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EXTENDED OBSTRUCTION TENSORS AND RENORMALIZED VOLUME COEFFICIENTS C. ROBIN GRAHAM

1. Introduction In recent years there has been a great deal of progress on the so-called σk -Yamabe problem. In [CF], Alice Chang and Hao Fang have suggested that a variant of this problem might also be fruitful to study. The main goal of this paper is to investigate the algebraic structure under conformal transformation of the renormalized volume coefficients, the curvature quantites considered by Chang-Fang. A key ingredient in the investigation is the introduction of “extended obstruction tensors”, which are anticipated to be of independent interest. These are natural tensors associated to a pseudo-Riemannian metric g which turn out to be building blocks for the expansion of the ambient or Poincar´e metric determined by g and thus also for the renormalized volume coefficients. The σk -Yamabe problem was introduced by Jeff Viaclovsky in [V]. Let 1 R Pij = Rij − gij n−2 2(n − 1) denote the Schouten tensor of a metric g on a manifold M of dimension n ≥ 3, and let g −1P denote the endomorphism P i j obtained by raising an index. For 1 ≤ k ≤ n, the σk -Yamabe problem is to find a metric in a given conformal class for which σk (g −1 P ) is constant, where σk (A) denotes the k-th elementary symmetric function of the eigenvalues of an endomorphism A. We set σk = 0 for k > n. For k = 1, σ1 (g −1 P ) is a multiple of the scalar curvature of g, so this is the Yamabe problem. For 2 ≤ k ≤ n, σk (g −1 P ) is a second order fully nonlinear operator in the conformal factor. Variational methods have played an important role in the study of this problem. In dimensions n > 2, the Yamabe equation R R = c is the Euler-Lagrange equation for the total scalar curvature functional M R dvg under conformal variations subject to the constraint Volg (M) = 1. Of course, this fails when n = 2 because of the Gauss-Bonnet Theorem. For k = 2 the R analogous special dimension is n = 4. In this dimension, the total σ2 curvature M σ2 (g −1 P ) dvg is a conformal invariant. If n 6= 4, the natural generalization of the variational characterization of the Yamabe equation holds: the equation σ2 (g −1 P ) = c is the Euler-Lagrange equation for the Partially supported by NSF grant # DMS 0505701.

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R functional M σ2 (g −1 P ) dvg under conformal variations subject to the constraint Volg (M) = 1. Both of these properties fail for general metrics when 3 ≤ R k ≤ n. The special dimension is now n = 2k. But it is no longer true that M σk (g −1 P ) dvg is a conformal invariant in dimension 2k. Nor is it true for n 6= 2k that the equation σk (g −1 P ) = c is the constrainted Euler-Lagrange equation for the total σk functional. Viaclovsky did show that both properties hold if g is locally conformally flat. But Branson and Gover proved in [BG] that if 3 ≤ k ≤ n and g is not locally conformally flat, then the equation σk (g −1 P ) = c is not the Euler-Lagrange equation of any functional. The renormalized volume coefficients of g, denoted here by vk (g), arose in the late ’90’s in the physics literature in the context of the AdS/CFT correspondence. A mathematical discussion is contained in [G]. They are defined in terms of the expansion of the ambient or Poincar´e metric associated to g in the sense of [FG1]. One searches for a smooth 1-parameter family of metrics hr on M so that h0 = g and so that the metric dr 2 + hr (1.1) g+ = r2 on M × (0, ǫ) is an asymptotic solution to Ric(g+ ) = −ng+ at r = 0. This together with the condition that hr be even in r uniquely determines hr to infinite order if n is odd, however only to order n if n is even, at which point there is a formal obstruction to finding a solution to the next order. The trace part of the Taylor coefficient at order n is determined but the determination of the trace-free part is obstructed by a trace-free symmetric 2-tensor called the ambient obstruction tensor. Since hr is even in r, it is natural to introduce a new variable ρ = − 12 r 2 and set gρ = hr . The ambient metric coefficients are the determined Taylor coefficients ∂ρk gρ |ρ=0 . These are given locally in terms of the initial metric g0 = g; each of them can be written as a polynomial natural tensor expressible in terms of the curvature tensor of g and its covariant derivatives. The renormalized volume coefficients are defined by the expansion of the volume form: 1/2 ∞ X det gρ (1.2) ∼1+ vk ρk . det g0 k=1 If n is odd, vk (g) is defined for all k ≥ 1. If n is even, vk (g) is defined only for k ≤ n/2 for general g, although vk (g) is defined for all k ≥ 1 also in even dimensions if g is Einstein or locally conformally flat. A more detailed discussion is contained in §2. The insight of Chang-Fang is to consider the vk (g) in the context of the properties satisfied by the σk (g −1P ). Just comparing the formulae for these quantities shows that vk (g) = σk (g −1P ) if k = 1 or 2. In [GJ] it is shown that this holds also for k ≥ 3

