondorder operator R = −δ(B#d) + (B, P) is an obstruction to the existence of a conformally covariant cube of the Laplacian. It would be interesting to interpret this result from the tractor point of view and to relate the obstructions. Problem 6.21.3. Characterize in geometric terms the conformally invariant condition B2N (g) = 0, 2N > n on g. Does the vanishing only depend on the metric i∗ (g) on Σ? Is the operator B2N (g) induced (in a sense) by the residue Res− n2 +N (ΘT2N (λ)) of the semi-holonomic family? T (M, Σ; g; λ) is regular at the argument If B2N (g) = 0, then the family D2N n λ = − 2 +N and the conformally covariant operator

n T T M, Σ; g; − +N : C ∞ (M ) → C ∞ (Σ) P2N (g) = D2N 2

is well deﬁned. Examples of such situations are provided by the group-equivariant families considered in Chapter 5. Example 6.21.1. The simplest case concerns the order 6 family D6T (λ). For D6 (λ), 3 T we have the relation D6 (− n2 + 3) = (Δ− n ) . The semi-holonomic lift D6 (λ) has T n poles at λ = −n+6 and λ = −n+5. Thus, using D6 (·, Σ ; g; λ), a construction of a curved analog P6 of Δ3 is problematic if − n2 + 3 = −n + 6, i.e., n = 6 and if − n2 + 3 = −n + 5, i.e., n = 4. According to Conjecture 6.21.1, the case n = 6 is not problematic. Therefore, it remains to analyze the vanishing of the obstruction operator B6 (g) : C ∞ (M 5 ) → C ∞ (Σ4 ). What is the relation between B6 and R?

418

Chapter 6. Conformally Covariant Families

Finally, we introduce the tractor Q-polynomial. The following deﬁnition generalizes the Q-polynomial in Deﬁnition 6.6.3. We restrict here to the critical case. Deﬁnition 6.21.5 (Tractor Q-polynomial). For (M n+1 , Σn ), we deﬁne the associated critical tractor Q-polynomial by QTn (M, Σ; g; λ) = DnT (M, Σ; g; λ)(1). We do not discuss here the quality of the function QTn (g; λ), i.e., whether it actually is a polynomial. It remains an open problem to embed the full theory of the Q-polynomials which are derived from the residue families into an analogous theory for tractor Q-polynomials. We have used homomorphisms of semi-holonomic Verma modules to suggest the existence of curved versions in terms of tractor D-operators. However, we expect that there is also a direct induction machine so that the rational family ΘN (λ) : Nλ−N (gn ) → Nλ (gn+1 ) induces

T DN (M n , Σn−1 ; g; λ) : C ∞ (M ) → C ∞ (Σ).

Such an induction machine should rest on conformal Cartan connections together with reductions of structures from M to Σ. Note that the Willmore curvature H 2 − K of a surface Σ ⊂ R3 appears in the curvature of the reduction of the Cartan connection of (R3 , gc ) to Σ ([216], p.314) as well as in the Q-polynomial Q2 (R3 , Σ2 ; gc ; λ) (see 6.3.1). In order to deﬁne an induction machine, it would be crucial to describe the operator ΠΣ in terms of Cartan’s principal bundles.

6.22 Some results on tractor families In the present section, we illustrate the perspective outlined in Section 6.21 by a series of results. First of all, we conﬁrm Conjecture 6.21.1 in the cases n = 2 and n = 4. For n = 2, the regularity of D2T (·, Σ2 ; g; λ) at λ = 0 follows from D2T (·, Σ2 ; g; λ) =

1 λD2 (·, Σ2 ; g; λ) (−2 + 2λ)(1 + 2λ)

(6.22.1)

(Theorem 6.24.1) with the holomorphic family D2 (λ). For n = 4, the proof of the analogous result requires some more work. Theorem 6.22.1. D4T (M 5 , Σ4 ; g; 0) = 0. Proof. We have to prove that CΣ (−3) ◦ {CΣ (−2)ΠΣ DM (−1)} ◦ DM (0) = 0. First, we observe that, up to irrelevant non-vanishing coeﬃcients, ⎞ ⎛ ⎞ ⎛1 u− 3 ΔM u CΣ (−3) ⎝ ω ⎠ = −4δΣ ω − ΔΣ u+ , DM (0)u = ⎝ du ⎠ u+ 0

6.22. Some results on tractor families

419

(see (6.20.6) and (6.20.2)). Hence ⎛1 ΠΣ DM (0)u = ⎝

⎞ − HN u ⎠. dΣ u 0

3 ΔM u

Now we use the fact that CΣ (−2) ◦ ΠΣ ◦ DM (−1) = (ΔTΣ −JΣ ) ◦ ΠΣ = TΣ ◦ ΠΣ , where ΔTΣ is the tractor Laplacian on Σ. This is a tractor generalization (Theorem 6.24.2) of Theorem 6.24.1. Next, we need an explicit formula for ΔTΣ . The deﬁnition of ∇T implies the following formula for ∇TX ∇TY : ⎛

∇X ∇Y −P(X, Y ) ⎝ −X ∇Y −∇X (Y ·) 2(X, Y )

∇X (P(Y ), ·) + (P(X), ∇Y ·) ∇X ∇Y −P(X)Y, ·−X (P(Y ), ·) −2X, ∇Y ·−2∇X Y, ·

1 (P(X), P(Y )) 2 1 (∇ (P(Y )·)+P(X)∇Y X 2 ∇X ∇Y −P(X, Y )

⎞ ·)⎠ .

Now applying Lemma 6.20.3 and the Weitzenb¨ ock formula tr(∇LC ∇LC ) = Δ + Ric # for the Bochner-Laplacian on 1-forms, we ﬁnd ⎛ Δ−J −(dJ, ·) + 2δ(P#·) ΔT = ⎝ −2d Δ + Ric # − 2P# 2n −4δ Since for n = 4, Ric −2P = Jg, we obtain ⎛ ∗ ΔΣ −JΣ ΔΣ + JΣ ΔTΣ = ⎝ −2dΣ 8 −4δΣ

1 2 2 |P|

⎞

+ P#d⎠ . Δ−J

1 2 dJ

1 2 2 |PΣ |

(6.22.2)

⎞

∗ ⎠. ΔΣ −JΣ

Hence the desired composition equals ⎞ ∗ CΣ (−3) ⎝− 32 dΣ ΔM u + 2dΣ (HN u) + ΔΣ dΣ u⎠ 8 3 ΔM u − 8HN u − 4ΔΣ u 8 = ΔΣ ΔM u − 8ΔΣ (HN u) − 4δΣ ΔΣ dΣ u 3 8 − ΔΣ ΔM u + 8ΔΣ (HN u) + 4Δ2Σ u = 0. 3 ⎛

The proof is complete.

420

Chapter 6. Conformally Covariant Families

Remark 6.22.1. Corollary 6.20.3 says that CΣ (−3)(ΔTΣ − JΣ )DΣ (0) = 0

(6.22.3)

on a 4-manifold Σ. Although that identity resembles Theorem 6.22.1, it is important to emphasize the diﬀerent nature of both assertions. In fact, while Theorem 6.22.1 says that ⎛1 ⎞ 3 ΔM u − HN u ⎠ ∈ ker CΣ (−3)TΣ , dΣ u ΠΣ DM (0)u = ⎝ 0 (6.22.3) states that ⎛1 DΣ (0)u =

⎞

2 ΔΣ u ⎝ dΣ u ⎠

∈ ker CΣ (−3)TΣ

0 which follows from ⎞

⎛1

(ΔTΣ −

2 ΔΣ u JΣ ) ⎝ dΣ u ⎠

0

⎛

∗ = ⎝−2dΣ 8

∗ ΔΣ −4δΣ

⎞ ⎛ ⎞ ⎞ ⎛1 ∗ ∗ 2 ΔΣ u ∗⎠ ⎝ dΣ u ⎠ = ⎝0⎠ . 0 ∗ 0

Remark 6.22.2. The curved translation D4T (M, Σ; g; 0) = CΣ (−3)CΣ (−2)ΠΣ DM (−1)DM (0) of i∗ : C ∞ (M ) → C ∞ (Σ) is the curved analog of Θ4 (0). Therefore, we consider the vanishing D4T (0) = 0 (Theorem 6.22.1) as the statement that the non-vanishing element 2 Θ4 (0) (Δ− 4 ) − L4 ∈ ker π (see (6.18.16)) induces (via Cartan geometry) a vanishing operator C ∞ (M 5 ) → 2 C ∞ (Σ4 ). On the other hand, (Δ− 4 ) − L4 also appears in the curved translation of the Yamabe operator. In fact, according to Theorem 6.18.1, it is a multiple of the Zuckerman translate of Δ− 4 . The curved version of that construction is CΣ (−3)TΣ DΣ (0). Its vanishing (Corollary 6.20.3) is to be considered as the statement that the non2 vanishing element (Δ− 4 ) − L4 induces (via Cartan geometry) a vanishing operator ∞ 4 ∞ 4 C (Σ ) → C (Σ ). We emphasize that the two respective inductions are of different nature. While the latter one uses intrinsic geometry of Σ the former one involves reductions from M to Σ. The observations in Remark 6.22.1 actually indicate the diﬀerent nature of both vanishing results.

6.22. Some results on tractor families

421

Next, we conﬁrm Conjecture 6.21.2 in the special case N = 2. In particular, assuming n = 4, we prove that D4T (·, Σn ; g; −(n−4)/2) only depends on the pullback of the metric to Σ. The critical case n = 4 is more complicated, and will be discussed below. n ∗ Theorem 6.22.2. D4T (M n+1 , Σn ; g; − n−4 2 ) = P4,n (Σ ; g)i if 3 ≤ n = 4.

Proof. We present the details for n = 3. The general case is analogous. We ﬁrst prove that the left-hand side does not depend on the embedding. For that purpose, we have to analyze the composition (−3+2λ)(−5+2λ)(2+2λ)2λD4T (λ) = (−5+2λ)(2+2λ)CΣ(λ−3) (−3+2λ)2λ CΣ (λ−2)ΠΣ DM (λ−1) DM (λ) = 3λ(λ−1)D4T (λ) at λ = 12 , i.e., D4T

1 5 1 1 T = 8CΣ − D2 M, Σ; g; − DM 2 2 2 2 1 1 5 T = 8CΣ − ΔΣ − JΣ ΠΣ DM . 2 2 2

Here we have used 1 1 D2T (λ−1) = (2λ−3)(2λ) D2T (λ−1) = (2λ−3)2λ CΣ (λ−2)ΠΣ DM (λ−1) λ λ and Theorem 6.24.2. Now we apply the explicit formulas ⎞ ⎛1 1 3 ΔM + 2 JM u 1 ⎠ , u ∈ C ∞ (M ), dM u DM u=⎝ 2 u 1 5 (−8)CΣ − u = −2u− − 4δΣ ω − ΔΣ u+ − JΣ u+ , u ∈ Γ(T ∗ Σ) 2 2 and (using Ric −2P = −P + J) ⎛ ΔΣ − 32 JΣ −(dΣ JΣ , ·) + 2δΣ (PΣ #·) 1 T ΔΣ − PΣ + 12 JΣ ΔΣ − JΣ = ⎝ −2dΣ 2 6 −4δΣ Hence

⎞

1 2 2 |PΣ | 1 ⎠. 2 dΣ JΣ + PΣ #dΣ ΔΣ − 32 JΣ

⎛1 ⎞ 1 1 2 3 ΔM + 2 JM u − HN u + 4 H u 1 ⎠. dΣ u ΠΣ DM u = i∗ ⎝ 2 u

422

Chapter 6. Conformally Covariant Families

Now observe that ⎛ ⎞ ∗ 1 ⎝0⎠ ∈ ker CΣ − 5 . ΔTΣ − JΣ 2 2 0

(6.22.4)

In fact, the operator in (6.22.4) is of the form 3 1 −2 ΔΣ − JΣ + 8δΣ dΣ − 6 ΔΣ + JΣ , ∗, ∗ = (0, ∗, ∗). 2 2 It follows that D4T ( 12 ) lives on Σ. It remains to calculate the composition of ⎛ ∗ −(dJ, ·) + 2δ(P#·) − ⎝∗ Δ − P + 12 J ∗ −4δ with

⎞⎛

⎞ 0 1 ⎠ ⎝du⎠ , u ∈ C ∞ (Σ) 2 dJ + P#d 3 u Δ− 2 J 1 2 2 |P|

1 u → −2u− − 4δω − Δ + J u+ , u ∈ Γ(T ∗ Σ); 2

here all constructions are with respect to Σ. The above matrix product yields ⎛ ⎞ −(dJ, du) + 2δ(P#du) + 12 |P|2 u ⎝ ⎠. Δdu + 12 Jdu + 12 u(dJ) 3 −3Δu − 2 Ju Thus we ﬁnd 2(dJ, du) − 4δ(P#du) − |P|2 u

1 3 − 4δ(Δdu) − 2δ(udJ) − 2δ(Jdu) + Δ + J 3Δ + J u 2 2 3 1 = −Δ2 u − 4δ(P#du) + (du, dJ) + JΔu − |P|2 u + J2 u − uΔJ 4 2 1 3 2 2 2 = −Δ u + δ(Jg − 4P)#du + J −2|P| −ΔJ u 2 2 = −P4 u

by (4.1.7). The proof is complete.

Remark 6.22.3. The extrinsic construction of the Paneitz operator P4 (Σn ; g) in Theorem 6.22.2 is an analog of the intrinsic construction n n

n −1 JΣ ◦ DΣ − +2 : C ∞ (Σ) → C ∞ (Σ) CΣ − −1 ◦ ΔTΣ − 2 2 2

6.22. Some results on tractor families

423

by curved translation (Corollary 6.20.3). The individual factors in the latter composition are given by the formulas ⎞ ⎛1 n n

2 ΔΣ −( 2 −2)JΣ ⎠, dΣ u DΣ − +2 = ⎝ 2 −(n−4)u n

n CΣ − +1 ∼ 2(n−4)u− − 4δω − ΔΣ − −2 JΣ u+ 2 2 and ⎛

ΔΣ − n2 JΣ T ΔΣ − −1 JΣ = ⎝ −2dΣ 2 2n n

−(dΣ JΣ , ·) + 2δΣ (PΣ #·) ΔΣ + n2 −2 (2PΣ −JΣ ) −4δΣ

⎞

1 2 2 |PΣ | 1 d J + P Σ #dΣ ⎠ 2 Σ Σ ΔΣ − n2 JΣ .

Now a calculation (see the proof of Theorem 6.20.7) shows that the composition is a multiple of (n − 4)P4 on Σn . In particular, it vanishes on Σ4 . In contrast, the operator D4T (·; Σ4 ; 0) does not vanish and reproduces P4 i∗ (together with a Paneitz-type operator of the embedding). Finally, note that the composition n

n −1 JΣ ΠΣ DM − −2 ΔΣ − 2 2 depends on the embedding. Next, we consider the Q-curvatures QT2 (·, Σ2 ; g) and QT4 (·, Σ4 ; g) more closely. Theorem 6.24.1 implies that for n ≥ 3, D2T (·, Σn ; dr2 +hr ; λ) = D2 (·, Σn ; dr2 +hr ; λ) for the family D2 (λ) constructed in Section 6.4. On the other hand, by Theorem 6.7.1, the residue of ε rλ uϕ vol(dr2 +hr ), −Δr−2 (dr2 +hr ) u = μ(n−μ)u 0

Σ

at λ = −μ−3 yields the family δ2 (h; λ) = −

1 D2 (dr2 +hr ; λ). 2(n−2+2λ)

Renormalization as in (6.6.7) gives the residue family D2res (h; λ) = −2(n−2+2λ)δ2(h; λ) = D2 (dr2 +hr ; λ), i.e., D2res (Σ; h; λ) = D2T (M, Σ; dr2 +hr ; λ).

(6.22.5)

424

Chapter 6. Conformally Covariant Families

Note that

n

n −1 J i∗ = P2 (h)i∗ . D2 dr2 +hr ; − +1 = Δ− 2 2 This is the non-critical case of the holographic duality for order 2 families with Σ of dimension n ≥ 3. For n = 2, the tractor family D2T (λ) is not deﬁned but the above arguments show that in this case the family D2 (λ) from Section 6.3 should be used as a substitute. Example 6.22.1. For n = 2, we have D2T (·, Σ2 ; g; λ) = D2 (·, Σ2 ; g; λ) by (6.22.1). Hence D˙ 2T (·, Σ2 ; g; 0)(1) = D˙ 2 (·, Σ2 ; g; 0)(1) = Q2 (Σ2 ; g) by Corollary 6.3.1. In particular, Qe2 (·, Σ2 ; g) = 0 for all metrics g. Example 6.22.2. Let n = 4. We determine the tractor Q-polynomial QT4 (M, Σ4 ; g; λ) (see Deﬁnition 6.21.5) if M = (0, ε) × Σ4 and g = dr2 + (h + r2 h(2) + r4 h(4) ) are the ﬁrst terms in the Taylor series of the Poincar´e-Einstein metric r−2 (dr2 + hr ) associated to h. The fact that h(4) is not fully determined by h = h(0) will have no inﬂuence. In fact, tr(h(4) ) is determined, and this is the only information on h(4) which enters into QT4 (g; λ). By Deﬁnition 6.21.1, we have D4T (M 5 , Σ4 ; g; λ) = CΣ (λ−3) ◦ {CΣ (λ−2) ◦ ΠΣ ◦ DM (λ−1)} ◦ DM (λ). Now D4T (·, Σ4 ; g; λ) is determined by the factorization 3

1 1 1 1 λ(λ+1)D4T (·, Σ4 ; g; λ) (2λ−4) (2λ−2) (2λ+1) (2λ+3)

(see Deﬁnition 6.21.3) of this family. We use the identity CΣ (λ−1)ΠΣ DM (λ) =

λ+2 DT (M, Σ; g; λ) 2λ(2λ+3) 2

with D2T (M, Σ; g; λ)

3 T = −(2λ+2)i ΔM − JM + (2λ+3) ΔTΣ − JΣ i∗ 2 ∗

(see (6.24.1)). Hence 6λD4T (g; λ) is given by the composition of ⎛ ⎞ u− ⎝ ω ⎠ → −4λ(λ−1)u− + 4(λ−1)δΣ ω − (ΔΣ −λJΣ )u+ u+

6.22. Some results on tractor families

and

425

3 −2λi∗ ΔTM − JM + (2λ+1) ΔTΣ −JΣ i∗ 2

acting on

⎛

⎞ (ΔM +λJM )u ⎝ (2λ+3)dM u ⎠ . 2λ(2λ+3)u

Thus 6λD4T (g; λ)(1) = 6λQT4 (g; λ) is given by the application of −4λ(λ−1)u− + 4(λ−1)δΣ ω − (ΔΣ −λJΣ )u+ to ⎛ ΔM − 52 JM ∗⎜ − 2λi ⎝ −2dM 10

∗ ∗ ∗ +

⎞

⎛ ⎞ 1 2 λJM 2 |PM | ⎟⎝ 1 ⎠ 0 2 dM JM ⎠ 5 2λ(2λ+3) ΔM − 2 JM ⎛ 1 2 ΔΣ − 2JΣ ∗ 2 |PΣ | ⎜ 1 (2λ+1) ⎝ −2dΣ ∗ 2 dΣ JΣ 8

∗

⎞⎛

⎞ λi∗ JM ⎟⎝ ⎠. 0 ⎠ 2λ(2λ+3) ΔΣ − 2JΣ

Here we have used formula (6.22.2) for the tractor Laplacian. Thus we ﬁnd that 6QT4 (g; λ) is given by " 5 − 4λ(λ−1) (−2λ)i∗ (ΔM − JM )JM + (2λ+3)|PM |2 2 #

+ (2λ+1) (ΔΣ −2JΣ )i∗ JM + (2λ+3)|PΣ |2 + 4(λ−1)δΣ {−2λ(2λ+1)i∗dM JM + (2λ+1)(−2dΣ (i∗ JM ) + (2λ+3)dΣ JΣ )} − (ΔΣ −λJΣ ) {10λ(2λ+1)i∗JM + (2λ+1)(8i∗ JM − 4(2λ+3)JΣ )} . Now using i∗ JM = JΣ (see Lemma 6.11.1), the latter sum simpliﬁes to " 5 − 4λ(λ−1) (−2λ)i∗ (ΔM − JM )JM + (2λ+3)|PM |2 2 # + (2λ+1) (ΔΣ −2JΣ )JΣ + (2λ+3)|PΣ |2 + 4(λ−1)(2λ+1)ΔΣJΣ − 2(2λ+1)(λ−2)(ΔΣ −λJΣ )JΣ . Hence the coeﬃcients of λ0 and λ1 are −4ΔΣ JΣ + 4ΔΣ JΣ = 0

426

Chapter 6. Conformally Covariant Families

and 4 (ΔΣ −2JΣ )JΣ + 3|PΣ |2 − 4ΔΣ JΣ + 6ΔΣ JΣ − 4J2Σ = 12|PΣ |2 − 12J2Σ + 6ΔΣ JΣ = −6Q4 (Σ; h), respectively. The latter results mean that QT4 (g; 0) = 0

and

− Q˙ T4 (g; 0) = Q4 (Σ; h).

