The Logarithmic Singularities of the Green Functions of the Conformal Powers of the Laplacian
aa r X i v : . [ m a t h . DG ] A ug THE LOGARITHMIC SINGULARITIES OF THE GREEN FUNCTIONS OFTHE CONFORMAL POWERS OF THE LAPLACIAN
RAPHA¨EL PONGE
Abstract.
Green functions play an important role in conformal geometry. In this paper, weexplain how to compute explicitly the logarithmic singularities of the Green functions of theconformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators,as well as the conformal fractional powers of the Laplacian arising from scattering theory forPoincar´e-Einstein metrics. The results are formulated in terms of Weyl conformal invariantsarising from the ambient metric of Fefferman-Graham. As applications we obtain characteriza-tions in terms of Green functions of locally conformally flat manifolds and a spectral theoreticcharacterization of the conformal class of the round sphere.
Introduction
Motivated by the analysis of the singularity of the Bergman kernel of a strictly pseudoconvexdomain Ω ⊂ C n , Fefferman [Fe3] launched the program of determining all local invariants of astrictly pseudoconvex CR structure. This program was subsequently extended to deal with localinvariants of other parabolic geometries, including conformal geometry [FG1, BEG]. It has sincefound connections with various areas of mathematics and mathematical physics such as geometricPDEs, geometric scattering theory and conformal field theory. For instance, the Poincar´e-Einsteinmetric of Fefferman-Graham [FG1, FG3] was a main impetus for the AdS/CFT correspondance.Green functions of conformally invariant operators plays a fundamental role in conformal ge-ometry. Parker-Rosenberg [PR] computed the logarithmic singularity of Yamabe operator in lowdimension. In [Po1, Po2] it was shown that the logarithmic singularities of Green functions ofconformally invariant ΨDOs are linear combinations of Weyl conformal invariants. Those invari-ants are obtained from complete metric constructions of the covariant derivatives of the curvaturetensor of ambient metric of Fefferman-Graham [FG1, FG3]. The approach of [PR] was based onresults of Gilkey [Gi3] on heat invariants for Laplace-type operators. It is not clear how to extendthis approach to higher order GJMS operators, leave aside conformal fractional powers of theLaplacian.Exploiting the invariant theory for conformal structures, the main result of this paper is anexplicit, and surprinsingly simple, formula for the logarithmic singularities of the Green functionsof the conformal powers of the Laplacian in terms of Weyl conformal invariants obtained from theheat invariants of the Laplace operator (Theorem 7.3). Here by conformal powers we mean theoperators of Graham-Jenne-Mason-Sparling [GJMS] and, more generally, the conformal fractionalpowers of the Laplacian [GZ]. These operators include the Yamabe and Paneitz operators.Granted this result, it becomes straightforward to compute the logarithmic singularities ofthe Green functions of the conformal powers of the Laplacian from the sole knowledge of the heatinvariants of the Laplace operators (see Theorem 7.8 and Theorem 7.9). These results have severalimportant consequences.In dimension n ≥ k -th conformal powerof the Laplacian with k = n − The research of this paper was partially supported by NSERC discovery grant 341328-07 (Canada), JSPS grantin aid 30549291(Japan), research resettlement and foreign faculty research grants from Seoul National University,and NRF grant for basic research 2013R1A1A2008802 (South Korea). he Bergman kernel. This also enables us to obtain a spectral theoretic characterization of theconformal class of the round sphere amount compact simply connected manifolds of dimension ≥ H DOs are linear combinations of Weyl CR invariants. We may also expectrelate the logarithmic singularities of the Green functions of the CR invariant powers of the sub-Laplacian [GG] to the heat invariants of the sub-Laplacian [BGS]. However, the heat invariantsfor the sub-Laplacian are much less known than that for the Laplace operator. In particular,in order to recapture the Chern tensor requires computing the 3rd coefficient in the heat kernelasymptotics for the sub-Laplacian. In fact, for our purpose it would be enough to carry out thecomputation in the special case of circle bundles over Ricci-flat K¨ahler manifolds.This paper is organized as follows. In Section 1, we recall the main facts on the heat kernelasymptotics for the Laplace operator. In Section 2, we recall the geometric description of thesingularity of the Bergman kernel of a strictly pseudoconvex complex domain and the constructionof local CR invariants by means of Fefferman’s ambient K¨ahler-Lorentz metric. In Section 3,we recall the construction of the Fefferman-Graham’s ambient metric and GJMS operators. InSection 4, we recall the construction of local conformal invariants by means of the ambient metric.In Section 5, we explain the construction of conformal fractional powers of the Laplacian and itsconnection with scattering theory and the Poincar´e-Einstein metric. In Section 6, we gather basicfacts about Green functions and their relationship with heat kernels. In Section 7, we state themain result and derive various consequences. In Section 8, we give an outline of the proof of themain result.
Acknowledgements.
It is a pleasure to thank Pierre Albin, Charles Fefferman, Rod Gover, RobinGraham, Colin Guillarmou, Kengo Hirachi, Dmitry Jakobson, Andreas Juhl, Andras Vasy, and MaciejZworski for various discussions related to the subject matter of this paper. Part of the research of thispaper was carried out during visits to Kyoto University, McGill University, MSRI, UC Berkeley, and theUniversity of Tokyo. The author wish to thank these institutions for their hospitality. The Heat Kernel of the Laplace Operator
The most important differential operator attached to a closed Riemannian manifold ( M n , g ) isthe Laplace operator, ∆ g u = − p det g ( x ) X i,j ∂ i (cid:16) g ij ( x ) p det g ( x ) ∂ j u (cid:17) . his operator lies at the interplay between Riemannian geometry and elliptic theory. On the onehand, a huge amount of geometric information can be extracted from the analysis of the Laplaceoperator. For instance, if 0 = λ (∆ g ) < λ (∆ g ) ≤ λ (∆ g ) ≤ · · · are the eigenvalues of ∆ g countedwith multiplicity, then Weyl’s Law asserts that, as k → ∞ ,(1.1) λ k (∆ g ) ∼ ( ck ) n , c := (4 π ) − n Γ (cid:16) n (cid:17) − Vol g ( M ) , where Vol g ( M ) is the Riemannian volume of ( M, g ). On the other hand, Riemannian geometryis used to describe singularities or asymptotic behavior of solutions of PDEs associated to theLaplace operator. This aspect is well illustrated by the asymptotics of the heat kernel of ∆ g .Denote by e − t ∆ g , t >
0, the semigroup generated by ∆ g . That is, u ( x, t ) = ( e − t ∆ g u ) ( x ) issolution of the heat equation,( ∂ t + ∆ g ) u ( x, t ) = 0 , u ( x, t ) = u ( x ) at t = 0 . The heat kernel K ∆ g ( x, y ; t ) is the kernel function of the heat semigroup, e − t ∆ g u ( x ) = Z M K ∆ g ( x, y ; t ) u ( y ) v g ( y ) , where v g ( y ) = p g ( x ) | dx | is the Riemannian volume density. Equivalently, K ∆ g ( x, y ; t ) providesus with a fundamental solution for the heat equation (1). Furthermore,(1.2) X e − tλ j (∆ g ) = Tr e − t ∆ g = Z M K ∆ g ( x, x ; t ) u ( x ) v g ( x ) . Theorem 1.1 ([ABP, Gi1, Mi]) . Let ( M n , g ) be a closed manifold. Then (1) As t → + , (1.3) K ∆ g ( x, x ; t ) ∼ (4 πt ) − n X j ≥ t j a j (∆ g ; x ) , where the asymptotics holds with respect to the Fr´echet-space topology of C ∞ ( M ) . (2) Each coefficient a j (∆ g ; x ) is a local Riemannian invariant of weight j .Remark . There is an asymptotics similar to (1.3) for the heat kernel of any selfadjoint differ-ential operators with positive-definite principal symbol (see [Gi3, Gr]).
Remark . A local Riemannian invariant is a (smooth) function I g ( x ) of x ∈ M and the metrictensor g which, in any local coordinates, has an universal expression of the form,(1.4) I g ( x ) = X a αβ ( g ( x )) (cid:0) ∂ β g ( x ) (cid:1) α , where the sum is finite and the a αβ are smooth functions on Gl n ( R ) that are independent of thechoice of the local coordinates. We further say that I g ( x ) has weight w when I λ g ( x ) = λ − w I g ( x ) ∀ λ > . These definitions continue to make sense for pseudo-Riemannian structures of nonpositive signa-ture. Note also that with our convention for the weight, the weight is always an even nonnegativeinteger.Examples of Riemannian invariants are provided by complete metric contractions of the tensorproducts of covariant derivatives of the curvature tensors R ijkl = h R ( ∂ i , ∂ j ) ∂ k , ∂ l i . Namely,Contr g ( ∇ k R ⊗ · · · ⊗ ∇ k l R ) . Such an invariant has weight w = ( k + · · · + k l ) + 2 l and is called a Weyl Riemannian invariant .Up to a sign factor, the only Weyl Riemannian invariants of weight w = 0 and w = 2 are theconstant function 1 and the scalar curvature κ := R ijji respectively. In weight w = 4 we obtainthe following four invariants, κ , | Ric | := Ric ij Ric ij , | R | := R ijkl R ijkl , ∆ g κ, here Ric ij := R kijk is the Ricci tensor. In weight w = 6 they are 17 such invariants; the onlyinvariants that do not involve the Ricci tensor are |∇ R | := ∇ m R ijkl ∇ m R ijkl , R klij R ijpq R pqkl , R ijkl R i kp q R pjql . Theorem 1.4 (Atiyah-Bott-Patodi [ABP]) . Any local Riemannian invariant of weight w , w ∈ N , is a universal linear combination of Weyl Riemannian invariants of same weight.Remark . By using normal coordinates the proof is reduced to determining all invariant poly-nomial of the orthogonal group O( n ) (or O( p, q ) in the pseudo-Riemannian case). They aredetermined thanks to Weyl’s invariant theory for semisimple groups.Combining Theorem 1.1 and Theorem 1.4 we obtain the following structure theorem for theheat invariants. Theorem 1.6 (Atiyah-Bott-Patodi [ABP]) . Each heat invariant a j (∆ g ; x ) in (1.3) is a universallinear combination of Weyl Riemannian invariants of weight j .Remark . There is a similar structure result for the heat invariants of any selfadjoint ellipticcovariant differential operator with positive principal symbol.In addition, the first few of the heat invariants are computed.