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if g is locally conformally flat. Moreover, vk (g) always satisfies the two properties discussed above which failed for σk (g −1 P ) for 3 ≤ k ≤ n for general metrics. One of the R first important properties established of the vk was that in dimension n = 2k, v (g) dvg is a conformal invariant for general metrics (a proof is given in [G]). M k And the new result of Chang-Fang is that the variational characterization holds for vk (g): for nR 6= 2k, the equation vk (g) = c is the Euler-Lagrange equation for the functional M vk (g) dvg under conformal variations subject to the constraint Volg (M) = 1. This collection of facts suggests a strong parallel between the vk (g) and the σk (g −1P ), and even that from some points of view the vk (g) have better properties. However, study of the vk (g) involves significant challenges not shared by the σk (g −1 P ). Firstly, for k ≥ 3, vk (g) depends on derivatives of the curvature of g. In fact, for k ≥ 2, vk (g) depends on derivatives of curvature of order up to 2k − 4. Secondly, the vk (g) are defined via an indirect, highly nonlinear, inductive algorithm: first one solves the Einstein equation formally to determine gρ and then expands its volume form to obtain vk (g). They are algebraically complicated and no explicit formula is known for general k. A formula for v3 was given in [GJ]; it is not difficult to carry out the algorithm explicitly by hand to this order. The result is: 1 (1.3) v3 (g) = σ3 (g −1P ) + P ij Bij , 3(n − 4) where Bij denotes the Bach tensor of g. It is well-known that under conformal change b g = e2ω g, the transformation law of the Bach tensor involves just first derivatives of the conformal factor. Thus an immediate consequence of (1.3) and the conformal tranformation law (1.4) Pbij = Pij − ωij + ωi ωj − 1 ωk ω k gij 2

of the Schouten tensor is the fact that the transformation law of v3 involves at most second order derivatives of ω. Thus for a fixed metric g, the equation v3 (e2ω g) = c is second order in ω. It is this equation that Chang-Fang propose to study by analogy with the σk -Yamabe problem. In this paper, it is proved that the conformal transformation law involves at most second order derivatives of ω for all the vk , as well as for all the ambient metric coefficients.

Theorem 1.1. Under conformal change b g = e2ω g, the conformal transformation k laws of the ∂ρ gρ |ρ=0 and the vk involve at most second derivatives of ω. If n is odd, this is true for all k. If n is even, it is true for ∂ρk gij |ρ=0 for 1 ≤ k ≤ n/2 − 1, and n/2 for g ij ∂ρ gij |ρ=0 and vk for 1 ≤ k ≤ n/2. We give two different proofs of Theorem 1.1, each of which yields further information. The first proof proceeds by establishing that each of the determined

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ambient metric coefficients ∂ρk gij |ρ=0 can be written in terms of simpler building blocks, each of which has a conformal transformation law involving at most second derivatives of ω. The building blocks consist of the Schouten tensor and a (k) family Ωij of trace-free symmetric natural 2-tensors which we call the extended (k) obstruction tensors. The Ωij are defined for all k ≥ 1 if n is odd, but only for 1 ≤ k ≤ n/2 − 2 if n is even. The name derives from the fact that when the (k) dimension is viewed as a formal parameter, Ωij has a simple pole at dimension n = 2k + 2 whose residue is a multiple of the obstruction tensor in that dimension. For example, 1 (1) (1.5) Ωij = Bij , 4−n and the obstruction tensor in dimension 4 is the Bach tensor Bij . The result asserting that the ambient metric coefficients can be written in terms of the building blocks is the following. (k) Theorem 1.2. Let k ≥ 1. There is a linear combination Gij P, Ω(1) , . . . , Ω(k−1) of partial contractions with respect to g −1 of the Schouten tensor P and the Ω(l) , (k) 1 ≤ l ≤ k − 1, such that the coefficients of Gij are independent of n, and such that the ambient metric coefficients in dimension n are given by: (k) (1.6) ∂ρk gij |ρ=0 = Gij P, Ω(1) , . . . , Ω(k−1) ,