We continue with the calculation of the remaining coeﬃcients of 6QT4 (g; λ). First of all, the coeﬃcient of λ4 is 16|PM |2 − 16|PΣ |2 = 0 by Lemma 6.11.2. For the coeﬃcients of λ2 and λ3 , we ﬁnd (using Lemma 6.11.2) the respective formulas −8¨JM + 6J2Σ − 4|PΣ |2 and 8(¨JM − |PΣ |2 ), where

1 ¨JM def = (∂/∂r)2 1r=0 (JM ).

We omit the details. Hence we obtain the formula

−6QT4 (g; λ) = 6λQ4 (h) + λ2 8¨JM − 6J2Σ + 4|PΣ |2 + λ3 −8¨JM + 8|PΣ |2

2 2 ¨JM , (6.22.6) = 6Qres (h; λ) + 8λ (λ−1) |P | − Σ 4 where Qres 4 (h; λ) is the Q-polynomial given by (6.6.44). In order to complete the evaluation of (6.22.6), we apply Lemma 6.11.1. Thus we have completed the proof of Theorem 6.22.3. In terms of the notation in Example 6.22.2, −QT4 (g; λ) = Qres 4 (h; λ). In other words, for the background metric g = dr2 + hr which corresponds to a Poincar´e-Einstein metric r−2 (dr2+hr ), the tractor Q-polynomial QT4 (g; λ) is given by the Q-polynomial Qres 4 (h; λ) (see (6.6.44)). The following result shows that for the theory of Q4 the trivial embedding is less natural than the one given by the Poincar´e-Einstein extension.

6.22. Some results on tractor families

427

Lemma 6.22.1. In the situation of Example 6.22.2, i.e., n = 4 and the background metric g = dr2 + hr so that r−2 (dr2 + hr ) is Einstein, we have Qe4 = 0, i.e., for such a background metric the Q-curvature QT4 (g) only depends on the Q-curvature of h. On the other hand, for the trivial embedding, i.e., for g = dr2 + h on R × Σ, we have Qe4 = −ΔJ. Proof. The ﬁrst assertion is contained in Theorem 6.22.3. For the convenience of the reader, we repeat the direct argument. In fact, since i∗ (PM ) = PΣ and H = 0 e (by Lemma 6.11.2) we ﬁnd J = 0, i.e., Q4 = 0. For the trivial embedding, we ﬁnd 0 0 L = 0, Ricg = and τg = τh . Therefore, 0 Rich 1 τh

1 τh

Rich − h , PΣ = Rich − h , i∗ (PM ) = 3 8 2 6 i.e., 1 1 1 1 J = i∗ (PM ) − PΣ = − Rich + τh h = − PΣ + Jh h. 6 24 3 12 Lemma 4.2.7 implies 1 1 1 δJ = − dJh + dJh = − dJh . 3 12 4 But since tr(J ) = 0, by Lemma 6.22.4, we ﬁnd Qe4 = 4δ(δJ ) − 2Δ(tr J ) = −ΔJ. The proof is complete.

The following result establishes factorization identities for D4T (dr2+hr ; λ) for general h. Theorem 6.22.4. In the situation of Example 6.22.2, D4T (dr2 +hr ; 0) = P4 (h)i∗ , 3 7 T 2 2 D4 dr +hr ; − = D2 dr +hr ; − ◦ P2 (dr2 +hr ), 2 2 D4T (dr2 +hr ; 1) = P2 (h) ◦ D2 (dr2 +hr ; 1). Proof. We begin with the proof of the ﬁrst identity. We recall from the discussion in Example 6.22.2 that the projection ΠΣ is given by pull-back i∗ and the family 6λD4T (g; λ) is the composition of u → −4λ(λ−1)u− + 4(λ−1)δΣ ω − (ΔΣ −λJΣ )u+ and

3 −2λi∗ ΔTM − JM + (2λ+1) ΔTΣ −JΣ i∗ 2

(6.22.7)

428

Chapter 6. Conformally Covariant Families

acting on

⎞ (ΔM +λJM )u ⎝ (2λ+3)dM u ⎠ . 2λ(2λ+3)u ⎛

We have to prove that the coeﬃcient of λ in this decomposition coincides with 6P4 (h) (recall that, by Theorem 6.22.1, the coeﬃcient of λ0 vanishes). For that purpose, we write (6.22.7) in the explicit form ⎞ ⎛ ⎞⎛ 1 2 ∗ ΔM − 52 JM (ΔM +λJM )u 2 |PM | ⎠ ⎝ (2λ+3)dM u ⎠ ΔM +PM #− 12 JM ∗ − 2λi∗ ⎝ −2dM 10 −4δM 2λ(2λ+3)u ΔM − 52 JM ⎞ ⎛ 1 2 ΔΣ −2JΣ −(dΣ JΣ , ·)+2δΣ (PΣ #·) 2 |PΣ | 1 ⎠ ΔΣ + (2λ+1) ⎝ −2dΣ 2 dΣ JΣ +PΣ #dΣ 8 −4δΣ ΔΣ −JΣ ⎛ ⎞ (ΔM +λJM )u i∗ ⎝ (2λ+3)dM u ⎠ . 2λ(2λ+3)u Here we have used the relations 3 1 (RicM −2PM )#− JM = PM #− JM 2 2

and (RicΣ −2PΣ )−JΣ = PΣ

and formula (6.22.2) for the tractor Laplacian on page 419. For the coeﬃcient of λ, we obtain the sum of ⎞ ΔM u 4 (ΔΣ −2JΣ , −(dΣ JΣ , ·)+2δΣ (PΣ #·), ∗) i∗ ⎝3dM u⎠ 0 ⎛ ⎞ ΔM u 1 + 8 −2ΔΣ , δΣ i∗ (ΔM +PM #− JM ), ∗ ⎝3dM u⎠ 2 0 ⎛

(6.22.8)

and the coeﬃcient of λ in 4(λ−1)(2λ+1) −2ΔΣ (ΔM +λJM )u 1 + (2λ+3)δΣ ΔΣ dΣ u + 2λ(2λ+3)δΣ (dΣ JΣ )u + PΣ #dΣ u 2

5 − (ΔΣ −λJΣ )2λ − 10(ΔM +λJM )u + 4(2λ+3)ΔM u − 2λ(2λ+3) ΔM − JM u 2

− (ΔΣ −λJΣ )(2λ+1) 8(ΔM +λJM )u − 4(2λ+3)ΔΣ u + 2λ(2λ+3) ΔΣ −2JΣ u .

6.22. Some results on tractor families

429

The latter sum yields the contributions 8ΔΣ ΔM u + 8ΔΣ (JM u) − 20Δ2Σ u − 24

1 δΣ ((dΣ JΣ )u) + δΣ (PΣ #dΣ u) 2

+ 20ΔΣ ΔM u − 24ΔΣ ΔM u − ΔΣ [16ΔM u + 8JM u − 32ΔΣ u + 6(ΔΣ −λJΣ )u] + JΣ (8ΔM u − 12ΔΣ u) = −12ΔΣ ΔM u + 6Δ2Σ u − 24δΣ (PΣ #dΣ u) − 12(ΔΣ JΣ )u − 12(dΣ JΣ , du) + 12ΔΣ (JΣ u) − 12JΣ ΔΣ u + 8JΣ ΔM u. The contributions in (6.22.8) are 4ΔΣ ΔM u − 8JΣ ΔM u + 12(−(dΣ JΣ , dΣ u) + δΣ (PΣ #dΣ u)) 1 − 16ΔΣ ΔM u + 24ΔΣ ΔM u + 24δΣ PM − JM dM u. 2 Hence we ﬁnd 6Δ2Σ u + 24δΣ (PΣ #dΣ u) − 12δΣ (JΣ dΣ u) = 6 Δ2Σ u − δΣ (2JΣ − 4PΣ #)dΣ u = 6P4 u. This proves the ﬁrst relation. For the proof of the second factorization, we observe that the formulas in Example 6.22.2 show that (−9)D4T (− 23 )u is given by the composition of ⎛ ⎞ u− ⎝ ω ⎠ → −15u− − 10δΣ ω − ΔΣ + 3 JΣ u+ 2 u +

and

acting on

⎛

ΔM − 52 JM ⎜ 3i∗ ⎝ −2dM 10

⎞ ⎛ ΔΣ −2JΣ ∗ ∗ ⎟ ⎜ ∗ ∗⎠ − 2 ⎝ −2dΣ ∗ ∗ 8

⎞ ∗ ∗ ⎟ ∗ ∗ ⎠ i∗ ∗ ∗

⎞ ⎛ ⎞ (ΔM − 32 JM )u P2 (dr2 +hr )u ⎠=⎝ ⎠. ⎝ 0 0 0 0 ⎛

In order to verify the explicit formula, we observe that 5 − 15 3i∗ ΔM − JM − 2(ΔΣ −2JΣ )i∗ − 10δΣ (−6i∗ dM + 4dΣ i∗ ) 2 ∂2 3 7 − 14 ΔΣ + JΣ = (−9) 5i∗ 2 + ΔΣ − JΣ i∗ . 2 ∂r 2

430

Chapter 6. Conformally Covariant Families

Similarly, it follows that 6D4T (1)u is given by −(ΔΣ −JΣ ) = −P2 (h) applied to the last component of 3 −2i∗ ΔTM − JM + 3 ΔTΣ − JΣ i∗ 2 acting on

⎛

⎞ (ΔM +JM )u ⎝ 5dM u ⎠ . 10u

A calculation shows that this component is given by ∂2 −6 −4 2 + (ΔΣ + JΣ ) = −6D2 (1). ∂r

The proof is complete. Moreover, we have

Theorem 6.22.5. In the situation of Example 6.22.2, the family D4T (dr2 +hr ; λ) is polynomial of degree 2. Proof. 6λD4T (dr2 + hr ; λ) is given by the composition of ⎛ ⎞ u− ⎝ ω ⎠ → −4λ(λ−1)u− + 4(λ−1)δΣ ω − (ΔΣ −λJΣ )u+ u+ and ⎛ ΔM − 52 JM ⎜ −2λi∗ ⎝ −2dM 10 acting on

⎞ ⎛ 1 2 ∗ ΔΣ −2JΣ 2 |PM | ⎟ ⎜ 1 ∗ 2 dM JM ⎠ + (2λ+1) ⎝ −2dΣ ∗ ΔM − 52 JM 8

∗ ∗ ∗

1 2 2 |PΣ | 1 2 dΣ JΣ

⎞ ⎟ ∗ ⎠i

ΔΣ −2JΣ

⎛

⎞ (ΔM +λJM )u ⎝ (2λ+3)dM u ⎠ . 2λ(2λ+3)u

Thus D4T (λ) is polynomial of degree ≤ 4. For the coeﬃcients of λ5 and λ4 in the product 6λD4T (λ)u, we ﬁnd the respective formulas 4 i∗ (|PM |2 u) − |PΣ |2 i∗ (u) 4 = 0

6.22. Some results on tractor families

431

and − 2|PΣ |2 4i∗ (u) 5 ∗ ∗ + JΣ −2i ΔM − JM u + 2(ΔΣ −2JΣ )i (u) 4 2 5 ∗ ∗ − 4 −2i ΔM − JM (JM u) + 2(ΔΣ −2JΣ )i (JM u) 2 2 1 2 ∗ 2 ∗ ∗ ∂ = −8|PΣ | i (u) − 8 JΣ i − JΣ i u ∂r2 2 1 2 ∗ ∗ ∗ ∗ ∗ + 8 i ΔM (JM )u − ΔΣ (JΣ )i (u) + JΣ i (ΔM u) − JΣ (ΔΣ i (u)) − JΣ i (u) 2 2 2 1 2 ∗ 1 2 ∗ 2 ∗ ∗ ∂ 2 ∗ ∗ ∂ = −8|PΣ | i (u) − 8 JΣ i − JΣ i u + 8 |PΣ | i + JΣ i − JΣ i u ∂r2 2 ∂r2 2 =0 using Lemma 6.11.2 and Lemma 6.11.1.

As a corollary of Theorem 6.22.5, we ﬁnd Theorem 6.22.6. Let n = 4, h any metric and r−2 (dr2 + hr ) the (approximate) Poincar´e-Einstein metric with conformal inﬁnity h: hr = h + r2 h(2) + r4 h(4) . Then D4T (dr2 +hr ; λ) = D4res (h; λ). Proof. The tractor family D4T (dr2 + hr ; λ) is polynomial of degree 2 (Theorem 6.22.5). Hence it is completely determined by the three factorization identities in Theorem 6.22.4. On the other hand, D4res (h; λ) satisﬁes an analogous system of three factorization identities. The structure of these identities and the relation D2res (h; λ) = D2T (dr2 +hr ; λ) (see (6.22.5)) imply the assertion.

Theorem 6.22.4 implies that D4T (dr2 +hr ; 0)(1) = 0. Although a formula for for general g is rather complicated, it is easy to prove the analogous vanishing result. D4T (g; 0)

Lemma 6.22.2. D4T (·, Σ4 ; g; 0)(1) = 0. Proof. We note that the quantity D4T (·; Σ4 ; g; 0)(1) is given by ⎛ ⎞ JM CΣ (−3)(ΔTΣ − JΣ )ΠΣ ⎝ 0 ⎠ . 6

432

Chapter 6. Conformally Covariant Families

⎞ u− But for u = ⎝ ω ⎠ ∈ Γ(T ∗ (Σ)) CΣ (−3)u is a multiple of −4δΣ ω − ΔΣ u+ . Moreu+ over, ⎞ ⎛∗ ⎞ ⎛ i JM + 32 H 2 ∗ ⎠ = ⎝−2dΣ (i∗ JM + 3 H 2 ) + 3dΣ JΣ ⎠ 0 (ΔTΣ − JΣ ) ⎝ 2 8i∗ JM + 12H 2 − 12JΣ 6 ⎛

using (6.22.2). Hence the desired composition is 3 8ΔΣ (i∗ JM + H 2 ) − 12ΔΣ JΣ − 8ΔΣ (i∗ JM ) − 12ΔΣ H 2 + 12ΔΣ JΣ = 0. 2

The proof is complete.

P4e (·, Σ4 ; g)

For a general background metric g, the extrinsic contribution does not vanish. However, it is not so easy to see that it is a diﬀerential operator which is tangential to Σ, i.e., lives on Σ. In fact, we recall that translation yields the rational family < = 1 − 2 2 2 − 2 − − 2 − − 2 − (Δ ) + L4 − 4 Δ4 (Y5 ) − (Δ4 ) − L4 + Yi (Y5 ) Yi + ··· λ 3 4 3 i 2 T (see (6.18.16)). Assuming that the residue (Δ− 4 ) − L4 induces Res0 (D4 ) = 0 (Theorem 6.22.1) (see also Remark 6.22.2), it is natural to expect that − 2 − − 2 − − 2 − (Δ4 ) − 4 Δ4 (Y5 ) − Yi (Y5 ) Yi (6.22.9) i

induces D4T (0). Since the second sum contains two normal derivatives (Y5− )2 , it seems to be natural to expect that the extrinsic part of D4T (0) contains (at most) two normal derivatives. But note that − 2 Yi− (Y5− )2 Yi− = Yi− [Yi− , Y5− ]Y5− + Y5− [Yi− , Y5− ]) . Δ− 4 (Y5 ) − i

i

This formula suggests that we should expect (at most) one normal derivative (combined with tangential derivatives and curvature). We will see later that the correct operator actually is tangential, i.e., of normal order 0. Next, we discuss the decomposition −QT4 = Q4 + Qe4 under the additional assumption H = 0. In particular, the following result gives an explicit formula for the extrinsic Q4 -curvature of (X 5 , M 4 , g) if H = 0. It implies that D4T (g; 0), acting on ker i∗ ∇N , is tangential to Σ if H = 0 (Theorem 6.22.8), i.e., D4T (g; 0) has normal order ≤ 1.

6.22. Some results on tractor families

433

Theorem 6.22.7. We consider Σ4 → M 5 with the background metric g. Assume that H = 0. Then QT4 (g) = D˙ 4T (g; 0)1 is given by −QT4 (g) = Q4 (i∗ (g)) + Qe4 (g), where

2 Qe4 = 4δ(δθ) − 2ΔΣ (tr θ) = 4δ(δθ) − ΔΣ (tr ∧2 L0 ) 3

and

θ = i∗ (PM ) − PΣ . def

In particular,

Qe4

(6.22.10)

= 0 if θ = 0.

Theorem 6.22.7 re-proves that Qe4 = 0 in Example 6.22.2 since in that case L = 0 and θ = 0. Theorem 6.22.7 conﬁrms Conjecture 6.21.3 in the case n = 4 and H = 0. In the case (S 5 , S 4 ) with the round metric, we have L = 0, PS 5 = 12 gS 5 and PS 4 = 12 gS 4 . Hence θ = 0. Thus Qe4 = 0 by Theorem 6.22.7. This corresponds to the observation that the family D4c (λ) : C ∞ (S 5 ) → C ∞ (S 4 ) calculates Q4 (S 4 ; gc ) (see Lemma 6.1.2). For the proof of Theorem 6.22.7, we write the Gauß equation in the form Lemma 6.22.3. For (M n+1 , Σn , g), i∗ JM − JΣ = i∗ PM (N, N ) +

1 tr(∧2 L). n−1

Proof. The Gauß equation (see (6.4.13)) τM = τΣ + 2 RicN N + tr(L2 ) − tr(L)2 is equivalent to 2nJM = 2(n−1)JΣ + 2 ((n−1)PM (N, N ) + JM ) + tr(L2 ) − tr(L)2 , i.e., 2(n−1)JM = 2(n−1)JΣ + 2(n−1)PM (N, N ) + 2 tr(∧2 L).

This yields the assertion. We continue with the proof of Theorem 6.22.7.