Theorem 1.8 ([BGM, MS, Gi2]) . The first four heat invariants a j (∆ g ; x ) , j = 0 , . . . , are givenby the following formulas, a (∆ g ; x ) = 1 , a (∆ g ; x ) = − κ, a (∆ g ; x ) = 1180 | R | − | Ric | + 172 κ −
130 ∆ g κ, (1.5) a (∆ g ; x ) = 19 · (cid:0) |∇ R | + 64 R klij R ij pq R pqkl + 352 R ijkl R i kp q R pjql (cid:1) + Weyl Riemannian invariants involving the Ricci tensor . Remark . We refer to Gilkey’s monograph [Gi3] for the full formula for a (∆ g ; x ) and formulasfor the heat invariants of various Laplace type operators, including the Hodge Laplacian on forms.We mention that Gilkey’s formulas involve a constant multiple of −h R, ∆ g R i = g pq R ijkl R ijkl ; pq ,but this Weyl Riemannian invariant is a linear combination of other Weyl Riemannian invariants.In particular, using the Bianchi identities we find that, modulo Weyl Riemannian involving theRicci tensor, −h R, ∆ g R i = R klij R ijpq R pqkl + 4 R ijkl R i kp q R pjql . This relation is incorporated into (1.5).
Remark . Polterovich [Po] established formulas for all the heat invariants a j (∆ g ; x ) in termsof the Riemannian distance function. Remark . Combining the equality a (∆ g ; x ) = 1 with (1.2) and (1.3) shows that, as t → + , X e − tλ j (∆ g ) ∼ (4 πt ) − n Z M v g ( x ) = (4 πt ) − n Vol g ( M ) . We then can apply Karamata’s Tauberian theorem to recover Weyl’s Law (1.1).
Remark . A fundamental application of the Riemannian invariant theory of the heat kernelasymptotics is the proof of the local index theorem by Atiyah-Bott-Patodi [ABP] (see also [Gi3]).2.
The Bergman Kernel of a Strictly Pseudoconvex Domain
A fundamental problem in several complex variables is to find local computable biholomorphicinvariants of a strictly pseudoconvex domain Ω ⊂ C n . One approach to this issue is to look at the oundary singularity of the Bergman metric of Ω or equivalently its Bergman kernel. Recall thatthe Bergman kernel is the kernel function of the orthogonal projection, B Ω : L (Ω) −→ L (Ω) ∩ Hol(Ω) ,B Ω u ( z ) = Z K Ω ( z, w ) u ( w ) dw. For instance, in the case of the unit ball B n = {| z | < } ⊂ C n we have B B n ( z, w ) = n ! π n (1 − zw ) − ( n +1) . The Bergman kernel lies at the interplay of complex analysis and differential geometry. On theone hand, it provides us with the reproducing kernel of the domain Ω and it plays a fundamentalrole in the analysis of the ∂ -Neuman problem on Ω. On the other hand, it provides us with abiholomorphic invariant K¨ahler metric, namely, the Bergman metric, ds = X ∂ ∂z j ∂z k log B ( z, z ) dz j dz k . In what follows we let ρ be a defining function of Ω so that Ω = { ρ < } and i∂∂ρ > Theorem 2.1 (Fefferman, Boutet de Monvel-Sj¨ostrand) . Near the boundary ∂ Ω = { ρ = 0 } , K Ω ( z, z ) = ϕ ( z ) ρ ( z ) − ( n +1) − ψ ( z ) log ρ ( z ) , where ϕ ( z ) and ψ ( z ) are smooth up to ∂ Ω . Motivated by the analogy with the heat asympototics (1.3), where the role of the time variable t is played by the defining function ρ ( z ), Fefferman [Fe3] launched the program of giving a geo-metric description of the singularity of the Bergman kernel similar to the description provided byTheorem 1.1 and Theorem 1.6 for the heat kernel asymptotics.A first issue at stake concerns the choice of the defining function. We would like to make abiholomorphically invariant choice of defining function. This issue is intimately related to thecomplex Monge-Amp`ere equation on Ω:(2.1) J ( u ) := ( − n det (cid:18) u ∂ z k u∂ z j u ∂ z j ∂ z k u (cid:19) , u | ∂ Ω = 0 . A solution of the Monge-Amp`ere equation is unique and biholomorphically invariant in the sensethat, given any bilohomorphism Φ : Ω → Ω, we have u (Φ( z )) = | det Φ ′ ( z ) | n +1 u ( z ) . We mention the following important results concerning the Monge-Amp`ere equation.
Theorem 2.2 (Cheng-Yau [CY]) . Let Ω ⊂ C n be a strictly pseudoconvex domain. (1) There is a unique exact solution u ( z ) of the complex Monge-Amp`ere equation (2.1). (2) The solution u ( z ) is C ∞ on Ω and belongs to C n + − ǫ (cid:0) Ω (cid:1) for all ǫ > . (3) The metric ds = X ∂ ∂z j ∂z k log (cid:16) u ( z ) − ( n +1) (cid:17) dz j dz k is a K¨ahler-Einstein metric on Ω . Theorem 2.3 (Lee-Melrose [LM]) . Let Ω ⊂ C n be a strictly pseudoconvex domain with definingfunction ρ ( z ) . Then, near the boundary ∂ Ω = { ρ = 0 } , the Cheng-Yau solution to the Monge-Amp`ere equation has a behavior of the form, (2.2) u ( z ) ∼ ρ ( z ) X k ≥ η k ( z ) (cid:0) ρ ( z ) n +1 log ρ ( z ) (cid:1) k , where the functions η k ( z ) are smooth up to the boundary. The Cheng-Yau solution is not smooth up to the boundary, but if we only seek for asymptoticsolutions then we do get smooth solutions. heorem 2.4 (Fefferman [Fe1]) . Let Ω ⊂ C n be a strictly pseudoconvex domain with definingfunction ρ ( z ) . Then there are functions in C ∞ (cid:0) Ω (cid:1) that are solutions to the asymptotic Monge-Amp`ere equation, (2.3) J ( u ) = 1 + O( ρ n +1 ) near ∂ Ω , u | ∂ Ω = 0 . Remark . As it can be seen from (2.2), the error term O( ρ n +1 ) cannot be improved in generalif we seek for a smooth approximate solution.Any smooth solution u ( z ) of (2.3) is a defining function for Ω and it is asymptotically biholo-morphically invariant in the sense that, for any biholomorphism Φ : Ω → Ω, u (Φ( z )) = | det Φ ′ ( z ) | n +1 u ( z ) + O( ρ ( z ) n +1 )The complex structure of Ω induces on the boundary ∂ Ω a strictly pseudoconvex CR structure.As a consequence of a well-known result of Fefferman [Fe2] there is one-to-one correspondancebetween biholomorphisms of Ω and CR diffeomorphisms of ∂ Ω. Therefore, boundary values ofbiholomorphic invariants of Ω gives rise to CR invariants of ∂ Ω. We refer to [Fe3] for the precisedefinition of a local CR invariant. Let us just mention that a local CR invariant I ( z ) has weight ω when, for any biholomorphism Φ : Ω → Ω, we have I (Φ( z )) = | det Φ ′ ( z ) | − wn +1 I ( z ) on ∂ Ω . Proposition 2.6 (Fefferman [Fe3]) . Let Ω ⊂ C n be a strictly pseudoconvex domain. Then (1) Near the boundary ∂ Ω , K Ω ( z, z ) = u ( z ) − ( n +1) n +1 X j =0 I ( j ) u ( z ) u ( z ) j + O (log u ( z )) , where u ( z ) is a smooth solution of (2.3). (2) For each j , the boundary value of I ( j ) u ( z ) is a local CR invariant of weight j . This leads us to the geometric part of program of Fefferman: determining all CR invariants ofstrictly pseudoconvex domain so as to have an analogue of Theorem 1.4 for CR invariants. How-ever, the CR invariant theory is much more involved than the Riemannian invariant theory. Thegeometric problem reduces to the invariant theory for a parabolic subgroup P ⊂ SU( n +1 , P is not semisimple, Weyl’s invariant theory does not apply anymore. The correspondinginvariant theory was developed by Fefferman [Fe3] and Bailey-Eastwood-Graham [BEG].The Weyl CR invariants are constructed as follows. Let u ( z ) be a smooth approximate solutionto the Monge-Amp`ere equation in the sense of (2.3). On the ambient space C ∗ × Ω consider thepotential, U ( z , z ) = | z | u ( z ) , ( z , z ) ∈ C ∗ × Ω . Using this potential we define the K¨ahler-Lorentz metric,˜ g = X ≤ j,k ≤ n ∂ U∂z j ∂z k dz j dz k . A biholomorphism Φ : Ω → Ω acts on the ambient space by˜Φ( z , z ) = (cid:16) | det Φ ′ ( z ) | − n +1 , Φ( z ) (cid:17) , ( z , z ) ∈ C ∗ × Ω . This is the transformation law under biholomorphisms for K n +1 Ω , where K Ω is the canonical linebundle of Ω. Theorem 2.7 (Fefferman [Fe1]) . (1) The K¨ahler-Lorentz metric ˜ g is asymptotically Ricci flat and invariant under biholomor-phisms in the sense that ˜Φ ∗ ˜ g = ˜ g + O (cid:0) ρ ( z ) n +1 (cid:1) and Ric(˜ g ) = O ( ρ ( z ) n ) near C ∗ × ∂ Ω , where Φ is any biholomorphism of Ω . The restriction of ˜ g to S × ∂ Ω is a Lorentz metric g whose conformal class is invariantunder biholomorphisms.Remark . The above construction of the ambient metric is a special case of the ambient metricconstruction associated to a conformal structure due to Fefferman-Graham [FG1, FG3] (see alsoSection 3).
Remark . We refer to [Le] for an intrinsic construction of the Fefferman circle bundle S × ∂ Ωand its Lorentz metric g for an arbitrary nondegenerate CR manifold.It follows from Theorem 2.7 that, given a Weyl Riemannian invariant I ˜ g ( z , z ) constructedout of the covariant derivatives of the curvature tensor of ˜ g , the boundary value of I ˜ g ( z , z ) isa local CR invariant provided that it does not involved covariant derivatives of too high order.This condition is fullfilled if I ˜ g ( z , z ) has weight w ≤ n , in which case its boundary value is aCR invariant of same weight. Such an invariant is called a Weyl CR invariant . We observe thatthe Ricci-flatness of the ambient metric kills all Weyl invariants involving the Ricci tensor of ˜ g .Therefore, there are much fewer Weyl CR invariants than Riemannian Weyl invariants.The following results are the analogues of Theorem 1.4 and Theorem 1.6 for the Bergman kernel. Theorem 2.10 (Fefferman [Fe3], Bayley-Eastwood-Graham [BEG]) . Any local CR invariant ofweight w ≤ n is a linear combination of Weyl CR invariants of same weight. Theorem 2.11 (Fefferman [Fe3], Bayley-Eastwood-Graham [BEG]) . Let Ω ⊂ C n be a strictlypseudoconvex domain. Then, near the boundary ∂ Ω , K Ω ( z, z ) = u ( z ) − ( n +1) n +1 X j =0 I ( j ) u ( z ) u ( z ) j + O (log u ( z )) , where u ( z ) is a smooth solution of (2.3) and, for j = 0 , . . . , n + 1 , I ( j ) u ( z ) = J ( j )˜ g ( z , z ) | z =1 , where J ( j )˜ g ( z , z ) is a linear combination of complete metric contractions of weight w of the co-variant derivatives of the K¨ahler-Lorentz metric ˜ g .Remark . We refer to [Hi] for an invariant description of the logarithmic singularity ψ ( z ) ofthe Bergman kernel.3. Ambient Metric and Conformal Powers of the Laplacian
Let ( M n , g ) be a Riemannian manifold. In dimension n = 2 the Laplace operator is conformallyinvariant in the sense that, if we make a conformal change of metric ˆ g = e g , Υ ∈ C ∞ ( M, R ),then ∆ ˆ g = e − ∆ g . This conformal invariance breaks down in dimension n ≥
3. A conformal version of the Lapla-cian is provided by the Yamabe operator, P ,g := ∆ g + n − n − κ,P ,e g = e − ( n +1 ) Υ P ,g e ( n − ) Υ ∀ Υ ∈ C ∞ ( M, R ) . The operator of Paneitz [Pa] is a conformally invariant operator with principal part ∆ g . Namely, P ,g := ∆ g + δV d + n − (cid:26) n −
1) ∆ g R g + n n − R g − | P | (cid:27) ,P ,e g = e − ( n +2 ) Υ P ,g e ( n − ) Υ ∀ Υ ∈ C ∞ ( M, R ) . Here P ij = n − (Ric g ij − R g n − g ij ) is the Schouten-Weyl tensor and V is the tensor V ij = n − n − R g g ij − P ij acting on 1-forms (i.e., V ( ω i dx i ) = ( V ji ω j ) dx i ). ore generally, Graham-Jenne-Mason-Sparling [GJMS] constucted higher order conformal pow-ers of the Laplacian by using the ambient metric of Fefferman-Graham [FG1, FG3]. This metricextends Fefferman’s K¨ahler-Lorentz metric to any ambient space associated to a conformal struc-ture. It is constructed as follows.Consider the ray-bundle, G := G x ∈ M (cid:8) t g ( x ); t > (cid:9) ⊂ S T ∗ M, where S T ∗ M is the bundle of symmetric (0 , G depends only the con-formal class [ g ]. Moreover, on G there is a natural family of dilations δ s , s >
0, given by(3.1) δ s (ˆ g ) = s ˆ g ∀ ˆ g ∈ G. Let π : G → M be the canonical submersion of G onto M (i.e., the restriction to G of thecanonical submersion of the bundle S T ∗ M ). Let dπ t : S T ∗ M → S T ∗ G be the differential of π on symmetric (0 , G carries a canonical symmetric (0 , g defined by(3.2) g ( x, ˆ g ) = dπ ( x ) t ˆ g ∀ ( x, ˆ g ) ∈ G. The datum of the representative metric g defines a fiber coordinate t on G such that ˆ g = t g ( x )for all ( x, ˆ g ) ∈ G . If { x j } are local coordinates on M , then { x j , t } are local coordinates on G andin these coordinates the tensor g is given by g = t g ij ( x ) dx i dx j , where g ij are the coefficients of the metric tensor g in the local coordinates { x j } .The ambient space is the ( n + 2)-dimensional manifold˜ G := G × ( − , ρ , where we denote by ρ the variable in ( − , G with the hypersurface G := { ρ = 0 } .We also note that the dilations δ s in (3.1) lifts to dilations on ˜ G . Theorem 3.1 (Fefferman-Graham [FG1, FG3]) . Near ρ = 0 there is a Lorentzian metric ˜ g , calledthe ambient metric, which is defined up to infinite order in ρ when n is odd, and up to order n when n is even, such that (i) In local coordinates { t, x j , ρ } , ˜ g = 2 ρ ( dt ) + t g ij ( x, ρ ) + 2 tdtdρ, where g ij ( x, ρ ) is a family of symmetric (0 , -tensors such that g ( x, ρ ) | ρ =0 = g ij ( x ) . (ii) The ambient metric is asymptotically Ricci-flat, in the sense that (3.3) Ric(˜ g ) = (cid:26) O( ρ ∞ ) if n is odd , O( ρ n ) if n is even . Remark . We note that (i) implies that(3.4) ˜ g | T G = g and δ ∗ s ˜ g = s ˜ g ∀ s > , where g is the symmetric tensor (3.2). Remark . The ambient metric is unique up to its order of definition near ρ = 0. Solving theequation (3.3) leads us to a system of nonlinear PDEs for g ij ( x, ρ ). When n is even, there isan obstruction to solve this system at infinite order which given by the a conformally invarianttensor. This tensor is the Bach tensor in dimension 4. Moreover, it vanishes on conformallyEinstein metrics. Example . Suppose that (
M, g ) is Einstein with Ric( g ) = 2 λ ( n − g . Then˜ g = 2 ρ ( dt ) + t (1 + λρ ) g ij ( x ) + 2 tdtdρ. Conformal powers of the Laplacian are obtained from the powers of the ambient metric Lapla-cian ˜∆ ˜ g on ˜ G as follows. roposition 3.5 (Graham-Jenne-Mason-Sparling [GJMS]) . Let k ∈ N and further assume k ≤ n when n is even. Then (1) We define a operator P k,g : C ∞ ( M ) → C ∞ ( M ) by (3.5) P k,g u ( x ) := t − ( n + k ) ˜∆ k ˜ g (cid:0) t n − k ˜ u ( x, ρ ) (cid:1)(cid:12)(cid:12)(cid:12) ρ =0 , u ∈ C ∞ ( M ) , where ˜ u is any smooth extension of u to M × R (i.e., the r.h.s. above is independent ofthe choice of ˜ u ). (2) P k,g is a differential operator of order k with same leading part as ∆ kg . (3) P k,g is a conformally covariant differential operator such that (3.6) P k,e g = e − ( n + k ) Υ ( P k,g ) e ( n − k ) Υ ∀ Υ ∈ C ∞ ( M, R ) . Remark . The homogeneity of the ambient metric ˜ g implies that the r.h.s. is idenpendent of t and obeys (3.6). Remark . The operator P k,g is called the k -th GJMS operator. For k = 1 and k = 2 we recoverthe Yamabe operator and Paneitz operator respectively. Remark . The GJMS operators are selfadjoint [FG2, GZ].