Proof. The discussion in Example 6.22.2 shows that 6QT4 (g; λ) is given by the application of ⎛ ⎞ u− ⎝ ω ⎠ → −4λ(λ−1)u− + 4(λ−1)δΣ ω − (ΔΣ −λJΣ )u+ (6.22.11) u+ to

⎛

⎞ JM ⎠, D2T (·, Σ4 ; g; λ−1) ⎝ 0 2λ(2λ+3)

434

Chapter 6. Conformally Covariant Families

where D2T (g; λ) is given by 2λ(2λ+3)CΣ (λ−1)ΠΣ DM (λ) = (λ+2)D2T (g; λ). Now the explicit form of D2T (g; λ) is more complicated than in Example 6.22.2. In fact, we ﬁnd 2(λ+2)D2T (g; λ)u = 4λ(2λ+3)CΣ (λ−1)ΠΣ DM (λ) T ∗ M,T i ∇ u = −4(λ+1)(λ+2)i∗(ΔTM + λJM )u + 4(λ+1)(2λ+3)δΣ

− 2λ(2λ+3)(ΔTΣ − (λ+2)JΣ )i∗ u. Notice that D2T (g; −1) = ΔTΣ − JΣ i∗

and D2T

3 3 g; − = i∗ ΔTM − JM . 2 2

These results are special cases of Theorem 6.24.2 for n = 5, H = 0. Now we rewrite D2T (g; λ−1) in the form 2(λ+1)D2T (g; λ−1) = −4λ(λ+1)i∗ (ΔTM + (λ−1)JM )u + 4λ(2λ+1)Ru + 4λ(2λ+1)ΔTΣ i∗ u − 2(λ−1)(2λ+1)(ΔTΣ − (λ+1)JΣ )i∗ u with

def T ∗ M,T − ∇Σ,T i∗ : Γ(T ∗ M ) → Γ(T ∗ Σ). R = δΣ i ∇

We obtain the formula D2T (g; λ−1) 2λ(2λ+1) Ru = − 2λΔTM u + (2λ+1)ΔTΣ u + (λ−1)(−2λJM + (2λ+1)JΣ )u + λ+1 2λ(2λ+1) = −2λ ΔTM + (λ−1)JM + (2λ+1) ΔTΣ + (λ−1)JΣ u + Ru. λ+1 In particular, the tractor family D2T (g; λ) is only rational in λ if R = 0. Now applying the explicit formulas for the tractor Laplacians, we ﬁnd ⎞ ⎛ ⎞ ⎡ ⎛ 1 2 JM ΔM −JM ∗ 2 |PM | ⎠ = ⎣−2λ ⎝ −2dM 0 ∗ 12 dM JM ⎠ D2T (g; λ−1) ⎝ 2(2λ+3) 10 ∗ ΔM −JM ⎛ ⎞⎤ ⎛ ⎞ 1 ΔΣ −JΣ ∗ 2 |PΣ |2 JM ⎠ + (2λ+1) ⎝ −2dΣ ∗ 12 dΣ JΣ ⎠⎦ ⎝ 0 2(2λ+3) 8 ∗ ΔΣ −JΣ ⎞ ⎞ ⎛ ⎛ JM JM 2λ(2λ+1) ⎠+ ⎠. 0 0 R⎝ + (λ−1) (−2λJM +(2λ+1)JΣ ) ⎝ λ+1 2(2λ+3) 2(2λ+3)

6.22. Some results on tractor families

435

Composing with (6.22.11) yields the sum of " − 4λ(λ−1) (−2λ) (ΔM −JM )JM + (2λ+3)|PM |2 # + (2λ+1) (ΔΣ −JΣ )JM + (2λ+3)|PΣ |2 + (λ−1)(−2λJM + (2λ+1)JΣ )JM + 4(λ−1)δΣ − 2(2λ+1)(λ+1)dΣJM + (2λ+1)(2λ+3)dΣJΣ − (ΔΣ −λJΣ ) − 4(λ+1)(λ−2)(2λ+1)JM + 2(2λ+1)(λ−2)(2λ+3)JΣ and the composition of (6.22.11) with

⎛ 2λ(2λ+1) ⎝ R λ+1

⎞ JM ⎠. 0 2(2λ+3)

We determine the coeﬃcients of λ0 and λ1 in the above sum. For λ0 , we ﬁnd immediately 8ΔΣ JM − 12ΔΣ JΣ − ΔΣ (8JM − 12JΣ ) = 0. This re-proves the special case H = 0 of Lemma 6.22.2. The coeﬃcient of λ in the above sum is given by 4 (ΔΣ − JΣ )JM + 3|PΣ |2 − JΣ JM + 8ΔΣ JM + 8ΔΣ JM − 20ΔΣ JΣ − ΔΣ (20JM − 26JΣ ) + JΣ (8JM − 12JΣ ) = 6ΔΣ JΣ + 12(|PΣ |2 − J2Σ ) = −6Q4 (Σ4 ; g). This yields the intrinsic Q4 -curvature of (Σ4 , g). Finally, we evaluate the extrinsic contribution of R. It does not contribute to the coeﬃcient of λ0 and its contribution to the coeﬃcient of λ1 is ⎛ ⎞ ⎛ ⎞ JM JM T i∗ ∇M,T − ∇Σ,T i∗ ⎝ 0 ⎠ . 2R ⎝ 0 ⎠ = 2δΣ 6 6 In order to make it explicit, we ﬁrst note that for X ∈ X (Σ) and u ∈ Γ(T ∗ M ), ∗ i∗ (∇M,T u) − ∇Σ,T X X (i u) ⎛ M ⎞⎛ ⎞ 0 ∇X (PM (X), ·) u− 1 ⎠⎝ ω ⎠ = i∗ ⎝−X ∇M X 2 PM (X) u+ 0 −2X, · ∇M X ⎞ ⎛ ⎞ ⎛ Σ 0 ∇X (PΣ (X), ·) u− 1 ⎠ i∗ ⎝ ω ⎠ − ⎝−X ∇Σ P (X) Σ X 2 u+ 0 −2X, · ∇Σ X ⎞ ⎛ (PM (X), ω) − (PΣ (X), i∗ (ω)) 1 ∗ Σ ∗ ⎠. = ⎝i∗ (∇M X ω) − ∇X i (ω) + 2 (i PM (X) − PΣ (X))u+ 0

436

Chapter 6. Conformally Covariant Families

Hence if ω = 0, then ⎞ ⎛ 0 1 ∗ ⎝(i∗ (PM (X)) − PΣ (X))u+ ⎠ . i∗ (∇M,T u) − ∇Σ,T X X (i u) = 2 0

(6.22.12)

Now let θ = i∗ PM − PΣ ,

θ(X) = i∗ (PM (X)) − PΣ (X) ∈ Ω1 (Σ), X ∈ X (Σ).

For a geodesic frame {Xi } of Σ, we ﬁnd ⎞ ⎛ ⎞ ⎛ JM 0 2R ⎝ 0 ⎠ = 6 ∇TXi ⎝θ(Xi )⎠ i 6 0 ⎛ Σ ∇Xi (PΣ (Xi ), ·) ⎝−Xi =6 ∇Σ Xi i 0 −2Xi , · ⎞ ⎛ ∗ ⎠. ⎝ ∇Σ =6 Xi θ(Xi ) i −2θ(Xi , Xi )

⎞ ⎞⎛ 0 0 1 ⎠ ⎝θ(Xi )⎠ 2 PΣ (Xi ) 0 ∇Σ Xi

Thus composing with (6.22.11) (for λ = 0) we ﬁnd the contribution Σ ∇Xi θ(Xi ) + 12ΔΣ (tr θ). −24δΣ i

Hence

6QT4 (g; λ) = −6Q4 (Σ4 ; g) − 24δΣ

∇Σ Xi θ(Xi )

+ 12ΔΣ (tr θ),

i

i.e., −QT4 (g; λ) = Q4 (Σ4 ; g) + 4δΣ (δθ) − 2ΔΣ (tr θ). Here we have used the fact that for the geodesic frame {Xi }, ∇Xi (θ)(Xi , Y ) = ∇Xi (θ(Xi )), Y . δθ(Y ) = i

i

Finally, we note that, by Lemma 6.22.3, tr θ =

4

PM (Xi , Xi ) − PΣ (Xi , Xi ) = JM − JΣ − PM (N, N ) =

i=1

The proof is complete.

1 tr(∧2 L0 ). 3

6.22. Some results on tractor families

437

Remark 6.22.4. The proof of Theorem 6.22.7 also shows that the family D2T (g; λ) : Γ(T ∗ M ) → Γ(T ∗ Σ) on tractors has a a simple pole at λ = −2 with residue 2R. In other words, the operator T ∗ M,T R = δΣ (i ∇ − ∇Σ,T i∗ )

deﬁnes a conformally invariant operator on weighted tractors (λ = −2). Note that, in the situation of Theorem 6.22.7, for closed Σ4 , Qe4 vol = 0. Σ

Next, we prove the fundamental identity for the extrinsic Q-curvature Qe4 . Theorem 6.22.8. In the situation of Theorem 6.22.7, assume that i∗ ∇N ϕ = 0. ˆ = 0 and Then H ˆ e4 = Qe4 + P4e (ϕ), e4ϕ Q where P4e = −4δ(θ#d), The operator P4e satisﬁes

θ = i∗ (PM ) − PΣ .

g ) = P4e (g) e4ϕ P4e (ˆ

for all ϕ as above. Note that the structure P4e = −4δ(·) is suggested by (6.22.9). Proof. Since Qe4 = 4δ(δθ) − 2Δ(tr θ), we have to determine

ˆ δˆθ) ˆ = e4ϕ δe ˆ −2ϕ e2ϕ δˆθˆ e4ϕ δ(

(6.22.13)

ˆ = e4ϕ Δe ˆ tr ˆ −2ϕ e2ϕ tr ˆ θ) ˆ θˆ . e4ϕ Δ(

(6.22.14)

and Now (2.5.9) implies (θˆ − θ)(X, Y )

1 Σ 2 = −g(∇M X (gradM (ϕ)), Y ) + g(∇X (gradΣ (ϕ)), Y ) − (∇N (ϕ)) g(X, Y ) 2 for X, Y ∈ X (Σ). But T ⊥ M g(∇M X (gradM (ϕ)), Y ) = g(∇X (gradM (ϕ) + gradM (ϕ)), Y ) M = g(∇M X (gradΣ (ϕ)), Y ) + g(∇X (∇N (ϕ)N ), Y )

= g(∇Σ X (gradΣ (ϕ)), Y ) + ∇N (ϕ)L(X, Y ).

438

Chapter 6. Conformally Covariant Families

Thus

1 θˆ − θ = −∇N (ϕ)L − (∇N (ϕ))2 g. 2

In particular, θˆ − θ = 0 if i∗ ∇N ϕ = 0. Moreover, we claim that for any symmetric bilinear form b, ˆ = δ(b) − b#dϕ − tr(b)dϕ. e2ϕ δ(b)

(6.22.15)

Now (6.22.15) implies e2ϕ δˆθˆ = δθ − θ#dϕ − tr(θ)dϕ. Therefore, by Lemma 4.2.1, the right-hand side of (6.22.13) equals δ (δθ − θ#dϕ − tr(θ)dϕ) . Moreover, by Lemma 4.2.1, the right-hand side of (6.22.14) simpliﬁes to Δ tr(θ) − 2δ(tr(θ)dϕ). Hence ˆ e = Qe − 4δ(θ#dϕ) − 4δ(tr(θ)dϕ) + 4δ(tr(θ)dϕ). e4ϕ Q 4 4 It remains to verify (6.22.15). But the transformation formula ˆ X ω = ∇X ω − X(ϕ)ω − ω, Xdϕ + (dϕ, ω)X , ∇

ω ∈ Ω1

implies ˆ = δ(b)

ˆ ˆ (b(X ˆ i )) ∇ Xi

i

=e

−2ϕ

∇Xi (b(Xi )) − 2Xi (ϕ)b(Xi ) − b(Xi , Xi )dϕ + (b(Xi ), dϕ)Xi

i

= e−2ϕ (δ(b) − b#dϕ − tr(b)dϕ) . ˆ −2ϕ = δ (Lemma 4.2.1), we ﬁnd Finally, using θˆ = θ and e4ϕ δe ˆ θˆ#du) ˆ e4ϕ δ( = δe2ϕ (θ#e−2ϕ du) = δ(θ#du). This proves the conformal covariance of P4e . The proof is complete. Corollary 6.22.1. Let H = 0. Then D4T (g; 0) = P4 (i∗ (g))i∗ + P4e (g) on ker i∗ ∇N .

6.22. Some results on tractor families

439

Proof. By the covariance of D4T (g; λ) and Lemma 6.22.2 we have g ; 0)(1) = D˙ 4T (g; 0)(1) − D4T (g; 0)(ϕ), e4ϕ D˙ 4T (ˆ i.e., g ) = QT4 (g) − D4T (g; 0)(ϕ). e4ϕ QT4 (ˆ Theorem 6.22.7 yields 4ϕ g ) − Q4 (g) + e4ϕ Qe4 (ˆ g ) − Qe4 (g) = D4T (g; 0)(ϕ). e Q4 (ˆ Hence, by Theorem 6.22.8 (and the fundamental identity for Q4 ), P4 (g)(ϕ) + P4e (g)(ϕ) = D4T (g; 0)(ϕ) for ϕ ∈ ker i∗ ∇N . The proof is complete.

Theorem 6.22.7 and Theorem 6.22.8 have analogs without the assumption H = 0. In the following, we only prove an extension of Theorem 6.22.8 in the sense that we ﬁnd a fundamental pair (P4e , Qe4 ). The detailed proof that these data actually are derived from D4T (λ) will be given elsewhere. Theorem 6.22.9. The operator P4e (g) : C ∞ (Σ4 ) u → −4δ (J #du) ∈ C ∞ (Σ4 ) with 1 1 J = θ + HL − H 2 g = θ + HL0 + H 2 g, 2 2

θ = i∗ (PM ) − PΣ

(6.22.16)

is conformally covariant e4ϕ P4e (ˆ g ) = P4e (g). Examples 6.22.1. (1) H = 0. Then P4e (g) reduces to δ(−4θ#d). It vanishes iﬀ θ = 0. (2) L0 = 0. Under this conformally invariant condition P4e (g) reduces to δ((−4θ− 2H 2 g)#d). It vanishes iﬀ θ + 12 H 2 g = 0. Moreover, we have Theorem 6.22.10. Let Qe4 = 4δ(δJ ) − 2Δ(tr(J )) ∈ C ∞ (Σ4 ), where J is deﬁned by (6.22.16). Then e4ϕ Qe4 (ˆ g ) = Qe4 (g) + P4e (g)(ϕ). We continue with the proof of Theorem 6.22.9.

440

Chapter 6. Conformally Covariant Families

Proof. The identity 1 θˆ = θ − ∇N (ϕ)L − (∇N (ϕ))2 g 2 (see the proof of Theorem 6.22.8) and the transformation formulas

(6.22.17)

ˆ = H + ∇N (ϕ) eϕ H ˆ = e−2ϕ #) ˆ 0 = L0 imply (using # and e−ϕ L

1 ˆ g) = e4ϕ δˆ −4Jˆ#d = δ − 4 θ − ∇N (ϕ)L − (∇N (ϕ))2 g e4ϕ Pˆ4e (ˆ 2

− 4(H + ∇N (ϕ))L0 − 2(H 2 + 2H∇N (ϕ) + (∇N (ϕ))2 )g #d.

Now L = L0 + Hg implies that the latter term simpliﬁes to δ −4θ − 4HL0 − 2H 2 g #d = −4δ(J #du).

The proof is complete.

Note that the above proof also shows the conformal invariance of J in all dimensions. We emphasize that observation in the following result. Theorem 6.22.11. For any isometric embedding i : (Σn , g) → (X n+1 , g) (n ≥ 3), the symmetric bilinear form 1 def J = θ + HL − H 2 g ∈ Γ(S 2 T ∗ Σ), 2

θ = i∗ (PM ) − PΣ

(6.22.18)

is conformally invariant: Jˆ = J . For the sake of convenience, we rewrite the argument. Proof. We use the formula J = θ + HL0 + 12 H 2 g. Then ˆ 2 gˆ ˆL ˆ0 + 1 H Jˆ = θˆ + H 2 1 1 2 = θ − ∇N (ϕ)L − (∇N (ϕ)) g + (H + ∇N (ϕ))L0 + (H + ∇N (ϕ))2 g 2 2 1 2 = θ + HL0 + H g − ∇N (ϕ)L + ∇N (ϕ)L0 + H∇N (ϕ)g 2 = J. The proof is complete.

Example 6.22.3. We calculate the invariant J for (X, Σ) = (Bn+1 , S n ) with the Euclidean metric which restricts to the round metric gc . In view of PX = 0 and PΣ = 12 gc , we ﬁnd θ = − 21 gc . Now L0 = 0 and H = 1 yields J = 0. The situation is conformally equivalent to (S n+1 , S n ) with the round metric. In that case θ = 0 and L = 0, i.e., J = 0.

6.22. Some results on tractor families

441

Some comments are in order. Lemma 6.23.2 implies that for a totally umbilic hypersurface in a conformally ﬂat background metric the J -invariant vanishes. Example 6.22.3 is a very special case. The natural second-order diﬀerential operator P4e (g) is tangential to Σ. But it is not a natural operator when regarded as a diﬀerential operator on Σ. In fact, its deﬁnition uses the pull-back i∗ (PM ) of the Schouten tensor of the metric g in a neighborhood of Σ (containing the curvature M e ∗ e RiN N j ), i.e., P4 (g) is not determined by i (g). Q4 is an exact divergence, i.e., its integral over a closed Σ vanishes. Although J is an invariant in all dimensions n ≥ 3, the operator P4e (g) is conformally covariant only for n = 4. P4e (g) and Qe4 (g) vanish if J (g) ≡ 0. This leads to the problem of characterizing the conformal submanifolds Σ → X with vanishing J . More generally, it is interesting to characterize in geometric terms the vanishing of the pair (P4e (g), Qe4 (g)). Now we prove Theorem 6.22.10. Proof. The proof is identical to that of Theorem 6.22.8 using the conformal invariance of J instead of θ. The following lemma shows that it is of interest to consider the trace of the linear map J on T (Σ) corresponding to the J -tensor. It implies that J is non-trivial. Lemma 6.22.4. tr(J ) = umbilic points.

n 2 2 (H

− τe ) =

1 2 2(n−1) |L0 |

≥ 0. Equality holds true at

Proof. As usual we shall use the same symbol for J and J . By deﬁnition of J , tr(J ) = tr(θ) +

n 2 H . 2

But tr(θ) = i∗ (JM ) − JΣ − i∗ PM (N, N ) =

1 tr(L2 ) − (tr(L))2 2(n−1)

by Lemma 6.22.3. Hence 1 tr(L2 ) − nH 2 2(n−1) ⎛ n 2 ⎞ n 1 1 ⎝ λ2 − λi ⎠ = 2(n−1) i=1 i n i=1

tr(J ) =

1 1 2 λi − λi λj . 2n i=1 n(n−1) i<j n

=

442

Chapter 6. Conformally Covariant Families

Here λi are the eigenvalues of L , i.e., the principal curvatures. On the other hand n 2 1 2 2 H − τe = 2 λi − λi λj n n(n−1) i<j i=1 =

n 1 2 2 1 λi − λi λj . 2 n i=1 n n(n−1) i<j

This proves the ﬁrst assertion. Now tr(J ) =

1 1 1 tr(L20 ) = |L0 |2 ≥ 0 tr(L2 ) − nH 2 = 2(n−1) 2(n−1) 2(n−1)

completes the proof.

The conformal invariance of J and Lemma 6.22.4 imply the conformal invariance of the symmetric bilinear form (H 2 − τe )g ([76]) (see also (6.4.12)). For n ≥ 3, the conformally invariant Chen-Willmore functional n CW = (H 2 − τe ) 2 vol Σ

is just a multiple of

n

tr(J ) 2 vol,

(6.22.19)

Σ

i.e., a multiple of the functional

|L0 |n vol Σ

(see [240], p. 256 and the references there describing the history of that functional). The critical points or the minima of CW are called Chen-Willmore submanifolds. Recall that for n = 2, |L0 |2 = H 2 − τe = H 2 − K + K (see page 197). J is not well deﬁned for n = 2 since PΣ is not. But (H 2 − τe ) vol is well deﬁned and reduces to the Willmore integrand 2 ¯ vol = λ1 − λ2 vol (H 2 − λ1 λ2 ) vol = (H 2 − K + K) 2 on the surface Σ (see (6.3.6)). The conformal invariant J can be used to introduce an analog of the ChenWillmore functional. Theorem 6.22.12. Let i : (Σn , g) → (X n+1 , g) be an isometric embedding (n ≥ 3). Then the integral n

|J (g)| 2 vol(g) ≥ 0

W(g) = Σ

is conformally invariant. W(g) vanishes iﬀ J (g) ≡ 0.

6.22. Some results on tractor families

443

ˆ 0 = L0 implies enϕ |L ˆ 0 |n = |L0 |n and Jˆ = J implies enϕ |Jˆ| n2 = In fact, e−ϕ L |J | . We recall that the vanishing of J (g) for n = 4 is equivalent to the vanishing of P4e (g). Of course, W gives rise to an SO(1, n+2)-invariant (i.e., M¨ obius-invariant) obius space functional for embeddings i : Σn → S n+1 of Σ (n ≥ 3) into the M¨ (S n+1 , gc ). Similarly, we can deﬁne other conformally invariant functionals in terms of the characteristic polynomials of the conformally invariant bilinear forms L0 and J . For instance, n tr(∧ 2 J ) vol (6.22.20) n 2

Σn

is a conformal invariant. Remark 6.22.5. For an isometric embedding i : Σm → M n of higher codimension, there is an analogous conformally invariant symmetric bilinear form J . Let ⊥ L(X, Y ) = −∇M X (Y )

be the T (Σ)⊥ -valued second fundamental form. In terms of an orthonormal basis {vi } of T (Σ)⊥ we decompose L = i Li vi and deﬁne the mean curvature vector 1 tr(Li )vi . H= m i Then

1 J = i∗ (PM ) − PΣ + (L, H) − |H|2 g 2 is conformally invariant. The above results suggest a series of questions. Is Pne tangential for n ≥ 6 and self-adjoint? Is Qen an exact divergence for n ≥ 6? Can one characterize the vanishing of Pne in terms of tensors? For n = 4, what is the geometric meaning of J ≡ 0 and Qe4 (g) ≡ 0? Can one prove the conformal invariance of J more directly by using conformally invariant calculus? Which symmetric bilinear forms arise as J ? Which functions arise as Qe4 (g)? Analyze the critical points of the functional W! For the convenience of the reader, we summarize the results for (M 5 , Σ4 ; g). • Let H = 0. Then the pairs (P4 , Q4 ) = Δ2 +δ(2Jg −4P)#d, 2(J2 −|P|2 )−ΔJ and (P4e , Qe4 ) = (δ(−4θ#d), 4δ(δθ)−2Δ(tr θ)) satisfy the fundamental identities ˆ 4 = Q4 + P4 (ϕ) e4ϕ Q if ϕ ∈ ker i∗ ∇N (Theorem 6.22.8).