Remark . The GJMS operator P k,g can also be obtained as the obstruction to extending afunction u ∈ C ∞ ( M ) into a homogeneous harmonic function on the ambient space. In addition,we refer to [FG4, GP, Ju1, Ju2] for various features and properties of the GJMS operarors. Remark . When the dimension n is even, the ambient metric construction is obstructed atfinite order, and so the GJMS construction breaks down for k > n . As proved by Graham [Gr1]in dimension 4 for k = 3 and by Gover-Hirachi [GH] in general, there do not exist conformallyinvariant operators with same leading part as ∆ kg for k > n when n is even. Remark . The operator P n ,g is sometimes called the critical GJMS operator . Note that for P n the transformation law (3.6) becomes P n ,e g = e − n Υ P n ,g ∀ Υ ∈ C ∞ ( M, R ) . It is tempting to use the critical GJMS operator as a candidate for a subsitute to the Laplacianto prove a version of the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula in dimension ≥
3. Werefer to [CJ] for results in this direction.
Example . Suppose that ( M n , g ) is a Einstein manifoldwith Ric g = λ ( n − g , λ ∈ R . Then, for all k ∈ N , we have(3.7) P k,g = Y ≤ j ≤ k (cid:18) ∆ g − λ ( n + 2 j − n − j ) (cid:19) . Local Conformal Invariants
In this section, we describe local scalar invariants of a conformal structure and explain how toconstruct them by means of the ambient metric of Fefferman-Graham [FG1, FG3].
Definition 4.1.
A local conformal invariant of weight w , w ∈ N , is a local Riemannian invariant I g ( x ) such that I e g ( x ) = e − w Υ( x ) I g ( x ) ∀ Υ ∈ C ∞ ( M, R ) . The most important conformally invariant tensor is the
Weyl tensor ,(4.1) W ijkl = R ijkl − ( P jk g il + P il g jk − P jl g ik − P ik g jl ) , where P jk = n − (Ric jk − κ g n − g jk ) is the Schouten-Weyl tensor. In particular, in dimension n ≥ M, g ) is locally conformally flat. Moreover, as the Weyltensor is conformally invariant of weight 2, we get scalar conformal invariants by taking completemetric contractions of tensor products of the Weyl tensor. et I g ( x ) be a local Riemannian invariant of weight w . By using the ambient metric, Fefferman-Graham [FG1, FG3] produced a recipe for constructing local conformal invariant from I g ( x ) asfollows.Step 1: Thanks to Theorem 1.4 we know that I g ( x ) is a linear combination of complete metriccontractions of covariant derivatives of the curvature tensor. Substituting into this complete metriccontractions the ambient metric ˜ g and the covariant derivatives of its curvature tensor we obtaina local Riemannian invariant I ˜ g ( t, x, ρ ) on the ambient metric space ( ˜ G, ˜ g ). For instance,Contr g ( ∇ k R ⊗ · · · ⊗ ∇ k l R ) −→ Contr ˜ g ( ˜ ∇ k ˜ R ⊗ · · · ⊗ ˜ ∇ k l ˜ R )where ˜ ∇ is the ambient Levi-Civita connection and ˜ R is the ambient curvature tensor.Step 2: We define a function on M by(4.2) ˜ I g ( x ) := t − w I ˜ g ( t, x, ρ ) (cid:12)(cid:12) ρ =0 ∀ x ∈ M. If n is odd this is always well defined. If n is even this is well defined provided only derivativesof ˜ g of not too high order are involved; this is the case if w ≤ n . Note also that thanks to thehomogeneity of the ambient metric the r.h.s. of (4.2) is always independent of the variable t . Proposition 4.2 ([FG1, FG3]) . The function ˜ I g ( x ) is a local conformal invariant of weight w .Remark . The fact that ˜ I g ( x ) transforms conformally under conformal change of metrics is aconsequence of the homogeneity of the ambient metric in (3.4).We shall refer to the rule I g ( x ) → ˜ I g ( x ) as Fefferman-Graham’s rule . When I g ( x ) is a WeylRiemannian invariant we shall call ˜ I g ( x ) a Weyl conformal invariant . Example . Let us give a few examples of using Fefferman-Graham’s rule (when n is even wefurther assume that the invariants I g ( x ) below have weight ≤ n ).(a) If I g ( x ) = Contr g ( ∇ k Ric ⊗∇ k R ⊗ · · · ⊗ ∇ k l R ), then the Ricci flatness of ˆ g implies that˜ I g ( x ) = 0 . (b) (See [FG1, FG3].) If I g ( x ) = Contr g ( R ⊗ · · · ⊗ R ) (i.e., no covariant derivatives areinvolved), then ˜ I g ( x ) is obtained by substituting the Weyl tensor for the curvature tensor,that is, ˜ I g ( x ) = Contr g ( W ⊗ · · · ⊗ W ) . (c) (See [FG1, FG3].) For I g ( x ) = |∇ R | , we have˜ I g ( x ) = Φ g ( x ) := | V | + 16 h W, U i + 16 | C | , where C jkl = ∇ l P jk − ∇ k P jl is the Cotton tensor and the tensors U and V are defined by V mijkl := ∇ s W ijkl − g im C jkl + g jm C ikl − g km C lij + g lm C kij ,U mjkl := ∇ m C jkl + g rs P mr W sjkl . Theorem 4.5 (Bailey-Eastwood-Graham [BEG], Fefferman-Graham [FG3]) . Let w ∈ N , andfurther assume w ≤ n when n is even. Then every local conformal invariant of weight w is alinear combination of Weyl conformal invariants of same weight.Remark . As in the Riemannian case, the strategy for the proof of Theorem 4.5 has two mainsteps. The first step is a reduction of the geometric problem to the invariant theory for a parabolicsubgroup P ⊂ O( n + 1 , P is developped in [BEG]. Remark . We refer to [GrH] for a description of the local conformal invariants in even dimensionbeyond the critical weight w = n . . Scattering Theory and Conformal Fractional Powers of the Laplacian
The ambient metric construction realizes a conformal structure as a hypersuface in the ambientspace. An equivalent approach is to realize a conformal structure as the conformal infinity of anasymptotically hyperbolic Einstein (AHE) manifold. This enables us to use scattering theory toconstruct conformal fractional powers of the Laplacian.Let ( M n , g ) be a compact Riemannian manifold. We assume that M is the boundary of somemanifold X with interior X . Note we always can realize the double M ⊔ M as the boundary of[0 , × M . We assume that X carries a AHE metric g + . This means that(1) There is a definining function ρ for M = ∂X such that, near M , the metric g + is of theform, g + = ρ − g ( ρ, x ) + ρ − ( dρ ) , where g ( ρ, x ) is smooth up ∂X and agrees with g ( x ) at ρ = 0.(2) The metric g + is Einstein, i.e., Ric( g + ) = − ng + . In particular (
X, g + ) has conformal boundary ( M, [ g ]), where [ g ] is the conformal class of g .The scattering matrix for an AHE metric g + is constructed as follows. We denote by ∆ g + theLaplacian on X for the metric g + . The continuous spectrum of ∆ g + agrees with [ n , ∞ ) and itspure point spectrum σ pp (∆ g + ) is a finite subset of (cid:0) , n (cid:1) (see [Ma1, Ma2]). DefineΣ = n s ∈ C ; ℜ s ≥ n , s n N , s ( n − s ) σ pp (∆ g + ) o . For s ∈ Σ, consider the eigenvalue problem,∆ g + − s ( n − s ) = 0 . For any f ∈ C ∞ ( M ), the above eigenvalue problem as a unique solution of the form, u = F ρ n − s + Gρ s , F, G ∈ C ∞ ( X ) , F | ∂X = f. The scattering matrix is the operator S g ( s ) : C ∞ ( M ) → C ∞ ( M ) given by S g ( s ) f := G | M ∀ f ∈ C ∞ ( M ) . Thus S g ( s ) can be seen as a generalized Dirichlet-to-Neuman operator, where F | ∂X represents the“Dirichlet data” and G | ∂X represents the “Neuman data”. Theorem 5.1 (Graham-Zworski [GZ]) . For z ∈ − n + Σ define P z,g := 2 z Γ( z )Γ( − z ) S g (cid:18) n + z (cid:19) . The family ( P z,g ) uniquely extends to a holomorphic family of pseudodifferential operators ( P z,g ) ℜ z> in such way that (i) The operator P z,g is a Ψ DO of order z with same principal symbol as ∆ zg . (ii) The operator P z,g is conformally invariant in the sense that (5.1) P z,e g = e − ( n + z ) Υ ( P z,g ) e ( n − z ) Υ ∀ Υ ∈ C ∞ ( M, R ) . (iii) The family ( P z,g ) z ∈ Σ is a holomorphic family of Ψ DOs and has a meromorphic extensionto the half-space ℜ z > with at worst finite rank simple pole singularities. (iv) Let k ∈ N be such that − k + n σ pp (∆ g + ) . Then lim z → k P z,g = P k,g , where P k,g is the k -th GJMS operator defined by (3.5).Remark . Results of Joshi-S´a Barreto [JS] show that each operator P z,g is a Riemannianinvariant ΨDO in the sense that all the homogeneous components are given by universal expressionsin terms of the partial derivatives of the components of the metric g (see [Po2] for the precisedefinition). emark . If k ∈ N is such that λ k := − k + n ∈ σ pp (∆ g + ), then (5.1) holds modulo a finiterank smoothing operator obtained as the restriction to M of the orthogonal projection onto theeigenspace ker (cid:0) ∆ g + − λ k (cid:1) . Remark . The analysis of the scattering matrix S g ( s ) by Graham-Zworski [GZ] relies on theanalysis of the resolvent ∆ g + − s ( n − s ) by Mazzeo-Melrose [MM]. Guillarmou [Gu1] establishedthe meromorphic continuation of the resolvent to the whole complex plane C . There are onlyfinite rank poles when the AHE metric is even. This gives the meromorphic continuation of thescattering matrix S g ( z ) and the operators P z,g to the whole complex plane. We refer to [Va1, Va2]for alternative approaches to these questions. Remark . We refer to [CM] for an interpretation of the operators P z,g as fractional Laplaciansin the sense of Caffarelli-Sylvestre [CS]. Example . Consider the round sphere S n seen as the boundary of the unit ball B n +1 ⊂ R n +1 equipped with its standard hyperbolic metric. Then P z,g = Γ (cid:16)q ∆ g + ( n − + 1 + z (cid:17) Γ (cid:16)q ∆ g + ( n − + 1 − z (cid:17) , z ∈ C . We refer to [GN] for an extension of this formula to manifolds with constant sectional curvature.A general conformal structure cannot always be realized as the conformal boundary of an AHEmanifold. However, as showed by Fefferman-Graham [FG1, FG3] it always can be realized as aconformal boundary formally.Let ( M n , g ) be a Riemannian manifold. Define X = M × (0 , ∞ ) and let X = M × [0 , ∞ ) be theclosure of X . We shall denote by r the variable in [0 , ∞ ) and we identify M with the boundary r = 0. Theorem 5.7 (Fefferman-Graham [FG1, FG3]) . Assume M has odd dimension. Then, nearthe boundary r = 0 , there is a Riemannian metric g + , called Poincar´e-Einstein metric, which isdefined up to infinite order in r and satisfies the following properties: (i) In local coordinates { r, x j } , g + = r − ( dr ) + r − g ij ( x, ρ ) , where g ij ( x, r ) is a family of symmetric (0 , -tensors such that g ( x, r ) | r =0 = g ij ( x ) . (ii) It is asymptotically Einstein in the sense that, near the boundary r = 0 , (5.2) Ric( g + ) = − ng + + O( r ∞ ) . Remark . When the dimension n is even there is also a Poincar´e-Einstein metric which isdefined up to order n − r and satisfies the Einstein equation (5.2) up to an O( r n − ) error. Remark . The constructions of the Poincar´e-Einstein metric and ambient metric are equivalent.We can obtain either metric from the other. For instance, the map φ : ( r, x ) → (cid:0) r − , x, − r (cid:1) isa smooth embedding of X into the hypersurface, H + = { ˜ g ( T, T ) = − } = (cid:8) ( t, xρ ); 2 ρt = − (cid:9) ⊂ ˜ G, where T = t∂ t is the infinitesimal generator for the dilations (3.1). Then g + is obtained as thepullback to X of ˜ g | T H . In particular, the metric g + is even in the sense that the tensor g ( x, r )has a Taylor expansion near r = 0 involving even powers of r only. Example . Assume that (
M, g ) is Einstein, with Ric( g ) = 2 λ ( n − g . Then g + = r − ( dr ) + r − (cid:18) − λr (cid:19) g ij ( x ) dx i dx j . Proposition 5.11.
Assume n is odd. Then there is an entire family ( P g,z ) z ∈ C of Ψ DOs on M uniquely defined up to smoothing operators such that (i) Each operator P g,z , z ∈ C , is a Ψ DO of order z and has same principal symbol as ∆ zg . ii) Each operator P g,z is a conformally invariant Ψ DO in the sense of [Po2] in such a waythat, for all Υ ∈ C ∞ ( M, R ) , (5.3) P z,e g = e − ( n + z ) Υ ( P z,g ) e ( n − z ) Υ mod Ψ −∞ ( M ) , where Ψ −∞ ( M ) is the space of smoothing operators on M . (iii) For all k ∈ N , the operator P z,g agrees at z = k with the k -th GJMS operator P k,g . (iv) For all z ∈ C , (5.4) P z,g P − z,g = 1 mod Ψ −∞ ( M ) . Remark . The equation (5.4) is an immediate consequence of the functional equation for thescattering matrix, S g ( s ) S g ( n − s ) = 1 . This enables us to extend the family ( P z,g ) beyond the halfspace ℜ z > Remark . When n is even, we also can construct operators P z,g as above except that eachoperator P z,g is unique and satisfies (5.3) modulo ΨDOs of order z − n . Remark . By using a polynomial continuation of the homogeneous components of the symbolsof the GJMS operators, Petterson [Pe] also constructed a holomorphic family of ΨDOs satisfyingthe properties (i)–(iv) above.
Example . Suppose that (
M, g ) is Ricci flat and compact. Then, forall z ∈ C , P g,z = ∆ zg mod Ψ −∞ ( M ) . As mentioned above, the constructions of the ambient metric and Poincar´e-Einstein metric areequivalent. Thus, it should be possible to define the GJMS operators directly in terms of thePoincar´e-Einstein metric, so as to interpret the GJMS operators as boundary operators. Thisinterpretation is actually implicit in [GZ]. Indeed, if we combine the definition (3.5) of the GJMSoperators with the formula on the 2nd line from the top on page 113 of [GZ], then we arrive atthe following statement.
Proposition 5.16 ([GJMS], [GZ]) . Let k ∈ N and further assume k ≤ n when n is even. Then,for all u ∈ C ∞ ( M ) , P k,g u = r − n − k k − Y j =0 (cid:18) ∆ g + + (cid:18) − n + k − j (cid:19) (cid:18) n + k − j (cid:19)(cid:19) ( r n − k u + ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 , where u + is any function in C ∞ ( X ) which agrees with u on the boundary r = 0 .Remark . Guillarmou [Gu2] obtained this formula via scattering theory. We also refer to [GW]for a similar formula in terms of the tractor bundle. In addition, using a similar approach as above,Hirachi [Hi] derived an analogous formula for the CR GJMS operators of Gover-Graham [GG].6.
Green Functions of Elliptic Operators
In this section, we describe the singularities of the Green function of an elliptic operator andits closed relationship with the heat kernel asymptotics.Let ( M n , g ) be a Riemannian manifold. Let P : C ∞ c ( M ) → C ∞ ( M ) be an elliptic pseudodif-ferential operator (ΨDO) of (integer) order m >
0. Its
Green function G P ( x, y ) (when it exists)is the fundamental solution of P , that is,(6.1) P x G P ( x, y ) = δ ( x − y ) . Equivalently, G P ( x, y ) is the inverse kernel of P , i.e., P x (cid:18)Z M G P ( x, y ) u ( y ) v g ( y ) (cid:19) = u ( x ) . n general, P may have a nontrivial kernel (e.g., the kernel of the Laplace operator ∆ g consistsof constant functions). Therefore, by Green function we shall actually mean a solution of (6.1)modulo a C ∞ -error, i.e., P x G P ( x, y ) = δ ( x − y ) mod C ∞ ( M × M ) . This means that using G P ( x, y ) we always can solve the equation P v = u modulo a smooth error.In other words, G P ( x, y ) is the kernel function of parametrix for P . As such it always exists.The Green function G P ( x, y ) is smooth off the diagonal y = x . As this is the kernel func-tion of a pseudodifferential operator of order − m (namely, a parametrix for P ), the theory ofpseudodifferential operators enables us to describe the form of its singularity near the diagonal. Proposition 6.1 (See [Ta]) . Suppose that P has order m ≤ n . Then, in local coordinates andnear the diagonal y = x , (6.2) G P ( x, y ) = | x − y | − n + m X ≤ j Assume that M is compact. Then the zeta function ζ ( P ; s ) = Tr P − s , ℜ s > nm ,has a meromorphic extension to the whole complex plane C with at worst simple pole singularities.Moreover, at s = 1 , we have (6.4) m Res s =1 Tr P − s = Z M γ P ( x ) v g ( x ) . Remark . We refer to [Se] for the construction of the complex powers P s , s ∈ C . The con-struction depends on the choice of a spectral cutting for both P and its principal symbol, but theresidues at integer points of the zeta function ζ ( P ; s ) = Tr P − s do not depend on this choice. Remark . Proposition 6.4 has a local version (see [Gui2, KV, Wo2]). If we denote by K P − s ( x, y )the kernel of P − s for ℜ s > nm , then the map s → K P − s ( x, x ) has a meromorphic extension to C with at worst simple pole singularities in such a way that(6.5) m Res s =1 K P − s ( x, x ) = γ P ( x ) . We also note that the above equality continues to hold if we replace P − s by any holomorphicΨDO family P ( s ) such that ord P ( s ) = − ms and P (1) is a parametrix for P . This way Eq. (6.5)continues to hold when M is not compact. uppose now that M is compact and P has a positive leading symbol, so that we can form theheat semigroup e − tP and the heat kernel K P ( x, y ; t ) associated to P . The heat kernel asymptoticsfor P then takes the form,(6.6) K P ( x, x ; t ) ∼ (4 πt ) − nm X j ≥ t jm a j ( P ; x ) + log t X k ≥ t k b j ( P ; x ) as t → + . We refer to [Wi, GS] for a derivation of the above heat kernel asymptotics for ΨDOs. The residuesof the local zeta function ζ ( P ; s ; x )( x ) = K P − s ( x, x ) are related to the coefficient in the heat kernelasymptotic for P as follows.By Mellin’s formula, for ℜ s > s ) P − s = Z ∞ t s − (1 − Π ) e − tP dt, ℜ s > , where Π is the orthogonal projection onto the nullspace of P . Together with the heat kernelasymptotics (6.6) this implies that, for ℜ s > nm ,Γ( s ) K P − s ( x, x ) = Z t s − K P ( x, x ; t ) dt + h ( x ; s )= (4 π ) − nm X ≤ j 0. Thus, for j = 0 , , . . . , n − 1, we haveΓ (cid:18) n − jm (cid:19) Res s = n − jm K P − s ( x, x ) = (4 π ) − nm a j ( P ; x ) . Combining this with (6.5) we arrive at the following result. Proposition 6.7 (Compare [PR]) . Let k ∈ m − N be such that mk − n ∈ N . Then, under theabove assumptions, we have (6.7) γ P k ( x ) = (4 π ) − nm m Γ( k ) a n − mk ( P ; x ) . The above result provides us with a way to compute the logarithmic singularities of the Greenfunctions of the powers of P from the knowledge of coefficients of the heat kernel asymptotics. Forinstance, in the case of the Laplace operator combining Theorem 1.8 and Proposition 6.7 gives thefollowing result. Proposition 6.8. For k = n , n − , n − and provided that k > , we have γ ∆ n g ( x ) = (4 π ) − n (cid:0) n (cid:1) , γ ∆ n − g ( x ) = (4 π ) − n (cid:0) n − (cid:1) . − κ,γ ∆ n − g ( x ) = (4 π ) − n (cid:0) n − (cid:1) (cid:18) | R | − | Ric | + 172 κ − 130 ∆ g κ (cid:19) . Green Functions and Conformal Geometry Green functions of the Yamabe operator and other conformal powers of the Laplacian play animportant role in conformal geometry. This is illustrated by the solution to the Yamabe problemby Schoen [Sc] or by the work of Okikoliu [Ok] on variation formulas for the zeta-regularizeddeterminant of the Yamabe operator in odd dimension. Furthermore, Parker-Rosenberg [PR]computed the logarithmic singularity of the Green function of the Yamabe operator in even lowdimension. Theorem 7.1 (Parker-Rosenberg [PR, Proposition 4.2]) . Let ( M n , g ) be a closed Riemannianmanifold. Then In dimension n = 2 and n = 4 we have γ P ,g ( x ) = 2(4 π ) − ( n = 2) and γ P ( x ) = 0 ( n = 4) . (2) In dimension n = 6 , γ P ,g ( x ) = (4 π ) − | W | , where | W | = W ijkl W ijkl is the norm-square of the Weyl tensor. (3) In dimension n = 8 , γ P ,g ( x ) = (4 π ) − · (cid:0) g + 64 W klij W ijpq W pqkl + 352 W ijkl W i kp q W pjql (cid:1) , where Φ g is given by (4.4).Remark . Parker and Rosenberg actually computed the coefficient a n − ( P ,g ; x ) of t − in theheat kernel asymptotics for the Yamabe operator when M is closed. We obtain the above formulasfor γ P ,g ( x ) by using (6.7). We also note that Parker and Rosenberg used a different sign conventionfor the curvature tensor.The computation of Parker and Rosenberg in [PR] has two main steps. The first step uses theformulas for Gilkey [Gi2, Gi3] for the heat invariants of Laplace type operator, since the Yamabeoperator is such an operator. This expresses the coefficient a n − ( P ,g ; x ) as a linear combinationof Weyl Riemannian invariants. For n = 8 there are 17 such invariants. The 2nd step consistsin rewriting these linear combinations in terms of Weyl conformal invariant, so as to obtain themuch simpler formulas above.It is not clear how to extend Parker-Rosenberg’s approach for computing the logarithmic sin-gularities γ P k,g ( x ) of the Green functions of other conformal powers of the Laplacian. This wouldinvolve computing the coefficients of the heat kernel asymptotics for all these operators, includingconformal fractional powers of the Laplacian in odd dimension.As the following result shows, somewhat amazingly, in order to compute the γ P k,g ( x ) the sole knowledge of the coefficients heat kernel asymptotics for the Laplace operator is enough. Theorem 7.3. Let ( M n , g ) be a Riemannian manifold and let k ∈ N be such that n − k ∈ N .Then the logarithmic singularity of the Green function of P k,g is given by γ P k,g ( x ) = 2Γ( k ) (4 π ) − n ˜ a n − k (∆ g ; x ) , where ˜ a n − k (∆ g ; x ) is the local conformal invariant obtained by applying Fefferman-Graham’s ruleto the heat invariant a n − k (∆ g ; x ) in (1.3).Remark . When n is odd the condition n − k ∈ N imposes k to be an half-integer, and so inthis case P k,g is a conformal fractional powers of the Laplacian.Using Theorem 7.3 and the knowledge of the heat invariants a j (∆ g ; x ) it becomes straighfor-ward to compute the logarithmic singularities γ P k,g ( x ). In particular, we recover the formulas ofParker-Rosenberg [PR] stated in Theorem 7.1. Theorem 7.5 ([Po1, Po2]) . Let ( M n , g ) be a Riemannian manifold of dimension n ≥ . Then γ P n ,g ( x ) = 2Γ (cid:0) n (cid:1) (4 π ) − n and γ P n − ,g ( x ) = 0 . Proof. We know from Theorem 1.8 that a (∆ g ; x ) = 1 and a (∆ g ; x ) = − κ . Thus ˜ a (∆ g ; x ) = 1.Moreover, as κ is the trace of Ricci tensor, from Example 4.4.(a) we see that ˜ a (∆ g ; x ) = 0.Combining this with Theorem 7.3 gives the result. (cid:3) Remark . The equality γ P n ( x ) = 2Γ (cid:0) n (cid:1) − (4 π ) − n can also be obtained by direct computationusing (6.3) and the fact that in this case p − n ( x, ξ ) is the principal symbol of a parametrix for P n ,g . As P n ,g has same principal symbol as ∆ n g , the principal symbol of any parametrix for P n ,g is equal to | ξ | − ng , where | ξ | g = g ij ( x ) ξ i ξ j . emark . The equality γ P n − ,g ( x ) = 0 is obtained in [Po1, Po2] by showing this is a conformalinvariant weight 1 and using the fact there is no nonzero conformal invariant of weight 1. Indeed,by Theorem 1.4 any Riemannian invariant of weight 1 is a scalar multiple of the scalar curvature,but the scalar curvature is not a conformal invariant. Therefore, any local conformal of weight 1must be zero. Theorem 7.8. Let ( M n , g ) be a Riemannian manifold of dimension n ≥ . Then (7.1) γ P n − ( x ) = (4 π ) − n Γ (cid:0) n − (cid:1) . | W | . Proof. By Theorem 1.8 we have a (∆ g ; x ) = 1180 | R | − | Ric | + 172 κ − 130 ∆ g κ. As Example 4.4.(b) shows, the Weyl conformal invariant associated to | R | by Fefferman-Graham’srule is | W | . Moreover, as | Ric | , κ , and ∆ g κ involve the Ricci tensor, the corresponding Weylconformal invariants are equal to 0. Thus,˜ a (∆ g ; x ) = 1180 | W | . Applying Theorem 7.3 then completes the proof. (cid:3) Theorem 7.9. Let ( M n , g ) be a Riemannian manifold of dimension n ≥ . Then γ P n − ( x ) = (4 π ) − n Γ (cid:0) n − (cid:1) · · (cid:0) g + 64 W klij W ijpq W pqkl + 352 W ijkl W i kp q W pjql (cid:1) , where Φ g ( x ) is given by (4.4).Proof. Thanks to Theorem 1.8 we know that a (∆ g ; x ) = 19 · (cid:0) |∇ R | + 64 R klij R ijpq R pqkl + 352 R ijkl R i kp q R pjql (cid:1) + I (3) g, ( x ) , where I (3) g, ( x ) is a linear combination of Weyl Riemannian invariants involving the Ricci tensor.Thus ˜ a (∆ g ; x ) = 0 by Example 4.4.(a). By Example 4.4.(b), the Weyl conformal invariantscorresponding to R klij R pqkl R ijpq and R i qjk R pki l R plj q are W klij W pqkl W ijpq and W i qjk W pki l W plj q respectively. Moreover, by Example 4.4.