ˆ e = Qe + P e (ϕ) and e4ϕ Q 4 4 4

444

Chapter 6. Conformally Covariant Families

• For general g, the fundamental identity holds true for the pair (P4e , Qe4 ) = (δ(−4J #d), 4δ(δJ )−2Δ(tr J )) (Theorem 6.22.10), where 1 1 J = i∗ (PM ) − PΣ + HL − H 2 g = i∗ (PM ) − PΣ + HL0 + H 2 g. 2 2 • The family D4T (g; λ) is regular at λ = 0 (Theorem 6.22.1) and D4T (g; 0) = P4 (i∗ (g)) + P4e (g) −QT4 (g)

∗

= Q4 (i (g)) +

(on ker ∇N by Corollary 6.22.1)

Qe4 (g)

(Theorem 6.22.7).

• Moreover, for the metric g = dr2 +hr (as in Example 6.22.2) D4T (dr2 +hr ; 0) = P4 (h)i∗

(Theorem 6.22.4)

−QT4 (dr2 +hr )

(Theorem 6.22.3).

= Q4 (h; λ)

In this case, the extrinsic pair (P4e , Qe4 ) vanishes (J = 0). In particular, the pair (P4 , Q4 ) can be extracted naturally • from the family D4T (dr2 +hr ; λ) and • from the eigenfunctions of the Laplacian for the Poincar´e-Einstein metric r−2 (dr2 +hr ). For a general background metric g, the pair (D4T (g; 0), D˙ 4T (g; 0)(1)) naturally decomposes as the sum of the intrinsic pair (P4 , Q4 ) and the extrinsic pair (P4e , Qe4 ) of the embedding. For the latter construction of (P4 , Q4 ), it is not necessary to solve the Einstein equation. In Section 6.23, we shall prove a formula for J in terms of L0 and the Weyl-tensor of the background. T (M, Σ; g; λ) If Σ is the boundary of a compact manifold M , the families DN can be used to deﬁne conformally covariant boundary value problems for GJMSoperators on M . More precisely, for such purposes, it is convenient to consider boundary value problems for the closely related operators n n

def T2N = CM − −N +1 ◦ · · · ◦ CM − −1 2 2 @A B ? N −1

n n

◦ ◦ DM − +2 ◦ · · · ◦ DM − +N : C ∞ (M ) → C ∞ (M ), 2 2 ? @A B T

N −1

where T is a tractor version of the Yamabe operator (see Section 6.20). T2N is conformally covariant, i.e., n ˆ T2N ◦ e(− n2 +N )ϕ = T2N , e( 2 +N )ϕ ◦

6.23. J and Fialkow’s fundamental forms

445

and is of the form cΔN M + LOT for a non-vanishing constant c except for 2N ≥ n (n even). The operators T2N were studied systematically in [116]. See also [109]. However, for even n ≥ 4, the operator Tn does not yield a non-trivial operator of that form. In the simplest case N = 2 (Corollary 6.20.3), n n

CM − −1 ◦ TM ◦ DM − +2 (n−4)P4 (M ). 2 2 Hence the composition degenerates to 0 for n = 4. For conformally ﬂat metrics, the operator T2N yields the GJMS-operator P2N ([116]). It suﬃces to verify that on Rn with the Euclidean metric the operator T2N coincides with a multiple of th ΔN translate of Δ− n . The latter fact is equivalent to the result that the N n is a N ) . This is a standard multiplicity one theorem. More precisely, multiple of (Δ− n we have the formula ([116]) c

N

N (n−2j)(Δ− n ) , N ≥ 2, c = 0.

j=2

The latter formula should be compared with N n

N πΘ2N − +N = c (n−2j)(Δ− n ) , N ≥ 1, c = 0 2 j=1

(see (6.18.25)). T (M, Σ; g; λ) : C ∞ (M n ) → C ∞ (Σn−1 ) for In [40], the boundary operators DL n−1 λ = − 2 + k, k ≥ 1 were used to construct conformally covariant self-adjoint elliptic boundary value problems for the operator T2N , 2N < n on M n . In particular, such boundary value problems give rise to conformally covariant operators on the boundary Σ. The observation that for even n, the tractor construction Tn on M n fails to produce the critical GJMS-operator Pn (M n ) can be seen as a reﬂection of the n 2 does not lift to a homomorphism of semiEastwood-Slovak theorem that (Δ− n) holonomic Verma modules ([92]). In the case n = 4, we have seen in Theorem 6.18.1 that semi-holonomic Zuckerman translation of Δ− n yields a multiple of 4L4,n − 2 − 2 ) . This is a lift of (Δ ) only for n = 4 since it obviously projects to the n(Δ− n n 2 2 holonomic (n−4)(Δ− ) . Hence translation can’t be used to lift (Δ− 4 4 ) , and actually do not exist by [92]. On the other hand, Theorem 6.18.5 yields a semi-holonomic 2 family D4T (λ) which lifts the family D4 (λ). The fact that D4 (0) = (Δ− 4 ) and T D4 (λ) has a pole at λ = 0 reﬂects the Eastwood-Slovak non-existence.

6.23 J and Fialkow’s fundamental forms In Section 6.21, we introduced the conformally invariant bilinear form J ∈ Γ(S 2 (T ∗ Σ))

446

Chapter 6. Conformally Covariant Families

which is associated to any hypersurface i : Σ → X of dimension ≥ 3. In the present section, we discuss its relation to Fialkow’s ([100]) fundamental forms and ﬁnd an explicit formula for J in terms of L0 and the Weyl tensor of the background. In the fundamental work [100], Fialkow characterizes conformally equivalent non-umbilic submanifolds of conformally ﬂat space in terms of conformal fundamental forms (see also [101] for the case of surfaces). For a modern presentation of the geometrical content of [101] we refer to [157]. See also [216] (Chapter 7) for the relation between Fialkow’s invariants and Cartan geometry. We describe the deﬁnition of the fundamental forms in the case of a hypersurface Σn → X n+1 . Fialkow starts with the observation that FI = |L0 |2 g is a conformally invariant bilinear form on Σ. In [100], FI is called the conformal measure tensor. The ﬁrst fundamental form FI is non-degenerate except at the umbilic points of Σ. In particular, it induces a conformally invariant connection which, in turn, gives rise to conformally invariant curvature etc. The second fundamental form is the conformally invariant bilinear form FII = |L0 |L0 . Moreover, Fialkow introduces the deviation tensor 1 2 1 def ∗ 2 (6.23.1) E = i (PX ) + HL0 + H g − Hess(v) − dv ⊗ dv + |dv| g , 2 2 where

def

v = log |L0 |

(6.23.2)

(see formulas (6.12), (7.1) and (13.9) in [100]). Note that in [100] v has a diﬀerent normalization and the curvature tensor has opposite sign. E is conformally invariant, i.e., Eˆ = E. Since 1 E = J + PΣ − Hess(v) − dv ⊗ dv + |dv|2 g , 2 the conformal invariance of E is a consequence of the conformal invariance of J and the following observation. Lemma 6.23.1. For v as in (6.23.2), 1 1 2 2 7 Hess(ˆ v ) − dˆ v ⊗ dˆ v + |dˆ v | gˆ − Hess(v) − dv ⊗ dv + |dv| g 2 2 1 ˆ Σ − PΣ . = − Hess(ϕ) + dϕ ⊗ dϕ − |dϕ|2 g = P 2 Proof. The transformation formulas 7 Hess(u) = Hess(u) − du ⊗ dϕ − dϕ ⊗ du + (du, dϕ)g for u ∈ C ∞ (Σ) and vˆ = v − ϕ, imply the ﬁrst identity. The second identity is (2.5.9).

6.23. J and Fialkow’s fundamental forms

447

An alternative proof of the conformal invariance of E using invariant calculus was given in [228]. The conformal invariance of J implies that the linear map J transforms as 2ϕ ˆ e J = J . Hence the traces σp (J ) of the induced maps ∧p (J ) : ∧p T (Σ) → ∧p T (Σ) transform according to e2pϕ σp (Jˆ ) = σp (J ). In particular, the volume form n

tr(∧ 2 J ) vol is conformally invariant and gives rise to the conformally invariant functional (6.22.20). obius space (conformal sphere), we For a submanifold Σn → S n+1 of the M¨ express J in terms of L0 . Lemma 6.23.2. For a hypersurface i : Σn → S n+1 (n ≥ 3) of the sphere (S n+1 , gc ), the invariant form J is given by 1 1 2 2 ∗ J = L0 − |L0 | i (gc ) . n−2 2(n−1) The form in brackets is conformally invariant for all n ≥ 2. Here L20 , by abuse of notation, denotes the bilinear form which corresponds to the operator (L0 )2 . The form |L0 |2 gc on Σ is Fialkow’s ﬁrst fundamental form. In the framework of M¨ obius geometry, it is sometimes called the M¨obius-form ([142]). Proof. We use the formulas RicS n+1 = ngc , PS n+1 = 12 gc , G = i∗ (gc ) = gΣ in the Gauß equations i∗ (RicM ) = RicΣ +G + L2 − nHL, 1 tr(L2 ) − (tr(L))2 i∗ (JM ) = JΣ + PM (N, N ) + 2(n−1) (see Lemma 6.22.3) for a hypersurface Σ → X, and ﬁnd RicΣ = (n−1)gΣ − L2 + nHL, 1 n tr(L2 ) − tr(L)2 . JΣ = − 2 2(n−1)

448

Chapter 6. Conformally Covariant Families

Hence 1 J = i∗ (PM ) − PΣ + HL − H 2 gΣ 2 1 1 1 (RicΣ −JΣ gΣ ) + HL − H 2 gΣ = gΣ − 2 n−2 2 n 1 1 1 2 2 2 − tr(L )−tr(L) = gΣ − gΣ (n−1)gΣ −L +nHL− 2 n−2 2 2(n−1) 1 + HL − H 2 gΣ 2 1 1 1 tr(L2 ) − tr(L)2 gΣ + HL − H 2 gΣ . = L2 − nHL − n−2 2(n−1) 2 Now we decompose L = L0 + HgΣ and ﬁnd 1 1 2 2 |L0 | gΣ . J = L0 − n−2 2(n−1)

The proof is complete.

Corollary 6.23.1. For a hypersurface Σn → S n+1 , the invariant J vanishes precisely at umbilic points. Thus J ≡ 0 iﬀ Σ is a totally umbilic hypersurface. Proof. By Lemma 6.23.2, J = 0 is equivalent to the relations 1 μ2 2(n−1) i i n

μ2i =

for the eigenvalues μi of L0 . Summation implies L0 = 0, i.e., L = Hg.

i

μ2i = 0, i.e., μi = 0, i.e.,

The following result extends Lemma 6.23.2 to the general case. Lemma 6.23.3. For a hypersurface i : Σn → X n+1 (n ≥ 3), the invariant form J is given by 1 1 2 2 Jij = |L0 | gij + CiN N j . (L0 )ij − n−2 2(n−1) The form in brackets is conformally invariant for all n ≥ 2. Proof. Similarly as in the proof of Lemma 6.23.2, we have RicΣ = i∗ (RicM ) − c23 (RM ) − L2 + nHL, 1 tr(L2 ) − (tr(L))2 JΣ = i∗ (JM ) − PM (N, N ) − 2(n−1)

6.23. J and Fialkow’s fundamental forms

449

by the Gauß equation, where c23 (R)ij = RiN N j . Hence 1 J = i∗ (PM ) − PΣ + HL − H 2 gΣ 2 1 1 ∗ (RicΣ −JΣ gΣ ) + HL − H 2 gΣ = i (PM ) − n−2 2 1 1 ∗ ∗ ∗ [i (RicM ) − i (JM )gΣ ] + [c23 (R) − PM (N, N )gΣ ] = i (PM ) − n−2 n−2 1 1 + tr(L20 ) − |L0 |2 gΣ n−2 2(n − 1) n−1 ∗ 1 i (PM ) + [c23 (R) − PM (N, N )gΣ ] = i∗ (PM ) − n−2 n−2 1 1 2 2 |L0 | gΣ . + tr(L0 ) − n−2 2(n − 1) But c23 (C)ij = CiN N j = RiN N j + (P g)iN N j = RiN N j − PM (N, N )gij − Pij implies that J is given by 1 1 1 ∗ 2 2 [−i (PM ) + c23 (R) − PM (N, N )gΣ ] + |L0 | g tr(L0 ) − n−2 n−2 2(n−1) 1 1 2 2 = |L0 | g + c23 (C) . tr(L0 ) − n−2 2(n−1)

The proof is complete. Note that Lemma 6.23.3 re-proves the relation (Lemma 6.22.4) tr(J ) =

1 |L0 |2 . 2(n−1)

In view of the analogy between the symmetric tensor 1 1 |L0 |2 g L20 − n−2 2(n−1) and the Schouten tensor 1 n−2

Ric −

1 τg , 2(n−1)

(6.23.3) should be viewed as an analog of tr(P) =

1 2(n−1) τ .

(6.23.3)

450

Chapter 6. Conformally Covariant Families

6.24 D2(g; λ) as a tractor family The following result relates the tractor construction D2T (M, Σ; g; λ) to the family D2 (M, Σ; g; λ) discussed in Section 6.4. In particular, we obtain an alternative proof of the conformal covariance of that family. Theorem 6.24.1. For n ≥ 4, let Σn−1 ⊂ M n be an (oriented) codimension 1 submanifold. Then the family D2T (M, Σ; g; λ) = CΣ (g; λ−1) ◦ ΠΣ ◦ DM (g; λ) : C ∞ (M ) → C ∞ (Σ) has the form 1 (n−3+λ)D2(M, Σ; g; λ). (n−5+2λ)(n−2+2λ) We regard the composition in Theorem 6.24.1 as being induced by the family Θ2 (λ) (Theorem 6.18.2). Proof. It follows immediately from the deﬁnitions that ΠΣ sends ⎛

⎞ (ΔM +λJM )u (n−2+2λ)DM (λ)u = ⎝ (n−2+2λ)du ⎠ 2λ(n−2+2λ)u to

⎞ ⎛ (ΔM +λJM )u + 12 (n−2+2λ)λH 2u − (n−2+2λ)HN u ⎠. ⎝ (n−2+2λ)i∗du 2λ(n−2+2λ)u

Hence the composition 2(n−5+2λ)(n−2+2λ)D2T (λ) is given by u → −2λ(n−2+2λ) (ΔΣ −(n−3+λ)JΣ) u + 2(n−2+2λ)(n−3+2λ)ΔΣu 1 2 − 2(n−3+2λ)(n−3+λ) (ΔM +λJM ) + λ(n−2+2λ)H − (n−2+2λ)HN u 2 = 2(n−2+2λ)(n−3+λ) (ΔΣ +λJΣ ) u − 2(n−3+2λ)(n−3+λ) (ΔM +λJM ) u 1 2 + 2(n−3+2λ)(n−3+λ) − (n−2+2λ)λH + (n−2+2λ)HN u. 2 It suﬃces to verify that the latter sum coincides with the product of 2(n−3+λ) with D2 (M, Σ; g; λ). Now using JM = Q2 (M ) and P2 (M ) = ΔM − n−2 2 Q2 (M ), we

6.24. D2 (g; λ) as a tractor family

451

ﬁnd − (n−3+2λ)(ΔM +λJM ) + (n−2+2λ)(ΔΣ +λJΣ ) 1 + (n−3+2λ)(n−2+2λ) HN − λ H 2 2 = −(n−3+2λ)P2(M ) + (n−2+2λ)P2(Σ) + (n−3+2λ)(n−2+2λ)HN n−2 n−3 + 2 λ+ λ+ (Q2 (Σ) − Q2 (M ) − λH 2 ). 2 2

The proof is complete.

In Section 6.21, we have used the following tractor generalization of Theorem 6.24.1. It can be regarded as the appropriate system of factorization identities. Theorem 6.24.2. Let M and Σ be as in Theorem 6.24.1. The conformally covariant family D2T (M, Σ; g; λ) deﬁned by (n−3+λ)D2T (M, Σ; g; λ) = (n−5+2λ)(n−2+2λ) {CΣ (λ−1)ΠΣ DM (λ)} : Γ(T ∗ M ) → Γ(T ∗ Σ) has the values n−2 T ΠΣ ΔM − JM , 2

n−3 T JΣ ΠΣ ΔΣ − 2

n−3 at λ = − n−2 2 and λ = − 2 , respectively.

Proof. For λ = − n2 +1 and u ∈ Γ(T ∗ M ), the product (n−2+2λ)DM (λ)u equals ⎞ ⎛ T ΔM −( n2 −1)JM u ⎝ ⎠ 0 0 (see (6.20.2)). ΠΣ maps this tractor into ⎞ ⎛ ΠΣ ΔTM −( n2 −1)JM u ⎝ ⎠ ∈ Γ(T ∗ Σ ⊗ T ∗ Σ) 0 0 (Theorem 6.20.6). Now, by (6.20.20), the product (n−5+2λ)CΣ(λ−1) sends the latter term into

n −1 JM u − (n−3+2λ)(n−3+λ)ΠΣ ΔTM − 2

n

−1 JM u. = (n−3+λ)ΠΣ ΔTM − 2

452

Chapter 6. Conformally Covariant Families

This proves the ﬁrst assertion. For the proof of the second one we notice that, by (6.20.20), 1 n−1 n−3 u= ΔTΣ − JΣ u+ , CΣ − 2 4 2 where u+ ∈ Γ(T ∗ Σ) is the last component of u ∈ Γ(T ∗ Σ ⊗ T ∗ Σ). Now the last ∗ component of the composition ΠΣ DM (− n−3 2 )u is −(n−3)ΠΣ u ∈ Γ(T Σ). Hence we ﬁnd the expression n−3 n−3 T − ΔΣ − ΠΣ u 4 2 n−3 for the composition CΣ (− n−1 2 )ΠΣ DM (− 2 )u, i.e.,

n−3 T D2 2

n−3 M, Σ; − 2 n−3 n−1 n−3 n−3 T = (−2)CΣ − ΠΣ DM − = ΔΣ − ΠΣ . 2 2 2 2

The proof is complete.

Although we do not discuss here an explicit formula for D2T (M, Σ; g; λ) for general g, one special case is not hard to get. We recall that for the Poincar´eEinstein metric r−2 (dr2 + hr ) on M n = (0, ε) × Σn−1 , the family D2 (dr2 + hr ; λ) takes the form n−2 n−3 ∗ −(n−3+2λ)i ΔM − JM + (n−2+2λ) ΔΣ − JΣ i∗ 2 2 with i∗ JM = JΣ . In particular, it is linear in λ. This result extends to the tractor generalization D2T (M, Σ; g; λ) : Γ(T ∗ M ) → Γ(T ∗ Σ) deﬁned in Theorem 6.24.2. In fact, by Lemma 6.11.2, the Schouten tensor of dr2 + hr pulls back to the Schouten tensor of h. Hence the tractor connections are compatible under pull-back. These observation imply that D2T (M, Σ; g; λ)

n−2

n−3

JM + (n−2+2λ) ΔTΣ − JΣ ΠΣ , (6.24.1) = −(n−3+2λ)ΠΣ ΔTM − 2 2

where ΠΣ is given by pull-back. In particular, the family is linear in λ. The following result provides an alternative tractor construction of the second-order family D2 (M, Σ; g; λ).