(c) applying Fefferman-Graham’s rule to |∇ R | yields theconformal invariant Φ g given by (4.4). Thus,˜ a (∆ g ; x ) = 19 · (cid:0) g + 64 W klij W ijpq W pqkl + 352 W ijkl W i kp q W pjql (cid:1) . Combining this with Theorem 7.3 proves the result. (cid:3) Remark . In the case of Yamabe operator (i.e., k = 1) the above results give back the resultsof Parker-Rosenberg [PR] as stated in Theorem 7.1.Let us present some applications of the above results. Recall that a Riemannian manifold( M n , g ) is said to be locally conformally flat when it is locally conformally equivalent to the flatEuclidean space R n . As is well known, in dimension n ≥ 4, local conformal flatness is equivalentto the vanishing of the Weyl tensor. Therefore, as an immediate consequence of Theorem 7.8 weobtain the following result. Theorem 7.11. Let ( M n , g ) be a Riemannian manifold of dimension n ≥ . Then the followingare equivalent: (1) ( M n , g ) is locally conformally flat. (2) The logarithmic singularity γ P n − ,g ( x ) vanishes identically on M . emark . A well known conjecture of Radamanov [Ra] asserts that, for a strictly pseudoconvexdomain Ω ⊂ C n , the vanishing of the logarithmic singularity of the Bergman kernel is equivalentto Ω being biholomorphically equivalent to the unit ball B n ⊂ C n . Therefore, we may seeTheorem 7.11 as an analogue of Radamanov conjecture in conformal geometry in terms of Greenfunctions of conformal powers of the Laplacian.In the compact case, we actually obtain a spectral theoretic characterization of the conformalclass of the round sphere as follows. Theorem 7.13. Let ( M n , g ) be a compact simply connected Riemannian manifold of dimension n ≥ . Then the following are equivalent: (1) ( M n , g ) is conformally equivalent to the round sphere S n . (2) R M γ P n − ,g ( x ) v g ( x ) = 0 . (3) Res s =1 Tr (cid:2) ( P n − ) − s (cid:3) = 0 .Proof. The equivalence between (2) and (3) is a consequence of (6.4). By Theorem 7.8, we have Z M γ P n − ,g ( x ) v g ( x ) = (4 π ) − n Γ (cid:0) n − (cid:1) . Z M | W ( x ) | v g ( x ) . As | W ( x ) | ≥ R M γ P n − ,g ( x ) v g ( x ) = 0 if and only if W ( x ) = 0 identically on M . As M is compact and simply connected, a well known result of Kuiper [Ku] asserts that the vanishingof Weyl tensor is equivalent to the existence of a conformal diffeomorphism from ( M, g ) onto theround sphere S n . Thus (1) and (2) are equivalent. The proof is complete. (cid:3) Proof of Theorem 7.3 In this section, we outline the proof of Theorem 7.3. The strategy of the proof is divided into9 main steps. Step 1. Let k ∈ N be such that n − k ∈ N . Then γ ∆ kg ( x ) = 2Γ( k ) − (4 π ) − n a n − k (∆ g ; x ) . This follows from Proposition 6.7. This is the Riemannian version of Theorem 7.3 and the mainimpetus for that result. Step 2. Let k ∈ N be such that n − k ∈ N . Then γ P k,g ( x ) is a linear combination of Weylconformal invariants of weight n − k . This step is carried out in [Po1, Po2]. This is a general result for the logarithmic singularities ofthe Green functions of conformally invariant ΨDOs. For the Yamabe operator, the transformationof γ P k,g ( x ) under conformal change of metrics was observed by Parker-Rosenberg [PR]. Theirargument extends to general conformally invariant operators. Step 3. Let I g ( x ) be a local Riemannian invariant of any weight w ∈ N if n is odd or of weight w ≤ n if n is even. If ( M, g ) is Ricci-flat, then I g ( x ) = ˜ I g ( x ) on M , where ˜ I g ( x ) is the local conformal invariant associted to I g ( x ) by Fefferman-Graham’s rule. This results seems to be new. It uses the fact that if ( M, g ) is Ricci flat, then by Example 3.4the ambient metric is given by ˜ g = 2 ρ ( dt ) + t g ij ( x ) + 2 tdtdρ. Step 4. Theorem 7.3 holds when ( M n , g ) is Ricci-flat. If ( M, g ) is Ricci-flat, then it follows from Example 3.12 and Example 5.15 that P k,g = ∆ kg , andhence γ P k,g ( x ) = γ ∆ kg ( x ). Combining this with Step 1 and Step 3 then gives γ P k,g ( x ) = γ ∆ kg ( x ) = 2Γ( k ) − (4 π ) − n a n − k (∆ g ; x ) = 2Γ( k ) − (4 π ) − n ˜ a n − k (∆ g ; x ) . This proves Step 4.In order to prove Theorem 7.3 for general Riemannian metrics we actually need to establishpointwise versions of Step 4 and Step 3. This is the purpose of the next three steps. tep 5. Let I g ( x ) = Contr g (cid:0) ∇ k R ⊗ · · · ⊗ ∇ k l R (cid:1) be a Weyl Riemannian invariant of weight < n and assume that Ric( g ) = O( | x − x | n − ) near a point x ∈ M . Then ˜ I g ( x ) = I g ( x ) at x = x . This step follows the properties of the ambient metric and complete metric contractions ofambient curvature tensors described in [FG3]. Step 6. Let k ∈ N be such that n − k ∈ N and assume that Ric( g ) = O( | x − x | n − ) near apoint x ∈ M . Then γ P k,g ( x ) = γ ∆ kg ( x ) at x = x . If P is an elliptic ΨDO, then (6.3) gives an expression in local coordinates for γ P ( x ) in terms ofthe symbol of degree − n of a parametrix for P . The construction of the symbol of a parametrixshows that this symbol is a polynomial in terms of inverse of the principal symbol of P and partialderivatives of its other homogeneous symbols. It then follows that if P is Riemannian invariant,then γ P ( x ) is of the form (1.4), that is, this is a Riemannian invariant (see [Po1, Po2]).Moreover, it follows from the ambient metric construction of the GJMS operator [GJMS, FG3]that, when k is an integer, P k,g and ∆ kg differs by a differential operator whose coefficients arepolynomials in the covariant derivatives of order < n − γ P k,g ( x )and γ ∆ kg ( x ) differ by a linear combination of complete metric contraction of tensor powers ofcovariant derivatives of the Ricci tensor. Thus γ P k,g ( x ) and γ ∆ kg ( x ) agree at x = x if Ric( g )vanishes at order < n − x . This is also true for the conformal fractional powers of theLaplacian thanks to the properties of the scattering matrix for Poincar´e-Einstein metric in [JS]and [GZ]. Step 7. Let I g ( x ) = Contr g (cid:0) ∇ k R ⊗ · · · ⊗ ∇ k l R (cid:1) be a Weyl Riemannian invariant of weight w < n . Then the following are equivalent: (i) ˜ I g ( x ) at x = 0 for all metrics of signature ( n, on R n . (ii) I g ( x ) = 0 at x = 0 for all metrics of signature ( n, on R n such that Ric( g ) = O (cid:0) | x | n − (cid:1) near x = 0 . (iii) I g ( x ) = 0 at x = 0 for all metrics of signature ( n + 1 , on R n +2 such that Ric( g ) =O (cid:0) | x | n − (cid:1) near x = 0 . The implication (i) ⇒ (ii) is an immediate consequence of Step 5. The implication (iii) ⇒ (i)follows from the construction of the conformal invariant ˜ I g ( x ), since the ambient metric is of theform given in (iii) if we use the variable r = √ ρ instead of ρ (cf. [FG3, Chapter 4]; see in particularTheorem 4.5 and Proposition 4.7). Therefore, the bulk of Step 7 is proving that (ii) implies (iii).It can be given sense to a vector space structure on formal linear combinations of complete metriccontractions of tensor without any reference to dimension or the signature of the metric. If theweight is less than n , then the formal vanishing is equivalent to the algebraic vanishing by inputingtensors in dimension n . The idea in the proof of the proof of the implication (ii) ⇒ (iii) is showingthat (ii) and (iii) are equivalent to the same system of linear equations on the space of formal linearcombinations of complete metric contractions of Ricci-flat curvature tensors. This involves provinga version of the “2nd main theorem of invariant theory” for Ricci-flat curvature tensors, ratherthan for collections of trace-free tensors satisfying the Young symmetries of Riemannian curvaturetensors. In other words, we need to establish a nonlinear version of [BEG, Theorem B.3]. Step 8. Proof of Theorem 7.3. By Step 2 we know that γ P k,g ( x ) is a linear combination of Weyl conformal invariants ofweight 2 n − k , so by Theorem 4.5 there is a Riemannian invariant I g ( x ) of weight 2 n − k such that γ P k,g ( x ) = 2Γ( k ) − (4 π ) − n ˜ I g ( x ) . We need to show that I g ( x ) = a n − k (∆ g ; x ). If ( M, g ) is Ricci flat, then by Step 3 and Step 4, I g ( x ) = ˜ I g ( x ) = 12 Γ( k )(4 π ) n γ P k,g ( x ) = 12 Γ( k )(4 π ) n γ ∆ kg ( x ) = a n − k (∆ g ; x ) . By Step 6 this result continues to hold at a point x ∈ M if Ric( g ) = O( | x − x | n − ) near x .Therefore, in this case the Riemannian invariant I g ( x ) − a n − k (∆ g ; x ) vanishes at x . Using Step 7 e then deduce that the associated conformal invariant vanishes. That is, γ P k,g ( x ) = 2Γ( k ) − (4 π ) − n ˜ a n − k (∆ g ; x ) . This proves Theorem 7.3. References [ABP] Atiyah, M., Bott, R., Patodi, V.: On the heat equation and the index theorem. Invent. Math. , 279–330(1973). Errata . Invent. Math. , 277–280 (1975).[BEG] Bailey, T.N.; Eastwood, M.G.; Graham, C.R.: Invariant theory for conformal and CR geometry. Ann.Math. (1994) 491–552.[BGS] Beals, R.; Greiner, P.C. Stanton, N.K.: The heat equation on a CR manifold . J. Differential Geom. (1984), no. 2, 343–387.[BGM] Berger, M.; Gauduchon, P.; Mazet, E.: Le spectre dune vari´et´e riemannienne . Lecture Notes in Mathe-matics, vol. 194, Springer-Verlag, 1971.[Br] Branson, T.: Sharp inequalities, the functional determinant, and the complementary series . Trans. Amer.Math. Soc. (1995), 3671–3742.[BG] Branson, T.; Gover, A.R.: Conformally invariant operators, differential forms, cohomology and a gener-alisation of Q -curvature . Comm. Partial Differential Equations (2005), no. 10-12, 1611–1669.