6.24. D2 (g; λ) as a tractor family

453

Theorem 6.24.3. (n−2+2λ)D2(M, Σ; g; λ)u

n−2 2 = −(DT (g; λ−1)DM (g; λ)u, N T ) − λ(n−1) λ+ H −τe u. 2

ˆ 2 − τˆe ) = H 2 − τe Note that H 2 − τe is a local conformal invariant, i.e., e2ϕ (H (see Section 6.4). Proof. We use the formula DT (λ−1)C(λ)u = ⎞ ⎛ (∇N −(λ−1)H)(Δ+λJ)u + (n−2+2λ)(P(N ), du) ⎝−N (ΔM +λJM )u + (n−2+2λ) (∇N −(λ−1)H) du + λ(n−2+2λ)P(N )u⎠ . −2(n−2+2λ)∇N u + 2λ(n−2+2λ)(∇N −(λ−1)H)u Hence for the scalar product we obtain (DT (λ−1)DM (λ)u, N T ) = −(ΔM +λJM )u + (n−2+2λ) (N, ∇N du−(λ−1)HN u) + λ(n−2+2λ)P(N, N )u + H [(n−2+2λ)∇N u − λ(n−2+2λ)(∇N −(λ−1)H)u] . Now let N be the geodesic normal ﬁeld. Then we ﬁnd for the scalar product − ΔM u + (n−2+2λ)N 2u − 2(λ−1)(n−2+2λ)HN u + −λJM + λ(n−2+2λ)P(N, N ) + λ(λ−1)(n−2+2λ)H 2 u = (n−3+2λ)N 2u − ΔΣ u − [2(λ−1)(n−2+2λ)+n−1]HN u + {· · · } = (n−3+2λ)N 2u − ΔΣ u − (n−3+2λ)(2λ−1)HN u + {· · · } . The latter sum can be reformulated as (n−3+2λ)ΔM u − (n−2+2λ)ΔΣu − (n−3+2λ)(n−2+2λ)HN u + {· · · } = (n−3+2λ)P2(M )u − (n−2+2λ)P2(Σ)u − (n−3+2λ)(n−2+2λ)HN u n−2 n−3 + (n−3+2λ) JM u − (n−2+2λ) JΣ u + {· · · } . 2 2 That result diﬀers from −D2 (M, Σ; g; λ) by the zeroth-order operator − λJM + λ(n−2+2λ)P(N, N ) + λ(λ−1)(n−2+2λ)H 2 n−2 n−3 (n−3+2λ)JM u − (n−2+2λ)JΣu + 2 2 n−2 n−3 + 2 λ+ λ+ Q2 (M ) − Q2 (Σ) − λH 2 . 2 2

454

Chapter 6. Conformally Covariant Families

The latter sum is a polynomial of order 3 in λ with absolute coeﬃcient n−2 n−3 (n−2)(n−3) (n−3)JM − (n−2)JΣ + (Q2 (M ) − Q2 (Σ)) = 0 2 2 2 and leading coeﬃcient 0. It remains to determine the respective coeﬃcients α and β of λ2 and λ. α and β satisfy the relations − JM + (n−2)P(N, N ) − (n−2)H 2 + (n−2)JM − (n−3)JΣ + (2n−5) (Q2 (Σ) − Q2 (M )) +

(n−2)(n−3) 2 H =β 2

(6.24.2)

and 2P(N, N ) + (n−4)H 2 + 2 (Q2 (Σ) − Q2 (M )) − (2n−5)H 2 = α.

(6.24.3)

But (6.24.2) is equivalent to −(n−2)JM + (n−2)JΣ −

(n−1)(n−2) 2 H + RicN −JM = β, 2

i.e., 1 1 (n−1)(n−2) 2 − τM + τΣ + RicN − H = β, 2 2 2 i.e., β=−

(n−1)(n−2) 2 H − τe 2

using τM = τΣ + 2 RicN −(n−1)(n−2)τe (see Section 6.4). Similarly, we ﬁnd that (6.24.3) is equivalent to 2 RicN −2JM + τΣ − 2(n−2)JM − (n−1)(n−2)H 2 = (n−2)α, i.e., 2 RicN +τΣ − τM − (n−1)(n−2)H 2 = (n−2)α, i.e.,

(n−1) τe − H 2 = α.

Hence the zeroth operator is given by n−2 2 n−2 2 2 −(n−1) λ +λ H − τe = −λ(n−1) λ+ H − τe . 2 2 The proof is complete

6.25. The family D3T (M, Σ; g; λ)

455

6.25 The family D3T (M, Σ; g; λ) In the present section, we analyze the tractor family D3T (λ) of order 3. The results will be used in Section 6.26 to relate this family to results of Chang and Qing [68]. The ﬁrst result concerns the obstruction. Theorem 6.25.1. The family D3T (M, Σ; g; λ) = CΣ (g; λ−2) ◦ ΠΣ ◦ DT (g; λ−1) ◦ DM (g; λ) : C ∞ (M 4 ) → C ∞ (Σ3 ) def

has the form 3 λDT (M, Σ; g; λ) (2λ−3)(2λ+2) 3 for a rational family D3T (·; λ) with a simple pole at λ = 0. Its residue is the operator def

R =

2 δΣ (L0 #dΣ ) i∗ : C ∞ (M 4 ) → C ∞ (Σ3 ). 3

R is conformally covariant: e3ϕ R(e2ϕ g) = R(g). Since D3T (M, Σ; g; λ) corresponds to D3T (λ) (Theorem 6.18.3), it is natural to regard the residue R as being induced by the residue (see (6.18.9)) 2 − − − Yi Y4 Yi − Yi− Y4− . 3 1 3

Res0 (D3T (λ)) = −

Proof. We ﬁrst determine the composition DT (g; λ−1) ◦ DM (g; λ) for general n. The deﬁnitions give ⎛ ⎞ T (ΔM +λJM )u T (n−2+2λ)D (λ−1)DM (λ)u = ∇N −(λ−1)H ⎝ (n−2+2λ)du ⎠ 2λ(n−2+2λ)u ⎞⎛ ⎞ ⎛ (P(N ), ·) 0 ∇N −(λ−1)H (ΔM +λJM )u 1 ⎠ ⎝ (n−2+2λ)du ⎠ =⎝ ∇N −(λ−1)H −N 2 P(N ) 2λ(n−2+2λ)u 0 −2N, · ∇N −(λ−1)H ⎛ ⎞ (∇N −(λ−1)H)(ΔM +λJM )u + (n−2+2λ)(P(N ), du) = ⎝−N (ΔM +λJM )u + (n−2+2λ)(∇N −(λ−1)H)du + λ(n−2+2λ)P(N )u⎠ . −2(n−2+2λ)∇N u + 2λ(n−2+2λ)(∇N −(λ−1)H)u The second ingredient is a formula for the composition CΣ (g; λ)◦ ΠΣ for general n.

456

Chapter 6. Conformally Covariant Families

The deﬁnitions yield ⎛

⎞ ⎞ ⎛ u− u− − Hω, N + 14 H 2 u+ ⎠ i∗ (ω) 2(n−3+2λ)CΣ(λ)ΠΣ ⎝ ω ⎠ = 2(n−3+2λ)CΣ(λ) ⎝ u+ u+ 1 = −2(n−1+2λ)(n−2+λ) u− − Hω, N + H 2 u+ 4 + 2(n−1+2λ)δΣi∗ (ω) − ΔΣ u+ + (n−2+λ)JΣ u+ , i.e., ⎛

⎞ u− 2(n−7+2λ)CΣ(λ−2)ΠΣ ⎝ ω ⎠ u+

1 = −2(n−5+2λ)(n−4+λ) u− − Hω, N + H 2 u+ 4 + 2(n−5+2λ)δΣi∗ (ω) − ΔΣ u+ + (n−4+λ)JΣ u+ .

It follows that the composition D3T (λ) has the structure 2(n−2+2λ)(n−7+2λ)D3T (λ) = (n−4+λ) {· · · } + 2(n−5+2λ)(n−2+2λ)δΣ (i∗ (∇N −(λ−1)H) du + λi∗ P(N )u) − 2(λ−1)(n−2+2λ)ΔΣi∗ (∇N −λH)u. For n = 4, we ﬁnd λ {· · · } 4(2λ−3)(λ+1) λ−1 2λ−1 δΣ (i∗ (∇N −(λ−1)H) du + λi∗ P(N )u) − ΔΣ i∗ (∇N −λH)u. + 2λ−3 2λ−3

D3T (λ) =

Now write D3T (λ) in the form D3T (λ) =

3 λDT (λ) (2λ−3)(2λ+2) 3

with a rational D3T (λ). Then D3T (λ) has a simple pole at λ = 0 with residue 2 δΣ {dΣ i∗ ∇N − i∗ ∇N d − HdΣ i∗ } . 3 In view of Lemma 6.25.1, the residue equals 2 2 [δΣ (L#dΣ ) − δΣ (Hg#dΣ )] i∗ = δΣ (L0 #dΣ )i∗ . 3 3 The proof is complete.

6.25. The family D3T (M,Σ;g;λ)

457

Lemma 6.25.1. dΣ i∗ ∇N u − i∗ ∇N du = L#dΣ i∗ u. Proof. Let N be the unit geodesic normal and X ∈ X (Σ). X will also denote an arbitrary extension of X to a neighborhood of Σ. Then ∇N du, X = −du, ∇N X + N du, X = −du, ∇X N − du, [N, X] + N du, X = −du, ∇X N + d(∇N u), X. Hence on Σ we have ∇N du, X − d(∇N u), X = −du, ∇X N = −dΣ u, ∇X N = −L(X, (dΣ u) ) = −L#dΣ u, X.

The proof is complete.

If the residue of D3T (λ) at λ = 0 vanishes, then its value P3T = D3T (0) and the curvature quantity QT3 = D˙ 3T (0)(1) are well deﬁned. Next, we determine these data. We begin with QT3 . Theorem 6.25.2. Assume that n = 4 and L0 = 0. Then for the critical Qpolynomial def 3 Q3,T (g; λ) = D3T (g; λ)(1) 2 we ﬁnd Q3,T (g; 0) = 0 and 1 1 def Q3,T = Q˙ 3,T (0) = N JM + JM H + JΣ H − HP(N, N ) − ΔΣ H + H 3 . 2 2 Q3,T is a convenient renormalization of QT3 (Deﬁnition 6.21.4). Proof. The formula in the proof of Theorem 6.25.1 implies ⎞ ⎛ ∇N −(λ−1)H)JM (2λ+2)DT (λ−1)DM (λ)(1) = λ ⎝−N JM + 2(λ+1)P(N )⎠ −4(λ−1)(λ+1)H Now 2(2λ−3)CΣ (λ−2)ΠΣ maps this tractor into the scalar " λ − 2(2λ−1)λ (∇N −(λ−1)H)JM − (λ2 −1)H 3 + JM H − 2(λ+1)HP(N, N ) # + 4(2λ−1)(λ+1)δΣi∗ P(N ) + (λ2 −1)ΔΣ H − 4λ(λ2 −1)JΣ H . It follows that Q3,T (0) is proportional to −4δΣ i∗ P(N ) − 4ΔΣ H.

458

Chapter 6. Conformally Covariant Families

By Lemma 6.25.2, that sum vanishes. This proves Q3,T (0) = 0. Next, using the relation 2(2λ+2)(2λ−3)D3(λ)(1) = 6λD3T (λ)(1), we read oﬀ 32 D˙ 3T (0)(1) from the above formula. We obtain 3 ˙T 1 D3 (0)(1) = 2 (∇N +H)JM +H 3 +JM H −2HP(N,N ) +4δΣ i∗ P(N )+4JΣ H 2 4 1 1 = N JM + JM H + JΣ H − HP(N,N ) + δΣ i∗ P(N ) + H 3 . 2 2

In order to complete the proof, we apply Lemma 6.25.2. Lemma 6.25.2. i∗ P(N ) =

1 n−2 δ(L0 )

− dΣ H.

Proof. An application of the Codazzi-Mainardi equation (R(X, Y )Z, N ) = ∇Y (L)(X, Z) − ∇X (L)(Y, Z) yields (n−2)i∗P(N ), Y = Ric(N, Y ) = (R(Y, ei )ei , N ) i

=

∇ei (L)(Y, ei ) − ∇Y (L)(ei , ei )

i

=

∇ei (L0 )(Y, ei ) +

i

= δ(L0 )(Y ) +

∇ei (gH)(Y, ei ) − ∇Y (gH)(ei , ei )

i

∇ei (H)g(Y, ei ) − ∇Y (H)

i

g(ei , ei )

i

= δ(L0 )(Y ) + ∇Y (H) − (n−1)∇Y (H) = δ(L0 )(Y ) − (n−2)dΣ H, Y . The proof is complete. Theorem 6.25.3. Let n = 4. Assume that L0 = 0. Then 3 T D (0) 2 3 1 3 N ΔM +ΔΣ N + HΔM − H∇N d·, N + (P(N ), d·) − JΣ N − H 2 N u. = 2 2

6.25. The family D3T (M, Σ; g; λ)

459

Proof. Similarly as in the proof of Theorem 6.25.1, we ﬁrst ﬁnd an explicit formula for the composition 2(n−2+2λ)(n−7+2λ)D3T (λ). It is the sum of " −2(n−5+2λ)(n−4+λ) (∇N −(λ−1)H)(ΔM +λJM )u + (n−2+2λ)(P(N ), du) n−2+2λ (λ−1)H 2 (∇N −λH)u 2 + H (ΔM +λJM )u − (n−2+2λ)∇N −(λ−1)H)du, N # − λ(n−2+2λ)P(N, N )u (6.25.1) +

and 2(n−5+2λ)(n−2+2λ)δΣ {i∗ (∇N −(λ−1)H)du + λi∗ P(N )u} − 2(n−2+2λ)(λ−1)ΔΣ(∇N −λH)u + 2(n−4+λ)(n−2+2λ)(λ−1)JΣ(∇N −λH)u.

(6.25.2)

For n = 4, Theorem 6.25.1 shows that this is a polynomial in λ with vanishing absolute coeﬃcient. We determine the coeﬃcient of λ. The above formula implies that this coeﬃcient is the sum of 2 ∇N ΔM u + 2(P(N ), du) − H 2 ∇N u + 2HΔM u − 2H((∇N + H)du, N ) = 2∇N ΔM u + 4HΔM u − 4H(((∇N + H)du, N ) + 4(P(N ), du) − 6H 2 N u, −4 (−δΣ (HdΣ u) + δΣ (i∗ P(N )u)) + 4δΣ (∇N du + Hdu),

(6.25.3)

−4ΔΣ (Hu) and −4JΣ N u. Now by Lemma 6.25.2, the ﬁrst two terms in (6.25.3) add up to 4ΔΣ (Hu). This sum cancels with −4ΔΣ (Hu). But by Lemma 6.25.1, the third term in (6.25.3) simpliﬁes to 4ΔΣ N u using L0 = 0. Thus we obtain 1 3 2 N ΔM u+ΔΣ N u+HδM u−H(∇N du, N )+(P(N ), du)−JΣN u− H N u . 4 2 2 In view of the relation 2(2λ + 2)(2λ − 3)D3T (λ) = 6λD3T (λ), the asserted formula for 32 D3T (0) follows. For general dimension, i.e., in the non-critical case n = 4, the situation is similar. Then we can write D3T (M n , Σn−1 ; g; λ) = (· · ·)(n−4+λ)D3T (M n , Σn−1 ; g; λ) for a rational family D3T (M n , Σn−1 ; g; λ) with a simple pole at λ = −n + 4 and a generally non-vanishing residue Res−n+4 (D3T (M, Σ; g; λ)) : C ∞ (M n ) → C ∞ (Σn−1 ).

460

Chapter 6. Conformally Covariant Families

However, for n = 4, the residue is more complicated than for n = 4. In particular, we have (up to a constant multiple)

n n −2 (· · ·). D3T ·; − +2 = Res−n+4 (D3T (·; λ)) + 2 2 Grant ([129]) found a conformally covariant operator Ln of the form

n −2 (· · ·). Ln = Res−n+4 (D3T ) + 2 Ln vanishes if L0 = 0. It follows that n

D3T − +2 − Ln = (n−4)P3 2 for a conformally covariant third-order operator P3 of the form 1 N ΔM + ΔΣ N + LOT. 2 P3 acts on the same space as the Paneitz operator P4 and deﬁnes a conformally covariant boundary condition for P4 . For n = 4 and L0 = 0 the relation between Grant’s P3 and the Chang-Qing operator P3 is not clear. However, for n = 4 and L0 = 0, we will clarify the relation in Section 6.26. An interesting observation is that both factorization identities for D3 (λ) and D3T (λ) continue to hold true for D3T (λ). More precisely, Lemma 6.25.3. D3T (M, Σ; g; λ) = D1 (M, Σ; g; λ−2) ◦ P2 (M, g)

for λ = − n−2 2

and D3T (M, Σ; g; λ) = P2 (Σ, g) ◦ D1 (M, Σ; g; λ)

for λ = − n−5 2 .

T Proof. For λ = − n−2 2 , the formula for D3 (λ) which was derived during the proof of Theorem 6.25.3 implies

2(n−2+2λ)(n−7+2λ)D3T (λ) = 6(n−4+λ)D3T (λ) = −2(n−5+2λ)(n−4+λ) {(∇N −(λ−1)H)(ΔM +λJM ) + H(ΔM +λJM )} = 6(n−4+λ)D1(λ−2)P2 (M ) using D1 (λ) = ∇N −λH. Similarly, for λ = − n−5 2 , the same formula yields 2(n−2+2λ)(n−7+2λ)D3T (λ) = 6(n−4+λ)D3T (λ) n−3 JΣ (∇N −λH) = 2(n−2+2λ)(λ−1) −ΔΣ (∇N −λH) + 2 = 6(n−4+λ)P2(Σ)D1 (λ). This completes the proof.

6.25. The family D3T (M, Σ; g; λ)

461

Remark 6.8.2 states the analogous factorization identities for order 3 residue families. Theorem 6.25.4. Let n = 4. Then D3T (dr2 +hr ; λ) = D3res (h; λ). Proof. The formulas (6.25.1) and (6.25.2) in the proof of Theorem 6.25.3 show that for the metric dr2 +hr , 6λD3T (λ) = 2(2λ+2)(2λ−3)D3T (λ) = −2(2λ−1)λ∇N (ΔM +λJM )u + 4λ(λ+1)ΔΣ (∇N u) + 4λ(λ+1)(λ−1)JΣ ∇N u. Here we have used that H = 0. Hence 6D3T (λ) = λ2 {−4∇N (JM u) + 4JΣ ∇N u} + λ {−4∇N (ΔM u) + 2∇N (JM u) + 4ΔΣ (∇N u)} + {2∇N (ΔM u) + 4ΔΣ (∇N u) − 4JΣ (∇N u)} . Now i∗ (JM ) = JΣ (Lemma 6.11.1) and i∗ ∇N (JM ) = 0 imply that the coeﬃcient of λ2 vanishes. For the coeﬃcient of λ, we ﬁnd 3 ∂ ∂3 ∂2 ∂ ∂ ∂ ∂ + 4Δ = −4 − J + Δ + 6JΣ −4 + 2J Σ Σ Σ Σ ∂r3 ∂r2 ∂r ∂r ∂r ∂r3 ∂r using arguments as in the proof of Lemma 6.8.1. Similar arguments yield 2∇N ΔM + 4ΔΣ ∇N − 4JΣ ∇N 3 ∂ ∂ ∂ ∂ ∂ + 2ΔΣ − 4JΣ = 2 3 − 2JΣ + 4ΔΣ ∂r ∂r ∂r ∂r ∂r ∂ 1 ∂3 + (ΔΣ − JΣ ) =6 3 ∂r3 ∂r for the coeﬃcient of λ0 . Hence 1 ∂3 ∂ 2 ∂3 ∂ + J + (Δ − J ) D3T (λ) = λ − + . Σ Σ Σ 3 ∂r3 ∂r 3 ∂r3 ∂r Since this formula coincides with (6.8.4), the proof is complete.

Remark 6.25.1. Theorem 6.25.4 continues to hold true for n ≥ 4. In fact, on (0, ε) × Σn−1 we have ∂3 1 ∂ D3res (λ) = − (n−5+2λ) 3 + (ΔΣ + (λ−1)JΣ ) 3 ∂r ∂r ∂ ∂ 2 ∂3 n−5 ∂ 3 + J + (Δ − J ) =λ − + − Σ Σ Σ 3 ∂r3 ∂r 3 ∂r3 ∂r

462

Chapter 6. Conformally Covariant Families

by (6.8.4). On the other hand, (6.25.1) and (6.25.2) yield 6D3T (λ) = −2(n−5+2λ)∇N (ΔM + λJM ) + 2(n−2+2λ)ΔΣ∇N + 2(n−2+2λ)(λ−1)JΣ∇N . Similar arguments as in the proof of Theorem 6.25.4 yield the assertion. Note that the crucial point in the proof are the identities i∗ (JM ) = JΣ

and

Now by construction the pair def

(P3,T (g), Q3,T (g)) =

i∗ ∇N (JM ) = 0.

3 T D3 (g; 0), D˙ 3T (g; 0)(1) 2

(6.25.4)

satisﬁes the fundamental identity e3ϕ Q3,T (e2ϕ g) = Q3,T (g) − P3,T (g)(ϕ). Thus we have proved Theorem 6.25.5. The natural pair 1 3 N ΔM + ΔΣ N + HΔM − H(∇N d·, N ) + (P(N ), d·) − JΣ N − H 2 N P3,T = 2 2 and

1 1 N JM + HJM + JΣ H − HP(N, N ) − ΔΣ H + H 3 2 2 satisﬁes the fundamental identity Q3,T =

ˆ 3,T = Q3,T − P3,T . e3ϕ Q Example 6.25.1. Let S 3 be the equator of S 4 with the round metric gc . Then τS 4 = 12, JS 4 = 2 and τS 3 = 6, JS 3 = 32 . Hence, by Theorem 6.25.3, 1 3 T 3 N ΔS 4 + ΔS 3 N + (P(N ), d·) − N. P3,T = D3 (0) = 2 2 2 Now P = 12 (Ric −2g) = 12 g, i.e., P(N ) = 12 N . Therefore, 1 2 2 N ΔS 4 + ΔS 3 N − N. 3 3 3 This formula coincides with D3c (0) for n = 4. D3T (0) =

Example 6.25.2. Let S 3 be the unit sphere in R4 with the ﬂat metric. For the inner normal N = −∂/∂r, we have H = −1. Thus P3,T =

∂ 3 T 1 ∂ D3 (0) = − ΔR4 − ΔS 3 − ΔS 3 . 2 2 ∂r ∂r

Therefore, D3T (0) = D3b (0) (see (5.4.14)). In Section 6.26, we will relate the latter result to results of Branson-Gilkey ([38]) and Chang-Qing ([68]). See also Corollary 5.4.1.

6.26. The pair (P3 , Q3 )

463

6.26 The pair (P3 , Q3) Let (M 4 , g) be a smooth compact four-manifold with boundary Σ3 = ∂M 4 . In [68], Chang and Qing introduced a natural pair (P3 , T ) P3 (M, Σ; g) : C ∞ (M ) → C ∞ (Σ),

T (M, Σ; g) ∈ C ∞ (Σ)

which satisﬁes the fundamental identity e3ϕ T (M, Σ; e2ϕ g) = T (M, Σ; g) + P3 (M, Σ; g)(ϕ).

(6.26.1)

In the present section, we discuss that pair under the conformally invariant assumption that the trace-free part L0 of the second fundamental form vanishes, i.e., L = Hg, i.e., the boundary is totally umbilic. In particular, we shall relate the pair (P3 , T ) to the tractor constructions in Section 6.25. Theorem 6.26.1 ([68]). Let N be the inner unit normal of ∂M . Assume that L0 (g) = 0. Then 1 def N ΔM + ΔΣ N + HΔΣ + (F − 2JM )N − (dH, d·) (6.26.2) P3 (M, Σ; g) = 2 and the scalar curvature quantity 1 1 def T (M, Σ; g) = − N JM − 3JM H + RicN H − 3H 3 + tr(L3 ) + ΔΣ H 2 3

(6.26.3)

satisfy the fundamental identity (6.26.1). Notice that P3 annihilates constants. All quantities are to be understood with respect to g, of course; we have suppressed the obvious restrictions i∗ and, as before, N denotes also the normal derivative ∇N . Finally, F is given by (2.5.19). In Theorem 6.26.1, all terms are written so that a direct comparison with [68] is easy. Notice, however, that our conventions diﬀer from those in [68]. In particular, in [68] the signs of Laplacians are opposite and their pair (N, L) satisﬁes ˆ = L − N (ϕ)g. eϕ L In [68], the proof of Theorem 6.26.1 uses the explicit formulas for the conformal variation of local invariants of order 3 given in [38]. Remark 6.26.1. In [68], there is also an analogous pair (P3 , T ) if no condition on L is posed. It is given by 1 N ΔM + ΔΣ N + {2HΔΣ − (L, HessΣ )} + (F − 2JM )N − (dH, d·) P3 = 2 and

1 1 T = − N JM − 3JM H + (G, L) − 3H 3 + tr(L3 ) + ΔΣ H. 2 3

464

Chapter 6. Conformally Covariant Families

The following result relates Theorem 6.26.1 to the tractor family of order 3. Theorem 6.26.2. Assume that L0 = 0. Then P3 = P3,T and

1 T = −Q3,T = − N JM − 2HJΣ + ΔΣ H + H 3 . 2 Proof. Lemma 6.25.2 implies that (P(N ), du) = P(N, N )N u + (i∗ P(N ), i∗ du) = P(N, N )N u − (dH, du).

Thus we have to prove that (6.26.2) coincides with P3,T u =

1 N ΔM u + ΔΣ N u + HΔM u 2 3 − H∇N du, N + P(N, N )N u − (dH, du) − JΣ N u − H 2 N u. 2

We assume that N is the geodesic unit normal ﬁeld. Then ΔM u = ∇N du, N + 3HN u + ΔΣ u. Hence P3,T u =

1 N ΔM u + ΔΣ N u + HΔM u 2 1 3 + H 2 N u + (RicN −JM )N u − (dH, du) − JΣ N u. 2 2

Therefore, it suﬃces to verify the relation 1 1 3 1 1 RicN − τM − τΣ + H 2 = RicN − τM . 2 6 4 2 3

(6.26.4)

But the Gauß equation and L0 = 0 imply τM = τΣ + 2 RicN − tr(L)2 + |L|2 = τΣ + 2 RicN −6H 2 .

(6.26.5)

Hence the left-hand side of (6.26.4) equals 1 1 1 1 1 RicN − τM − τM + RicN = RicN − τM . 2 12 4 2 3 This proves (6.26.4). The relation T = −Q3,T follows from the validity of the corresponding fundamental identities. An alternative direct argument is as follows. We recall (Theorem 6.25.2) that 1 1 −Q3,T = − N JM − JM H − JΣ H + ΔΣ H + P(N, N )H − H 3 . 2 2

6.26. The pair (P3 , Q3 )

465

Hence, for the proof of the relation T = −Q3,T , it is enough to prove that 1 1 H(P(N, N ) − JΣ ) − H 3 = −2JM H + RicN H − 3H 3 + tr(L3 ). 2 3 In view of tr(L3 ) = 3H 3 , that identity is equivalent to 1 1 1 1 H RicN − τM − τΣ − H 3 = −2JM H + H RicN −2H 3 . 2 12 4 2

(6.26.6)

Now (6.26.5) implies that the left-hand side of (6.26.6) equals 1 H RicN − τM − 2H 3 . 3 This proves (6.26.6) and hence T = −Q3,T . Finally, we ﬁnd 1 T = − N JM + H(RicN −3JM − 2H 2 ) + ΔΣ H 2 1 1 2 = − N JM + H − τΣ + H + ΔΣ H 2 2

using (6.26.5). The proof is complete. If L vanishes, the formulas for P3 and T further simplify to 1 1 N ΔM + ΔΣ N + RicN − τM N P3 (M, Σ; g) = 2 3 and T (M, Σ; g) = − The renormalization 2 P3 (M, Σ; g) = 3

(6.26.7)

1 N τM . 12

1 2 N ΔM + ΔΣ N 3 3

+

(6.26.8)

2 2 RicN − τM 3 9

N

ﬁts better with the normalizations used in the present work. In the light of Theorem 6.4.1, an obvious question is whether it is more natural to write this operator in the alternative form 1 2 N P2 (M, g) + P2 (Σ, g)N + LOT. 3 3 The following calculation shows that this is actually the case. We have to determine the lower order terms. We ﬁrst recall that P2 (M ) = ΔM − 16 τM and P2 (Σ) = ΔΣ − 18 τΣ . Hence 1 1 2 2 1 τM + τΣ − τM + RicN N + N τM . LOT = 18 12 9 3 18

466

Chapter 6. Conformally Covariant Families

The assumption L = 0 implies τM = τΣ + 2 RicN on Σ. Therefore, the above formula for LOT simpliﬁes to 1 1 1 − τΣ + RicN N + N τM . 12 3 18 Now we observe that Q2 (M ) − Q2 (Σ) =

1 1 1 1 τM − τΣ = − τΣ + RicN . 6 4 12 3

In other words, we have proved Lemma 6.26.1. If L form 1 N P2 (M ; g) + 3

= 0, then the operator

2 3 P3 (M, Σ; g)

can be written in the

2 P2 (Σ; g)N + (Q2 (M ; g) − Q2 (Σ; g)) N + Q3 (M, Σ; g), 3

where Q3 (M, Σ; g) = − 23 Q3,T = 13 N JM . Lemma 6.26.1 is similar in spirit to the formula for the family D2 (g; λ) in Theorem 6.4.1. In both cases, the algebraic theory (Verma modules) determines the top degree part by replacing the Lie-algebraic Laplacians Δ− by Yamabe operators. Then the lower orders terms are naturally formulated in terms of Qcurvatures and mean curvature. The condition L = 0 is rather restrictive and, in particular, not conformally invariant. Therefore, it is of interest to have the following generalization of Lemma 6.26.1. Theorem 6.26.3. Assume that L0 = 0. Then 1 2 2 2 P3 (M, Σ; g) = N P2 (M ; g) + P2 (Σ; g)N + (HP2 (Σ; g) − (dH, d·)) 3 3 3 3 2 + Q2 (M ; g) − Q2 (Σ; g) + 2H N 2 (P2 (Σ; g) − Q2 (Σ; g))H + H 3 , + Q3 (M, Σ; g) + 3 where Q3 (M, Σ; g) = − 23 Q3,T . Proof. We ﬁrst rewrite (6.26.2) in terms of P2 (M ) and P2 (Σ). We get 1 N P2 (M, g) + P2 (Σ, g)N + HP2 (Σ, g) − (dH, d·) P3 (M, Σ; g) = 2 1 1 1 1 1 + RicN − τM + τM + τΣ N + N τM + HτΣ . 3 12 8 12 8

6.26. The pair (P3 , Q3 )

467

Now (6.26.5) implies 1 1 1 1 3 RicN − τM + τΣ = RicN − τΣ + H 2 4 8 2 8 2 and 1 1 1 1 τM − τΣ = τΣ + 2 RicN −6H 2 − τΣ 6 4 6 4 2 1 1 2 RicN − τΣ − H . = 3 2 8

Q2 (M ) − Q2 (Σ) =

Hence

1 1 3 RicN − τM + τΣ = (Q2 (M ) − Q2 (Σ)) + 3H 2 . 3 8 2

Therefore, P3 (M, Σ; g) = +

1 N P2 (M, g) + P2 (Σ, g)N 2

+ HP2 (Σ, g) − (dH, d·)

1 1 3 (Q2 (M ) − Q2 (Σ)) + 2H 2 )N + N τM + HτΣ . 2 12 8

Thus it only remains to prove that 1 1 N τM + HτΣ = −Q3,T + (P2 (Σ) − Q2 (Σ))H + H 3 . 12 8 But this is a consequence of the formula for Q3,T in Theorem 6.26.2. The proof is complete. It it not known whether the curvature quantity T also has an interpretation in the spirit of a holographic formula for even order Q-curvature. Moreover, it would be interesting to remove the assumption L0 = 0 in Theorem 6.26.2. We ﬁnish with the description of the relation between Q4 , Q3 and the Euler characteristic of a four-manifold with a totally umbilic boundary. Theorem 6.26.4. Assume that the boundary Σ3 of M 4 is totally umbilic for g, i.e., L0 (g) = 0. Then 2 1 1 3 + 4Q Q3 volΣ (g). χ(M 4 ) = |C| vol(g) − 4 32π 2 M 4π 2 2 Σ Corollary 6.26.1. If L0 (g) = 0, then the sum Q4 vol(g) − 3 Q3 volΣ (g) M

is conformally invariant.

Σ

468

Chapter 6. Conformally Covariant Families

Theorem 6.26.4 shows that for L0 = 0, the boundary integral in terms of Q3 is directly related to the boundary integral in the Gauß-Bonnet formula. If L0 does not vanish, then the boundary integral in Gauß-Bonnet diﬀers from the integral of Q3 by the integral of a pointwise conformally invariant quantity. Proof. We use the formula ([38], formula (4.3)) 2 1 χ(M ) = τ − 4| Ric |2 + |R|2 vol 32π 2 M M 1 9τM H − 18F H − 6(T, L) + 54H 3 − 18H|L|2 + 4 tr(L3 ) volΣ . − 2 24π Σ Note that our conventions diﬀer from those in [38]: their Laplacian has opposite sign and L is the negative of ours. We recall from Section 6.14 that the ﬁrst integrand equals 8 Pf 4 , i.e., the ﬁrst integral is just 1 Pf 4 . (2π)2 M We have also seen in Section 6.14 that 1 Q4 = 2 Pf 4 − |C|2 − ΔJM 4 (here −Δ denotes the non-negative Laplacian). Thus the ﬁrst integral can be written in the form 2 1 |C| + 4Q4 + 4ΔJM vol . 2 32π M Next, we consider the boundary term. It simpliﬁes under the assumption L0 = 0. For the second integrand, we obtain 9τM H − 18H RicN −6H(τM − 2 RicN ) + 12H 3 = 3HτM − 6H RicN +12H 3. Using the relation τM = τΣ + 2 RicN −6H 2 , that term simpliﬁes to 3HτΣ − 6H 3 . Thus 2 1 1 1 3 Hτ |C| vol − + 4Q + 4ΔJ − H χ(M ) = volΣ . 4 M Σ 32π 2 M 4π 2 Σ 2 Now Green’s formula and Theorem 6.26.2 yield 2 1 1 1 1 3 χ(M ) = N JM + HτΣ − H volΣ |C| + 4Q4 vol − 2 32π 2 M 4π Σ 2 2 2 1 1 = Q3,T volΣ . |C| + 4Q4 vol + 2 32π 2 M 4π Σ The relation Q3,T = − 32 Q3 completes the proof.

Bibliography [1] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz. Large N ﬁeld theories, string theory and gravity. Phys. Rep., 323(3-4):183–386, 2000. arXiv:hep-th/9905111v3. [2] P. Albin. Renormalizing curvature integrals on Poincar´e-Einstein manifolds. arXiv:math/0504161v1. [3] S. Alexakis. The decomposition of global conformal invariants: On a conjecture of Deser and Schwimmer. arXiv:0711.1685v1. [4] S. Alexakis. On conformally invariant diﬀerential operators. arXiv:math/ 0608771v3. [5] S. Alexakis. On the decomposition of global conformal invariants I. Annals of Math., 2006. arXiv:math/0509571v3. [6] S. Alexakis. On the decomposition of global conformal invariants. II. Adv. Math., 206(2):466–502, 2006. arXiv:math/0509572v3. [7] S. Alexakis and R. Mazzeo. Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds. arXiv:math/0802.2250v2. ´ [8] E. Alvarez, J. Conde, and L. Hern´ andez. The Dirichlet obstruction in AdS/CFT. Internat. J. Modern Phys. D, 12(8):1415–1429, 2003. arXiv: hep-th/0303164v2. ´ [9] E. Alvarez, J. Conde, and L. Hern´ andez. Rudiments of holography. Internat. J. Modern Phys. D, 12(4):543–582, 2003. arXiv:hep-th/0205075v5. [10] M. Anderson. L2 curvature and volume renormalization of AHE metrics on 4-manifolds. Math. Res. Lett., 8(1-2):171–188, 2001. arXiv:math/ 0011051v2. [11] M. Anderson. Geometric aspects of the AdS/CFT correspondence. In AdS/CFT correspondence: Einstein metrics and their conformal boundaries, volume 8 of IRMA Lect. Math. Theor. Phys., pages 1–31. Eur. Math. Soc., Z¨ urich, 2005. arXiv:hep-th/0403087v2. [12] D. Anselmi. Quantum irreversibility in arbitrary dimension. Nuclear Phys. B, 567(1-2):331–359, 2000. arXiv:hep-th/9905005v3. [13] M.F. Atiyah and R. Bott. A Lefschetz ﬁxed point formula for elliptic complexes. I. Ann. of Math. (2), 86:374–407, 1967.

470

Bibliography

[14] M.F. Atiyah and R. Bott. A Lefschetz ﬁxed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968. [15] I.G. Avramidi. Covariant methods for the calculation of the eﬀective action in quantum ﬁeld theory and investigation of higher-derivative quantum gravity. PhD thesis, Moscow State Lomonosov University, 1986. arXiv: hep-th/9510140v3. [16] P. B¨ acklund. Families of equivariant diﬀerential operators and Anti-de Sitter spaces. 2007. arXiv:0806.4508v1. [17] T.N. Bailey, M. Eastwood, and A.R. Gover. Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math., 24(4):1191–1217, 1994. [18] F. Bastianelli, S. Frolov, and A. Tseytlin. Conformal anomaly of (2, 0) tensor multiplet in six dimensions and AdS/CFT correspondence. J. High Energy Phys., (2):Paper 13, 21, 2000. arXiv:hep-th/0001041v2. [19] R. Baston. Verma modules and diﬀerential conformal invariants. J. Diﬀerential Geom., 32(3):851–898, 1990. [20] R. Baston and M. Eastwood. Invariant operators. In Twistors in mathematics and physics, volume 156 of London Math. Soc. Lecture Note Ser., pages 129–163. Cambridge Univ. Press, 1990. [21] H. Bateman and A. Erdelyi. Higher transcendental functions, volume I. 1953. [22] H. Baum. The conformal analog of Calabi-Yau manifolds. In Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA-Series. Publishing House of EMS, 2007. (to appear). [23] W. Beckner. Sharp Sobolev inequalities on the sphere and the MoserTrudinger inequality. Ann. of Math. (2), 138(1):213–242, 1993. [24] A. Besse. Einstein manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1987. [25] O. Biquard and R. Mazzeo. Parabolic geometries as conformal inﬁnities of Einstein metrics. Arch. Math. (Brno), 42(suppl.):85–104, 2006. [26] C. Bishop and P. Jones. Hausdorﬀ dimension and Kleinian groups. Acta Math., 179(1):1–39, 1997. [27] M. Bonk, J. Heinonen, and E. Saksman. Logarithmic potentials, quasiconformal ﬂows, and Q-curvature. Duke Math. J., 142(2):197–239, 2008. [28] D. Borthwick. Spectral theory of inﬁnite-area hyperbolic surfaces, volume 256 of Progress in Mathematics. Birkh¨ auser Boston Inc., 2007. [29] R. Bousso. The holographic principle. Rev. Modern Phys., 74(3):825–874, 2002. [30] T.P. Branson. Diﬀerential operators canonically associated to a conformal structure. Math. Scand., 57(2):293–345, 1985.

Bibliography

471

[31] T.P. Branson. The functional determinant, volume 4 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, 1993. [32] T.P. Branson. Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc., 347(10):3671–3742, 1995. [33] T.P. Branson. An anomaly associated with 4-dimensional quantum gravity. Comm. Math. Phys., 178(2):301–309, 1996. [34] T.P. Branson. Spectral theory of invariant operators, sharp inequalities, and representation theory. In The Proceedings of the 16th Winter School “Geometry and Physics” (Srn´ı, 1996), number 46, pages 29–54, 1997. [35] T.P. Branson. Q-curvature and spectral invariants. Palermo (2) Suppl., 75:11–55, 2005.

Rend. Circ. Mat.