[CS] Caffarelli, L.; Silvestre, L.: An extension problem related to the fractional Laplacian . Comm. PartialDifferential Equations (2007), 1245–1260.[CM] Chang, S.-Y.A.; del Mar Gonz`alez, M.: Fractional Laplacian in conformal geometry . Adv. Math. (2011), no. 2, 1410–1432[CJ] Chen, L.; Jakobson, D.: Gaussian free fields and KPZ relation in R . E-print, arXiv, October 2012.[CY] Cheng, S.-Y.; Yau, S.-T.: On the regularity of the Monge-Amp`ere equation det( ∂ u/∂x i ∂x j ) = F ( x, u ).Comm. Pure Appl. Math. (1977), 41–68.[Fe1] Fefferman, C.: Monge-Amp`ere equations, the Bergman kernel, and geometry of pseudoconvex domains .Ann. of Math. (2) (1976), no. 2, 395–416.[Fe2] Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudo-convex domains . Invent. Math. (1974), 1–65.[Fe3] Fefferman, C.: Parabolic invariant theory in complex analysis . Adv. in Math. (1979), no. 2, 131–262.[FG1] Fefferman, C.; Graham, C.R.: Conformal invariants . ´Elie Cartan et les Math´ematiques d’Aujourd’hui,Ast´erisque, hors s´erie, (1985), 95–116.[FG2] Fefferman, C.; Graham., C.R.: Q -curvature and Poincar´e-Einsteinmetrics. Math. Res. Lett. (2002),139–151.[FG3] Fefferman, C.; Graham, C.R.: The ambient metric . Ann. Math. Studies 178, Princeton Univ. Press, 2012.[FG4] Fefferman, C.; Graham, C.R.: Juhls formulae for GJMS operators and Q -curvatures . To appear in J.Amer. Soc.. DOI: 10.1090/S0894-0347-2013-00765-1.[Gi1] Gilkey, P.B.: Curvature and the eigenvalues of the Laplacian for elliptic complexes . Adv. Math. (1973),344–382.[Gi2] Gilkey, P.B.: The spectral geometry of a Riemannian manifold . J. Differential Geom. (1975), 601–618.[Gi3] Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem . 2nd edition.Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.[Go] Gover, A.R.: Laplacian operators and Q -curvature on conformally Einstein manifolds. Math. Ann. (2006), 311–334.[GG] Gover, A.R.; Graham, C.R.: CR invariant powers of the sub-Laplacian. J. Reine Angew. Math. (2005), 1–27.[GH] A.R. Gover and K. Hirachi. Conformally invariant powers of the Laplacian – A complete non-existencetheorem . J. Amer. Math. Soc. (2004), 389–405.[GP] Gover, A.R.; Peterson, L.J.: Conformally Invariant Powers of the Laplacian, Q -Curvature, and TractorCalculus . Comm. Math. Phys. (2003), 339–378.[GW] Gover, A.R.; Waldron, A.: Boundary calculus for conformally compact manifolds . E-print, arXiv, April2011.[Gr1] C.R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence . J. London Math. Soc.(2) (1992), no. 3, 566–576.[Gr2] Graham, C.R.: Talk at the workshop Conformal structure in geometry, analysis, and physics . AIM, PaloAlto, Aug. 12–16, 2003.[GJMS] Graham, C.R.; Jenne, R.; Mason, L.J.; Sparling, G.A.: Conformally invariant powers of the Laplacian. I.Existence. J. London Math. Soc. (2) (1992), no. 3, 557–565.[GrH] Graham, C.R.; Hirachi, M.: Inhomogeneous ambient metrics . Symmetries and overdetermined systems ofpartial differential equations , 403–420, IMA Vol. Math. Appl., 144, Springer, New York, 2008.[GZ] Graham, C.R.; Zworki, M.: Scattering matrix in conformal geometry . Inv. Math. (2003), 89–118.[Gr] Greiner, P.: An asymptotic expansion for the heat equation . Arch. Rational Mech. Anal. (1971) 163–218. GS] Grubb, G.; Seeley, R.: Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundaryproblems . Inv. Math. (1995), 481–530.[Gu1] Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds . DukeMath. J. (2005), 1–37.[Gu2] Guillarmou, C.: Personal communication.[GMP1] Guillarmou, C.; Moroianu, S.; Park, J.: Eta invariant and Selberg zeta function of odd type over convexco-compact hyperbolic manifolds , Adv. Math. (2010), no. 5, 2464–2516.[GMP2] Guillarmou, C.; Moroianu, S.; Park, J.: Calderon and Bergman projectors on spin manifolds with bound-ary . J. Geom. Anal., doi:10.1007/s12220-012-9338-9.[GN] Guillarmou, C.; Naud, F.: Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds. Comm. Anal. Geom. (2006), no. 5, 945–967.[Gui1] Guillemin, V.W.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues . Adv.Math. (1985), no. 2, 131–160.[Gui2] Guillemin, V.: gauged Lagrangian distributions. Adv. Math. (1993), no. 2, 184–201.[Ju1] Juhl, A.: Families of conformally covariant differential operators, Q -curvature and holography . Progressin Mathematics, Birkh¨auser, Vol. 275, 2009, 500 pages.[Ju2] Juhl, A.: Explicit formulas for GJMS-operators and Q -curvatures . To appear in Geom. Funct. Anal.. DOI:10.1007/s00039-013-0232-9.[Hi] Hirachi, K.: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel .Ann. Math. (2000) 151–191.[Hi] Hirachi, K.: Personal communication.[JS] Joshi, M.S.; S´a Barreto, A.: Inverse scattering on asymptotically hyperbolic manifolds . Acta Math. (2000), 41–86.[KV] Kontsevich, M.; Vishik, S.: Geometry of determinants of elliptic operators. Functional analysis on the eveof the 21st century , Vol. 1 (New Brunswick, NJ, 1993), 173–197, Progr. Math., 131, Birkh¨auser Boston,Boston, MA, 1995.[Ku] Kuiper, N.H.: On conformally flat spaces in the large . Ann. of Math. (1949), 916–924.[MS] McKean, H.P.; Singer I.M.: Curvature and the eigenvalues of the Laplacian . J. Differential Geom. (1967),43–69.[Le] Lee, J.M.: The Fefferman metric and pseudo-Hermitian invariants . Trans. Amer. Math. Soc. (1986),no. 1, 411–429.[LM] Lee, J.M.; Melrose, M.: Boundary behaviour of the complex Monge-Ampre equation . Acta Math. (1982) 159–192.[Ma1] Mazzeo, R.: The Hodge cohomology of a conformally compact metric . J. Differential Geom. (1988),309–339.[Ma2] Mazzeo, R.: Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic man-ifolds . Amer. J. Math. (1991), 25–45.[MM] Mazzeo, R.; Melrose, R.: Meromorphic extension of the resolvent on complete spaces with asymptoticallyconstant negative curvature . J. Funct. Anal. (1987), 260–310.[Mi] Minakshisundaram, S. Eigenfunctions on Riemannian manifolds . J. Indian Math. Soc. (N.S.) (1953),159–165.[Ok] Okikiolu, K.: Critical metrics for the determinant of the Laplacian in odd dimensions. Ann. of Math. (2001), 471–531.[Pa] Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian man-ifolds . Preprint (1983). Reproduced as: SIGMA (2008), 036, 3 pages.[PR] Parker, T; Rosenberg, S.: Invariants of conformal Laplacians . J. Differential Geom. (1987), no. 2,199–222.[Pe] Peterson, L.J.: Conformally covariant pseudo-differential operators . Diff. Geom. Appl. (2000), 197–211.[Po] Polterovich, I.: Heat invariants of Riemannian manifolds . Israel J. Math. (2000), 239–252.[Po1] Ponge, R.: Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR struc-tures operators . E-print, arXiv, October 2007.[Po2] Ponge, R.: A microlocal approach to Fefferman’s program in conformal and CR geometry . Clay Math.Proc. 12, Amer. Math. Soc. Providence, RI, 2011, pp. 219–242.[Ra] Ramadanov, I.P.: A characterization of the balls in C n by means of the Bergman kernel . C. R. Acad.Bulgare Sci., (1981) 927–929.[Sc] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature . J. Diff. Geom. (1984), 479–495.[Se] Seeley, R.T.: Complex powers of an elliptic operator . Singular Integrals , Proceedings of Symposia in PureMathematics, Vol. 10, Amer. Math. Soc. Providence, RI, 1976) pp. 288–307. Corrections in Amer. J. Math.textbf91 (1969) 917–919.[Ta] Taylor, M.E: Partial differential equations. II. Qualitative studies of linear equations . Applied Mathemat-ical Sciences, 116. Springer-Verlag, New York, 1996. Gr:FCPDOBP Va1] Vasy, A.: Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates . Inverse problems and applications. Inside Out II , MSRI Publications, no. 60, Cambridge University Press,2012.[Va2] Vasy, A.: Analytic continuation and high energy estimates for the resolvent of the Laplacian on forms onasymptotically hyperbolic spaces. E-print, arXiv, June 2012.[Wi] Widom, H.: A complete symbolic calculus for pseudo-differerential operators . Bull. Sci. Math. (1980),19–63.[Wo1] Wodzicki, M.: Local invariants of spectral asymmetry . Invent. Math. (1984), no. 1, 143–177.[Wo2] Wodzicki, M.: Noncommutative residue. I. Fundamentals . K -theory, arithmetic and geometry (Moscow,1984–1986), 320–399, Lecture Notes in Math., 1289, Springer, Berlin-New York, 1987. Department of Mathematical Sciences, Seoul National University, Seoul, South Korea E-mail address : [email protected]@gmail.com