[36] T.P. Branson. Q-curvature, spectral invariants, and representation theory. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 090, 31, 2007. arXiv:0709.2471v1. [37] T.P. Branson, Sun-Yung A. Chang, and P. Yang. Estimates and extremals for zeta function determinants on four-manifolds. Comm. Math. Phys., 149(2):241–262, 1992. [38] T.P. Branson and P. Gilkey. The functional determinant of a fourdimensional boundary value problem. Trans. Amer. Math. Soc., 344(2):479– 531, 1994. [39] T.P. Branson, P. Gilkey, and J. Pohjanpelto. Invariants of locally conformally ﬂat manifolds. Trans. Amer. Math. Soc., 347(3):939–953, 1995. [40] T.P. Branson and A.R. Gover. Conformally invariant non-local operators. Paciﬁc J. Math., 201(1):19–60, 2001. [41] T.P. Branson and A.R. Gover. Origins, applications and generalizations of the Q-curvature. The American Institute of Mathematics, 2003. http: //www.aimath.org/WWN/confstruct/confstruct.pdf. [42] T.P. Branson and A.R. Gover. Conformally invariant operators, diﬀerential forms, cohomology and a generalisation of Q-curvature. Comm. Partial Diﬀerential Equations, 30(10-12):1611–1669, 2005. arXiv:math/0309085v2. [43] T.P. Branson and A.R. Gover. Origins, applications and generalisations of the Q-curvature. Acta Appl. Math., 102(2-3):131–146, 2008. [44] T.P. Branson and A.R. Gover. Variational status of a class of fully nonlinear curvature prescription problems. Calc. Var. Partial Diﬀerential Equations, 32(2):253–262, 2008. arXiv:math/0610773v1. [45] T.P. Branson and B. Ørsted. Conformal indices of Riemannian manifolds. Compositio Math., 60(3):261–293, 1986. [46] T.P. Branson and B. Ørsted. Conformal deformation and the heat operator. Indiana Univ. Math. J., 37(1):83–110, 1988.

472

Bibliography

[47] T.P. Branson and B. Ørsted. Conformal geometry and global invariants. Diﬀ. Geom. Appl., 1:279–308, 1991. [48] T.P. Branson and B. Ørsted. Explicit functional determinants in four dimensions. Proc. Amer. Math. Soc., 113(3):669–682, 1991. [49] S. Brendle. Global existence and convergence for a higher order ﬂow in conformal geometry. Ann. of Math. (2), 158(1):323–343, 2003. [50] U. Bunke and M. Olbrich. Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. of Math. (2), 149(2):627–689, 1999. [51] U. Bunke and M. Olbrich. The spectrum of Kleinian manifolds. J. Funct. Anal., 172(1):76–164, 2000. [52] F. Burstall and D. Calderbank. Submanifold geometry in generalized ﬂag manifolds. Rend. Circ. Mat. Palermo (2) Suppl., 72:13–41, 2004. [53] P. Buser. Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics. Birkh¨ auser Boston Inc., 1992. ˇ [54] A. Cap and A.R. Gover. Tractor calculi for parabolic geometries. Trans. Amer. Math. Soc., 354(4):1511–1548 (electronic), 2002. ˇ [55] A. Cap and A.R. Gover. Standard tractors and the conformal ambient metric construction. Ann. Global Anal. Geom., 24(3):231–259, 2003. arXiv: math/0207016v1. ˇ [56] A. Cap, A.R. Gover, and V. Souˇcek. Conformally invariant operators via curved Casimirs: examples. arXiv:0808.1978v1. ˇ [57] A. Cap, J. Slov´ ak, and V. Souˇcek. Invariant operators on manifolds with almost Hermitian symmetric structures. I. Invariant diﬀerentiation. Acta Math. Univ. Comenian. (N.S.), 66(1):33–69, 1997. ˇ [58] A. Cap, J. Slov´ ak, and V. Souˇcek. Invariant operators on manifolds with almost Hermitian symmetric structures. II. Normal Cartan connections. Acta Math. Univ. Comenian. (N.S.), 66(2):203–220, 1997. ˇ [59] A. Cap, J. Slov´ ak, and V. Souˇcek. Invariant operators on manifolds with almost Hermitian symmetric structures. III. Standard operators. Diﬀerential Geom. Appl., 12(1):51–84, 2000. ˇ [60] A. Cap, J. Slov´ ak, and V. Souˇcek. Bernstein-Gelfand-Gelfand sequences. Ann. of Math. (2), 154(1):97–113, 2001. [61] Cheng-Hung Chang and D. Mayer. The transfer operator approach to Selberg’s zeta function and modular and Maass wave forms for PSL(2, Z). In Emerging applications of number theory (Minneapolis, MN, 1996), volume 109 of IMA Vol. Math. Appl., pages 73–141. Springer, 1999. [62] Sun-Yung A. Chang. On zeta functional determinant. In Partial diﬀerential equations and their applications (Toronto, ON, 1995), volume 12 of CRM Proc. Lecture Notes, pages 25–50. Amer. Math. Soc., 1997. With notes taken by Jie Qing.

Bibliography

473

[63] Sun-Yung A. Chang. Non-linear elliptic equations in conformal geometry. Z¨ urich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z¨ urich, 2004. (reviewed by R. Mazzeo in Bull. AMS 44 (2) (2007), 323–330). [64] Sun-Yung A. Chang. Conformal invariants and partial diﬀerential equations. Bull. Amer. Math. Soc. (N.S.), 42(3):365–393 (electronic), 2005. [65] Sun-Yung A. Chang and Hao Fang. A class of variational functionals in conformal geometry. Intern. Math. Research Notices, 008, 2008. arXiv: 0803.0333v1. [66] Sun-Yung A. Chang, M. Gursky, and P. Yang. An equation of MongeAmp`ere type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2), 155(3):709–787, 2002. [67] Sun-Yung A. Chang, M. Gursky, and P. Yang. A conformally invariant ´ sphere theorem in four dimensions. Publ. Math. Inst. Hautes Etudes Sci., pages 105–143, 2003. [68] Sun-Yung A. Chang and J. Qing. The zeta functional determinants on manifolds with boundary. I. The formula. J. Funct. Anal., 147(2):327–362, 1997. [69] Sun-Yung A. Chang and J. Qing. The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set. J. Funct. Anal., 147(2):363–399, 1997. [70] Sun-Yung A. Chang, J. Qing, and P. Yang. On the renormalized volumes for conformally compact Einstein manifolds. ESI, 2004. (preprint 1513). [71] Sun-Yung A. Chang, J. Qing, and P. Yang. On the topology of conformally compact Einstein 4-manifolds. In Noncompact problems at the intersection of geometry, analysis, and topology, volume 350 of Contemp. Math., pages 49–61. Amer. Math. Soc., 2004. [72] Sun-Yung A. Chang and P. Yang. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2), 142(1):171–212, 1995. [73] Sun-Yung A. Chang and P. Yang. On a fourth order curvature invariant. In Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), volume 237 of Contemp. Math., pages 9–28. Amer. Math. Soc., 1999. [74] Sun-Yung A. Chang and P. Yang. Non-linear partial diﬀerential equations in conformal geometry. In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 189–207, 2002. [75] I. Chavel. Eigenvalues in Riemannian geometry, volume 115 of Pure and Applied Mathematics. Academic Press Inc., 1984. [76] B. Chen. Some conformal invariants of submanifolds and their applications. Boll. Un. Mat. Ital. (4), 10:380–385, 1974.

474

Bibliography

[77] B. Chow and D. Knopf. The Ricci ﬂow: an introduction, volume 110 of Mathematical Surveys and Monographs. American Mathematical Society, 2004. [78] A. Connes. Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math. (N.S.), 5(1):29–106, 1999. [79] G. D’Ambra and M. Gromov. Lectures on transformation groups: geometry and dynamics. In Surveys in diﬀerential geometry (Cambridge, MA, 1990), pages 19–111. Lehigh Univ., 1991. [80] S. de Haro, K. Skenderis, and S. Solodukhin. Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence. Comm. Math. Phys., 217(3):595–622, 2001. arXiv:hep-th/0002230v3. [81] S. Deser and A. Schwimmer. Geometric classiﬁcation of conformal anomalies in arbitrary dimensions. Phys. Lett. B, 309(3-4):279–284, 1993. arXiv: hep-th/9302047v1. [82] E. D’Hoker and D. Freedman. Supersymmetric gauge theories and the ADS/CFT correspondence. In Strings, branes and extra dimensions. TASI 2001, pages 3–158. World Sci. Publ., River Edge, NJ, 2004. arXiv:hep-th/ 0201253v2. [83] Z. Djadli. Op´erateurs g´eom´etriques et g´eom´etrie conforme. In Actes de S´eminaire de Th´eorie Spectrale et G´eom´etrie. Vol. 23. Ann´ee 2004–2005, volume 23 of S´emin. Th´eor. Spectr. G´eom., pages 49–103. Univ. Grenoble I, 2005. [84] Z. Djadli and A. Malchiodi. Existence of conformal metrics with constant Q-curvature. Ann. of Math. (2 ), 168(3):813–858, 2008. arXiv: math/0410141v3. [85] M. do Carmo. Riemannian geometry. Birkh¨ auser Boston Inc., 1992. [86] H. Donnelly. Heat equation and the volume of tubes. 29(29):239–243, 1975.

Invent. Math.,

[87] M.J. Duﬀ. Twenty years of the Weyl anomaly. Classical Quantum Gravity, 11(6):1387–1403, 1994. [88] M. Eastwood. Notes on conformal diﬀerential geometry. In The Proceedings of the 15th Winter School “Geometry and Physics” (Srn´ı, 1995), volume 43, pages 57–76, 1996. [89] M. Eastwood. Variations on the de Rham complex. Notices Amer. Math. Soc., 46(11):1368–1376, 1999. [90] M. Eastwood and J. Rice. Conformally invariant diﬀerential operators on Minkowski space and their curved analogues. Comm. Math. Phys., 109(2):207–228, 1987. [91] M. Eastwood and M. Singer. A conformally invariant Maxwell gauge. Physics Letters, 107A(2):73–74, 1985.

Bibliography

475

[92] M. Eastwood and J. Slov´ ak. Semiholonomic Verma modules. J. Algebra, 197(2):424–448, 1997. [93] Khalil El Mehdi. Prescribing Q-curvature on higher dimensional spheres. Ann. Math. Blaise Pascal, 12:259–295, 2005. [94] C. Falk. Konforme Berandung vollst¨ andiger Anti-deSitter- Mannigfaltigkeiten. Master’s thesis, Humboldt-Universit¨at Berlin, 2007. [95] C. Falk and A. Juhl. Universal recursive formulas for Q-curvature. arXiv: math/0804.2745v2. [96] C. Feﬀerman and C.R. Graham. The ambient metric. arXiv:0710.0919v1. [97] C. Feﬀerman and C.R. Graham. Q-curvature and Poincar´e metrics. Math. Res. Lett., 9(2-3):139–151, 2002. arXiv:math/0110271v1. [98] C. Feﬀerman and K. Hirachi. Ambient metric construction of Q-curvature in conformal and CR geometries. Math. Res. Lett., 10(5-6):819–831, 2003. arXiv:math/0303184v2. [99] C. Feﬀermann and C.R. Graham. Conformal invariants. Ast´erisque, Numero ´ Cartan (Lyon, Hors Serie:95–116, 1985. The mathematical heritage of Elie 1984). [100] A. Fialkow. Conformal diﬀerential geometry of a subspace. Trans. Amer. Math. Soc., 56:309–433, 1944. [101] A. Fialkow. Conformal classes of surfaces. Amer. J. Math., 67:583–616, 1945. [102] C. Frances. G´eom´etrie et dynamique Lorentziennes conformes. PhD thesis, ENS Lyon, 2002. http://www.umpa.ens-lyon.fr/~cfrances/. [103] C. Frances. The conformal boundary of anti-de Sitter space-times. In AdS/CFT correspondence: Einstein metrics and their conformal boundaries, volume 8 of IRMA Lect. Math. Theor. Phys., pages 205–216. Eur. Math. Soc., Z¨ urich, 2005. ´ [104] D. Fried. The zeta functions of Ruelle and Selberg. I. Ann. Sci. Ecole Norm. Sup. (4), 19(4):491–517, 1986. [105] I.M. Gelfand and G.E. Shilov. Generalized functions. Vol. 1. Academic Press, 1964 [1977]. Translated from the Russian by Eugene Saletan. [106] P. Gilkey. The spectral geometry of a Riemannian manifold. J. Diﬀerential Geometry, 10(4):601–618, 1975. [107] P. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1995. [108] A.R. Gover. q curvature prescription; forbidden functions and the GJMS null space. arXiv:0810.5604v1. [109] A.R. Gover. Aspects of parabolic invariant theory. Rend. Circ. Mat. Palermo (2) Suppl., 59:25–47, 1999. The 18th Winter School “Geometry and Physics” (Srn´ı, 1998).

476

Bibliography

[110] A.R. Gover. Laplacian operators and Q-curvature on conformally Einstein manifolds. Math. Ann., 336(2):311–334, 2006. arXiv:math/0506037v3. [111] A.R. Gover. Conformal Dirichlet-Neumann maps and Poincar´e-Einstein manifolds. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 100, 21, 2007. arXiv:0710.2585v2. [112] A.R. Gover and K. Hirachi. Conformally invariant powers of the Laplacian— a complete nonexistence theorem. J. Amer. Math. Soc., 17(2):389–405, 2004. arXiv:math/0304082v2. [113] A.R. Gover and P. Leitner. A class of compact Poincar´e-Einstein manifolds: properties and construction. arXiv:0808.2097v1. [114] A.R. Gover and P. Leitner. A sub-product construction of Poincar´e-Einstein metrics. Intern. J. Math., 2009. arXiv:math/0608044v2. [115] A.R. Gover and P. Nurowski. Obstructions to conformally Einstein metrics in n dimensions. J. Geom. Phys., 56(3):450–484, 2006. arXiv:math/ 0405304v3. [116] A.R. Gover and L. Peterson. Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus. Comm. Math. Phys., 235(2):339–378, 2003. arXiv:math-ph/0201030v3. [117] C.R. Graham. Extended obstruction tensors and renormalized volume coefﬁcients. arXiv:0810.4203v1. [118] C.R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence. J. London Math. Soc. (2), 46(3):566–576, 1992. [119] C.R. Graham. Volume and area renormalizations for conformally compact Einstein metrics. In The Proceedings of the 19th Winter School “Geometry and Physics” (Srn´ı, 1999), volume 63, pages 31–42, 2000. arXiv:math/ 9909042v1. [120] C.R. Graham. Conformal powers of the Laplacian via stereographic projection. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 121, 4, 2007. arXiv:0711.4798v2. [121] C.R. Graham. Jet isomorphism for conformal geometry. Arch. Math. (Brno), 43(5):389–415, 2007. arXiv:0710.1671v2. [122] C.R. Graham and K. Hirachi. The ambient obstruction tensor and Qcurvature. In AdS/CFT correspondence: Einstein metrics and their conformal boundaries, volume 8 of IRMA Lect. Math. Theor. Phys., pages 59–71. Eur. Math. Soc., Z¨ urich, 2005. arXiv:math/0405068v1. [123] C.R. Graham and K. Hirachi. Inhomogeneous ambient metrics. In Symmetries and overdetermined systems of partial diﬀerential equations, volume 144 of IMA Vol. Math. Appl., pages 403–420. Springer, New York, 2008. [124] C.R. Graham, R. Jenne, L.J. Mason, and G.A. J. Sparling. Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2), 46(3):557–565, 1992.

Bibliography

477

[125] C.R. Graham and A. Juhl. Holographic formula for Q-curvature. Adv. Math., 216(2):841–853, 2007. [126] C.R. Graham and J.M. Lee. Einstein metrics with prescribed conformal inﬁnity on the ball. Adv. Math., 87(2):186–225, 1991. [127] C.R. Graham and E. Witten. Conformal anomaly of submanifold observables in AdS/CFT correspondence. Nuclear Phys. B, 546(1-2):52–64, 1999. arXiv:hep-th/9901021v1. [128] C.R. Graham and M. Zworski. Scattering matrix in conformal geometry. Invent. Math., 152(1):89–118, 2003. arXiv:math/0109089v1. [129] D. Grant. A conformally invariant third order Neumann-type operator for hypersurfaces. Master’s thesis, University of Auckland, 2003. [130] A. Gray. Tubes, volume 221 of Progress in Mathematics. Birkh¨ auser Verlag, second edition, 2004. [131] C. Guillarmou. Generalized Krein formula, determinants and Selberg zeta function in even dimension. arXiv:math/0512173v3. [132] C. Guillarmou. Resonances and scattering poles on asymptotically hyperbolic manifolds. Math. Res. Lett., 12(1):103–119, 2005. arXiv:math/ 0403545v1. [133] C. Guillarmou and F. Naud. Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds. Commun. Anal. Geom., 14(5):945–967, 2006. arXiv:math.DG/0606223v1. [134] M. Gursky and J. Viaclovsky. Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana Univ. Math. J., 52(2):399–419, 2003. arXiv:math/0210303v2. [135] M. Gursky and J. Viaclovsky. Volume comparison and the σk -Yamabe problem. Adv. Math., 187(2):447–487, 2004. arXiv:math/0210302v2. [136] S. Hawking. Zeta function regularization of path integrals in curved spacetime. Comm. Math. Phys., 55:133–148, 1977. [137] S. Hawking. The path-integral approach to quantum gravity. In S. Hawking and W. Israel, editors, General relativity. An Einstein centenary survey. Cambridge University Press, 1979. [138] H. Hecht and W. Schmid. Characters, asymptotics and n-homology of Harish-Chandra modules. Acta Math., 151(1-2):49–151, 1983. [139] D.A. Hejhal. The Selberg trace formula for PSL(2, R). Vol. I. SpringerVerlag, Berlin, 1976. Lecture Notes in Mathematics, Vol. 548. [140] S. Helgason. Groups and geometric analysis, volume 113 of Pure and Applied Mathematics. Academic Press Inc., 1984. [141] M. Henningson and K. Skenderis. The holographic Weyl anomaly. JHEP, 7:Paper 23, 12 pp. (electronic), 1998. arXiv:hep-th/9806087v2.

478

Bibliography

[142] U. Hertrich-Jeromin. Introduction to M¨ obius diﬀerential geometry, volume 300 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2003. [143] M. Herzlich. A remark on renormalized volume and Euler characteristic for ACHE 4-manifolds. Diﬀerential Geom. Appl., 25(1):78–91, 2007. arXiv: math/0305134v2. [144] P. Hislop, P. Perry, and S.-H. Tang. CR-invariants and the scattering operator for complex manifolds with CR-boundary. C. R. Math. Acad. Sci. Paris, 342(9):651–654, 2006. arXiv:0709.1103v1. [145] J. Holland and G. Sparling. Conformally invariant powers of the ambient Dirac operator. arXiv:math/0112033v2. [146] G.T. Horowitz and J. Polchinski. Gauge/gravity duality. arXiv:gr-qc/ 0602037v3. [147] C. Imbimbo, A. Schwimmer, S. Theisen, and S. Yankielowicz. Diﬀeomorphisms and holographic anomalies. Classical Quantum Gravity, 17(5):1129– 1138, 2000. Strings ’99 (Potsdam). [148] H. Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2002. [149] R.W. Jenne. A construction of conformally invariant diﬀerential operators. PhD thesis, University of Washington, 1988. [150] A. Juhl. Invariant extension of automorphic functions, diﬀerential intertwining operators, and Jacobi polynomials. (in preparation). [151] A. Juhl. Cohomological theory of dynamical zeta functions, volume 194 of Progress in Mathematics. Birkh¨ auser Verlag, 2001. ¯ [152] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima, and M. Tanaka. Eigenfunctions of invariant diﬀerential operators on a symmetric space. Ann. of Math. (2), 107(1):1–39, 1978. [153] A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems, volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1995. [154] A. Kholodenko. Boundary conformal ﬁeld theories, limit sets of Kleinian groups and holography. J. Geom. Phys., 35(2-3):193–238, 2000. [155] S. Kichenassamy. On a conjecture of Feﬀerman and Graham. Adv. Math., 184(2):268–288, 2004. [156] A. Knapp. Representation theory of semisimple groups, volume 36 of Princeton Mathematical Series. Princeton University Press, 1986. An overview based on examples. [157] O. Kowalski. Partial curvature structures and conformal geometry of submanifolds. J. Diﬀerential Geometry, 8:53–70, 1973.

Bibliography

479

[158] C. Kozameh, E. Newman, and P. Nurowski. Conformal Einstein equations and Cartan conformal connection. Classical Quantum Gravity, 20(14):3029– 3035, 2003. [159] C. Kozameh, E. Newman, and K. Tod. Conformal Einstein spaces. Gen. Relativity Gravitation, 17(4):343–352, 1985. [160] K. Krasnov. Holography and Riemann surfaces. Adv. Theor. Math. Phys., 4:929–979, 2000. arXiv:hep-th/0005106v2. [161] K. Krasnov and J.-M. Schlenker. On the renormalized volume of hyperbolic 3-manifolds. Comm. Math. Phys., 279(3):637–668, 2008. arXiv: math/0607081v3. [162] R. Kulkarni. Conformal structures and M¨ obius structures. In Conformal geometry (Bonn, 1985/1986), Aspects Math., E12, pages 1–39. Vieweg, 1988. [163] P.D. Lax and R.S. Phillips. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal., 46(3):280–350, 1982. [164] C. LeBrun. H-space with a cosmological constant. Proc. Roy. Soc. London Ser. A, 380:171–185, 1982. [165] J.M. Lee and T.H. Parker. The Yamabe problem. Bull. Amer. Math. Soc. (N.S.), 17(1):37–91, 1987. [166] T. Leistner and P. Nurowski. Ambient metrics for n-dimensional pp-waves. arXiv:0810.2903v2. [167] J. Lewis and D. Zagier. Period functions for Maass wave forms. I. Ann. of Math. (2), 153(1):191–258, 2001. [168] M. Listing. Conformally invariant Cotton and Bach tensors in n dimensions. arXiv:math/0408224v1. [169] M. Lyubich and Y. Minsky. Laminations in holomorphic dynamics. J. Diﬀerential Geom., 47(1):17–94, 1997. [170] A. Malchiodi. Conformal metrics with constant Q-curvature. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 120, 11, 2007. arXiv: 0712.2123v1. [171] J. Maldacena. The large N limit of superconformal ﬁeld theories and supergravity. Adv. Theor. Math. Phys., 2(2):231–252, 1998. arXiv:hep-th/ 9711200v3. [172] J. Maldacena. Tasi 2003 lectures on AdS/CFT. 2004. arXiv:hep-th/ 0309246v5. [173] N. Mandouvalos. Scattering operator, Eisenstein series, inner product formula and “Maass-Selberg” relations for Kleinian groups. Mem. Amer. Math. Soc., 78:iv+87, 1989. [174] Y. Manin and M. Marcolli. Holography principle and arithmetic of algebraic curves. Adv. Theor. Math. Phys., 5(3):617–650, 2001.

480

Bibliography

[175] R. Manvelyan, K. Mkrtchyan, and R. Mkrtchyan. Conformal invariant powers of the Laplacian, Feﬀerman-Graham ambient metric and Ricci gauging. Phys. Lett. B, 657(1-3):112–119, 2007. arXiv:0707.1737v3. [176] R. Manvelyan and D.H. Tchrakian. Conformal coupling of the scalar ﬁeld with gravity in higher dimensions and invariant powers of the Laplacian. Phys. Lett. B, 644(5-6):370–374, 2007. arXiv:hep-th/0611077v2. [177] C. Margerin. G´eom´etrie conforme en dimension 4: ce que l’analyse nous apprend. Ast´erisque, 307:Exp. No. 950, ix–x, 415–468, 2006. S´eminaire Bourbaki. Vol. 2004/2005. [178] K. Matsuzaki and M. Taniguchi. Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, 1998. [179] R. Mazzeo. The Hodge cohomology of a conformally compact metric. J. Diﬀerential Geom., 28(2):309–339, 1988. [180] R. Mazzeo and R.B. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal., 75(2):260–310, 1987. [181] R. Melrose. Geometric scattering theory. Stanford Lectures. Cambridge University Press, 1995. [182] S. Miller and W. Schmid. Automorphic distributions, L-functions, and Voronoi summation for GL(3). Ann. of Math. (2), 164(2):423–488, 2006. arXiv:math/0408100v3. [183] S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda. Holomorphic dynamics, volume 66 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2000. [184] D. Mumford, C. Series, and D. Wright. Indra’s pearls. Cambridge University Press, 2002. The vision of Felix Klein. [185] S. Nayatani. Patterson-Sullivan measure and conformally ﬂat metrics. Math. Z., 225(1):115–131, 1997. [186] C. Ndiaye. Curvature ﬂows on four manifolds with boundary. arXiv:0708. 2029v1. [187] C.B. Ndiaye. Constant Q-curvature metrics in arbitrary dimension. J. Funct. Anal., 251(1):1–58, 2007. [188] P. Nicholls. The ergodic theory of discrete groups, volume 143 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1989. [189] K. Okikiolu. Critical metrics for the determinant of the Laplacian in odd dimensions. Ann. of Math. (2), 153(2):471–531, 2001. [190] M. Olbrich. Cohomology of convex cocompact groups and invariant distributions on limit sets. Habilitation G¨ ottingen, arXiv:math/0207301, 2001.

Bibliography

481

[191] B. Osgood, R. Phillips, and P. Sarnak. Extremals of determinants of Laplacians. J. Funct. Anal., 80(1):148–211, 1988. [192] S. Paneitz. A quartic conformally covariant diﬀerential operator for arbitrary pseudo-Riemannian manifolds (summary). SIGMA Symmetry Integrability Geom. Methods Appl., 4:Paper 036, 3, 2008. [193] I. Papadimitriou and K. Skenderis. AdS/CFT correspondence and geometry. In AdS/CFT correspondence: Einstein metrics and their conformal boundaries, volume 8 of IRMA Lect. Math. Theor. Phys., pages 73–101. Eur. Math. Soc., Z¨ urich, 2005. arXiv:hep-th/0404176v2. [194] T. Parker and S. Rosenberg. Invariants of conformal Laplacians. J. Diﬀerential Geom., 25(2):199–222, 1987. [195] S. Patterson and P. Perry. The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J., 106(2):321–390, 2001. Appendix A by Charles Epstein. [196] S.J. Patterson. Lectures on measures on limit sets of Kleinian groups. In Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984 ), volume 111 of London Math. Soc. Lecture Note Ser., pages 281–323. Cambridge Univ. Press, 1987. [197] S.J. Patterson. An introduction to the theory of the Riemann zeta-function, volume 14 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1988. [198] S. Paycha and S. Rosenberg. Conformal anomalies via canonical traces. In Analysis, geometry and topology of elliptic operators, pages 263–294. World Sci. Publ., Hackensack, NJ, 2006. [199] R. Penrose. Conformal treatment of inﬁnity. In Relativit´e, Groupes et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys., Univ. Grenoble), pages 565–584. Gordon and Breach, 1964. [200] P. Perry. The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix. J. Reine Angew. Math., 398:67–91, 1989. [201] P. Perry. The spectral geometry of geometrically ﬁnite hyperbolic manifolds. In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, volume 76 of Proc. Sympos. Pure Math., pages 289– 327. Amer. Math. Soc., 2007. [202] L. Peterson. Future directions of research in geometry: a summary of the panel discussion at the 2007 Midwest Geometry Conference. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 081, 7, 2007. http: //www.emis.de/journals/SIGMA/MGC2007.html. [203] A.M. Polyakov. Quantum geometry of bosonic strings. Phys. Lett. B, 103(3):207–210, 1981. [204] R. Ponge. Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structures. arXiv:0710.5783v1.

482

Bibliography

[205] J. Qing and D. Raske. Compactness for conformal metrics with constant Q-curvature on locally conformally ﬂat manifolds. Calc. Var. Partial Differential Equations, 26(3):343–356, 2006. [206] J. Ratcliﬀe. Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics. Springer, second edition, 2006. [207] D.B. Ray and I.M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7:145–210, 1971. [208] R. Riegert. A nonlocal action for the trace anomaly. Phys. Lett. B, 134(12):56–60, 1984. [209] D. Ruelle. Zeta-functions for expanding maps and Anosov ﬂows. Invent. Math., 34(3):231–242, 1976. [210] W. Schmid. Automorphic distributions for SL(2, R). In Conf´erence Mosh´e Flato 1999, Vol. I (Dijon), volume 21 of Math. Phys. Stud., pages 345–387. Kluwer Acad. Publ., 2000. [211] A. Schwimmer and S. Theisen. Diﬀeomorphisms, anomalies and the Feﬀerman-Graham ambiguity. JHEP, 8:Paper 32, 16, 2000. arXiv:hep-th/ 0008082v1. [212] A. Schwimmer and S. Theisen. Universal features of holographic anomalies. JHEP, 10:001, 15 pp. (electronic), 2003. arXiv:hep-th/0309064v1. [213] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.), 20:47–87, 1956. [214] N. Seshadri. Approximate Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds. arXiv:0707.0597v2. [215] N. Seshadri. Volume renormalization for complete Einstein-K¨ ahler metrics. Diﬀerential Geom. Appl., 25(4):356–379, 2007. arXiv:math/0404455v2. [216] R.W. Sharpe. Diﬀerential geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, 1997. Cartan’s generalization of Klein’s Erlangen program, with a foreword by S.S. Chern. [217] Wei-Min Sheng, N. Trudinger, and Xu-Jia Wang. The Yamabe problem for higher order curvatures. J. Diﬀerential Geom., 77(3):515–553, 2007. arXiv:math/0505463v1. [218] K. Skenderis. Asymptotically anti-de Sitter spacetimes and their stress energy tensor. In Strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI ), volume 16, pages 740–749, 2001. arXiv:hep-th/0010138v1. [219] K. Skenderis. Lecture notes on holographic renormalization. Classical Quantum Gravity, 19(22):5849–5876, 2002. European Winter School on the Quantum Structure of Spacetime (Utrecht, 2002).

Bibliography

483

[220] K. Skenderis and S. Solodukhin. Quantum eﬀective action from the AdS/CFT correspondence. Phys. Lett. B, 472(3-4):316–322, 2000. arXiv: hep-th/9910023v1. [221] J. Slov´ ak. Natural operators on conformal manifolds. PhD thesis, Brno, 1993. [222] J. Slov´ ak. Parabolic geometries, 1997. Habilitation, Masaryk University. [223] H.M. Smith. Principles of Holography. Wiley-Interscience, 1969. [224] D. Sullivan. The density at inﬁnity of a discrete group of hyperbolic motions. ´ Inst. Hautes Etudes Sci. Publ. Math., 50:171–202, 1979. [225] D. Sullivan. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. (N.S.), 6(1):57–73, 1982. [226] Dennis Sullivan. Conformal dynamical systems. In Geometric dynamics (Rio de Janeiro, 1981), volume 1007 of Lecture Notes in Math., pages 725–752. Springer, Berlin, 1983. [227] L. Susskind and J. Lindesay. An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe. World Scientiﬁc Publishing, 2004. [228] K. Takano and T. Imai. Note on the conformal theory of subspaces. Tensor (N.S.), 3:108–118, 1954. [229] L. Takhtajan and Lee-Peng Teo. Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography. Comm. Math. Phys., 239(1-2):183–240, 2003. arXiv:math/0204318v2. [230] E.C. Titchmarsh. The theory of the Riemann zeta-function. The Clarendon Press Oxford University Press, New York, second edition, 1986. Edited and with a preface by D.R. Heath-Brown. [231] W. Ugalde. A construction of critical GJMS operators using Wodzicki’s residue. Comm. Math. Phys., 261(3):771–788, 2006. [232] M. Urba´ nski. Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. (N.S.), 40(3):281–321 (electronic), 2003. [233] E. van den Ban and H. Schlichtkrull. Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces. J. Reine Angew. Math., 380:108–165, 1987. [234] D.V. Vassilevich. Heat kernel expansion: user’s manual. Phys. Rep., 388(56):279–360, 2003. [235] J. Viaclovsky. Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J., 101(2):283–316, 2000. [236] D. Vogan. Representations of real reductive Lie groups, volume 15 of Progress in Mathematics. Birkh¨ auser Boston, 1981. [237] X. Wang. Holography and the geometry of certain convex cocompact hyperbolic 3-manifolds. arXiv:math/0210164v2.

484

Bibliography

[238] X. Wang. On conformally compact Einstein manifolds. Math. Res. Lett., 8(5-6):671–688, 2001. [239] T.J. Willmore. Riemannian geometry. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1993. [240] T.J. Willmore. Surfaces in conformal geometry. Ann. Global Anal. Geom., 18(3-4):255–264, 2000. Special issue in memory of Alfred Gray (1939–1998). [241] E. Witten. Anti de Sitter space and holography. Adv. Theor. Math. Phys., 2(2):253–291, 1998. arXiv:hep-th/9802150v2. [242] V. W¨ unsch. On conformally invariant diﬀerential operators. Math. Nachr., 129:269–281, 1986. [243] G. Zuckerman. Tensor products of ﬁnite and inﬁnite dimensional representations of semisimple Lie groups. Ann. Math. (2), 106(2):295–308, 1977.

Index AdSn anti de Sitter space, 6 B2N (g) obstruction operator, 417 C ∞ (S m )λ , 72 CM (λ) tractor C-operator, 390 DW , 385 DT (λ), 398 b (λ), 172, 173 DN D2 (g; λ), 196, 201 DM (λ) tractor D-operator, 389 T DN (g; λ) tractor family, 407 T DN (g; λ), 405 c DN (λ) intertwining family, 133 res (h; λ) residue family, 217 DN nc DN (λ) intertwining family, 133 Dnres (h; λ) critical residue family, 215 E2n Euler form, 333 F , 86 Fn standard representation, 48 G G-tensor, 86 Gn , 63 H mean curvature, 46 H + upper hemisphere, 68 H0 , 64 Iλ intertwining operator, 3 Iλ (gn ) ideal deﬁned by ξλ , 130 K n , 64 L second fundamental form, 46 L total holographic anomaly, 225 L0 trace-free second fundamental form, 53 M n , 64 Mij , 64 NT normal tractor, 399 ± N , 65

P parabolic subgroup, 65 P3 Chang-Qing operator, 463 P3T , 457 P4 Paneitz operator, 11, 107 P4e , 53 Pc− contracting part of Poincar´e map, 4 Pn critical GJMS-operator, 88 Pne , 409 P2N = P2N,n GJMS-operator, 88 P4,n Paneitz operator, 107 Q-curvature critical, 90 extrinsic, 409 for Einstein metrics, 343 fundamental identity, 90 in dimension 4, 106 odd order, 410 subcritical, 90 Q-polynomial, 218 QT3 , 457 Q4 , 20, 106, 230 Qe4 , 53 Q6 , 28, 36, 255, 257 Qn critical Q-curvature, 90 QTn (g; λ) tractor Q-polynomial, 418 QTn (g) tractor Q-curvature, 408 Qen extrinsic Q-curvature, 409 G QF n , 411 Q2N = Q2N,n Q-curvature, 90 Qres Q-polynomial, 218 2N (λ) Q3,T , 457 Q4,n , 107 Q6,n , 261

486

R Riemannian curvature, 83 tractor curvature, 393 RT S Einstein tensor, 115 T -curvature, 463 T (g, ϕ) trivialization, 388 V (gE , h) renormalized volume, 309 W Weyl group, 64 X , 106 Y Yamabe operator, 11 Yj± , 64 ZΓ Selberg zeta function, 4 ZΓR Ruelle zeta function, 4 #, 106 A(g) semi-holonomic universal enveloping algebra, 357 B Bach tensor, 240 C Weyl tensor, 84 Δ Laplacian, 87 ΔW , 385 ΔT , 394 ΔT tractor Laplacian, 394 ΔTn , 387 Δ− Laplacian, 117 n Einn , 6 GM ray bundle, 88 H mean curvature vector, 443 H0 height, 67 Hess covariant Hessian form, 83 J, 84 Λ(Γ) limit set, 3 Mλ Verma module, 15 M(Vσ ) Verma module, 357 N (Vσ ) semi-holonomic Verma module, 357 Nλ semi-holonomic Verma module, 360 Ω(Γ) proper set, 3 Pλ Poisson transformation, 72, 76 ΠΣ projection, 400 P Schouten tensor, 84 P(X), 391 Ric Ricci tensor, 83 RicN , 86

Index

S(λ) scattering operator, 94 T ∗ M dual tractor bundle, 388 Tj (h; λ), 215 ΘN (λ), 367 U(g) universal enveloping algebra, 356 X (M ) vector ﬁelds on M , 83 C Cotton tensor, 393 a, 64 α restricted root, 129 α0 , 64 Bn ball model, 2, 66 C(λ), 15, 129 δN (h; λ), 218 δ(Γ) critical exponent, 4 extλ extension operator, 179 gn , 63 g, 88 Hn hyperbolic space, 68 J , 439 κ, 220 κ stereographic projection, 67 mn , 64 μP S Patterson-Sullivan measure, 5 n± n , 64 ∇ Levi-Civita connection, 82 ∇W , 385 tractor connection, 391 ∇T ω Cartan connection, 358 ω , 106 p parabolic subalgebra, 15, 129 π, 357 πΣ , 400 πλc compact model of principal series, 72 πλnc non-compact model of principal series, 75 resΩ(Γ) restriction to Ω(Γ), 180 , 106 T , 395 0 (λ), 118, 128 DN DN (λ), 118, 127 T D2N (λ), 378

Index

τ scalar curvature, 84 τe extrinsic scalar curvature, 207 θ, 433 ˜ N (λ), 367 Θ g˜ ambient metric, 89 CW Chen-Willmore functional, 442 ξλ character, 15, 129 ζ Riemann zeta function, 4 c(λ) c-function, 96 g-trace, 85 gN Nayatani metric, 59 i(λ), 364 p(λ), 364 vn holographic anomaly, 24 v2j holographic coeﬃcients, 24 w, 64 AdS/CFT duality, 28, 336, 343 ambient metric, 89 asymptotic volume, 309 Atiyah-Bott-Lefschetz formula, 8 automorphic distributions, 179 Bach tensor, 240 Bianchi identities, 112 Bochner-Weitzenb¨ ock formula, 112 Bruhat decomposition, 69 Cartan connection, 358 Chang-Qing operator, 463 Chen-Willmore functional, 442 Chen-Willmore quantity, 190 Chern-Gauß-Bonnet theorem, 333 Codazzi equation, 458 conformal compactiﬁcation, 92 index density, 30 inﬁnity, 92 Cotton tensor, 112, 393 critical exponent, 4 critical GJMS-operator, 88 critical residue family, 215 curvature tensor, 83 curved translation principle, 47

487

Deser-Schwimmer classiﬁcation, 334 Einstein’s static universe, 6 Einstein-Hilbert action, 29 Eisenstein series, 103 Euler characteristic, 27 Euler form, 333 extension operator, 179 extrinsic scalar curvature, 207 factorization identities for intertwining families, 139, 148 for residue families, 266 Feﬀerman-Graham expansion, 22 Feﬀerman-Graham obstruction, 240, 249 Fialkow’s fundamental forms, 446 fundamental identity, 90 fundamental pair, 191 G-tensor, 86 Gauß equation, 207 Gegenbauer polynomial, 118 geodesic ﬂow, 7 GJMS-operators, 88 Gover’s formula, 343 Harish-Chandra’s c-function, 96 Henningson-Skenderis test, 249, 342 Hessian, 83 holographic anomaly, 24 coeﬃcients, 24 duality, 412 renormalization, 29 holographic formula for Q4 , 230 for Q6 , 254 for critical Q-curvature, 26, 225 for subcritical Q-curvature, 300 hyperbolic space as symmetric space, 65 ball model, 66 upper half space model, 68

488

induced representation, 131 Iwasawa decomposition, 65 Koszul formula, 82 Kulkarni-Nomizu product, 84 Langlands decomposition, 15 Laplacian, 87 Lefschetz ﬁxed point formula, 8 Lichnerowicz Laplacian, 242 light cone, 65 limit set, 81 mean curvature, 46 mean curvature vector, 443

Index

standard tractor bundle, 388 stereographic projection, 67 subcritical GJMS-operator, 88 T-tensor, 86 tractor C-operator, 390 D-operator, 389 Q-polynomial, 418 Bochner-Laplacian, 394 connection, 391 curvature endomorphism, 393 scalar product, 393 tractor families, 407 triangle decomposition, 15

Nayatani metric, 59, 81 Newton’s relations, 246 normal tractor, 399

universal enveloping algebra, 356

Paneitz curvature, 20, 106 operator, 11, 107 quantity, 106 Patterson-Sullivan measure, 5 PBH transformations, 25 Pfaﬃan, 333 Poincar´e-Einstein metric, 93 Poisson transformation, 72, 76 Polyakov formula, 30, 32, 340 principal series representation, 72, 75 proper set, 3

Weyl tensor, 84 Weyl’s tube formula, 336 Willmore functional, 197

renormalized volume, 309 residue families, 217 Ricci tensor, 83 scalar curvature, 84 scattering operator, 94 Schouten tensor, 84 second fundamental form, 46 semi-holonomic Verma module, 357 spherical metric, 68

Verma module, 356

Yamabe equation, 91 operator, 11 zeta function Riemann, 4 Ruelle, 4 Selberg, 4 Zuckerman translation, 360