for all k ≥ 1 if n is odd and for 1 ≤ k ≤ n/2−1 if n iseven. Additionally, if k ≥ 2, there is a linear combination T (k) P, Ω(1) , . . . , Ω(k−2) of complete contractions of the indicated tensors whose coefficients are independent of n, such that in even dimension n, one has (1.7) g ij ∂ρn/2 gij |ρ=0 = T (n/2) P, Ω(1) , . . . , Ω(n/2−2) . A corollary is the analogous result for the renormalized volume coefficients.

Corollary 1.3. Let k ≥ 1. There is a linear combination Vk P, Ω(1) , . . . , Ω(k−2) of complete contractions with respect to g −1 of the Schouten tensor P and the Ω(l) , 1 ≤ l ≤ k − 2, such that the coefficients of Vk are independent of n, and such that the renormalized volume coefficients in dimension n are given by vk (g) = Vk P, Ω(1) , . . . , Ω(k−2) , for all k ≥ 1 if n is odd and for 1 ≤ k ≤ n/2 if n is even. For example, (1.3) and (1.5) give (1)

v3 (g) = σ3 (g −1 P ) − 31 P ij Ωij .

The proof of Theorem 1.2 gives a fairly simple, direct algorithm for the inductive (k) determination of the Gij which is independent of the formal solution of the Einstein

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(k)

equation. It is easy to carry this out to exhibit Gij for small k; we give the result for k ≤ 5. The more significant algebraic complexity occurs in the expressions for (k) the Ωij in terms of the curvature of g, for which solution of the Einstein equation is required and in which the dimension enters explicitly. The extended obstruction tensors are part of the theory of conformal curvature tensors developed in §6 of [FG2]; they are particular instances of conformal curvature tensors. In particular, each of them has the property shared by all conformal curvature tensors that its conformal transformation law can be written explicitly in terms of other conformal curvature tensors and first derivatives of the conformal factor. Thus Theorem 1.1 follows immediately from Theorem 1.2 and Corollary 1.3. Moreover, this shows that the only way second derivative terms in ω can arise in the conformal transformation law of vk (g) is from occurrences in Vk of the Schouten tensor. (k) A closer analysis of the conformal transformation law of the Ωij and of the form of the Vk gives the following result describing the structure of vk (e2ω g) as a second order fully nonlinear operator. Theorem 1.4. Let k ≥ 1 and suppose k ≤ n/2 if n is even. Then e2kω vk (b g ) = σk (g −1 Pb) +

k−2 X

m=0

rk,m (x, ∇ω, Pb),

where rk,m (x, ∇ω, Pb) is a polynomial in (ωi , Pbij ) which is homogeneous of degree m in Pb, of degree ≤ 2k − 2m − 2 in ∇ω, and with coefficients depending on g.

Here Pb is the conformally transformed Schouten tensor given by (1.4). Theorem 1.4 shows that σk (g −1 Pb) can be viewed as the leading term in vk (b g) from two points of view (at least for k ≤ n so that σk 6= 0). First, it has the highest homogeneity degree in Pb, which contains all the second derivative terms. Second, it contains all the terms with the highest total number of derivatives of ω. By this we mean that we expand e2kω vk (b g ) as a polynomial in ωi , ωij and add up the total number derivatives on ω in each monomial. For example, in (1.4), each of the terms ωij , ωi ωj and ωk ω k has a total of 2 derivatives of ω. Thus σk (g −1 Pb) contains terms with 2k derivatives of ω. Theorem 1.4 implies that each term in each rk,m(x, ∇ω, Pb) involves at most 2k − 2 derivatives of ω. It is tempting to speculate that requiring these two properties gives a reasonable definition of a “principal part” of a second order fully nonlinear operator depending polynomially on the derivatives. The second proof of Theorem 1.1 proceeds via a study of the linearization of vk (e2ω g) as a function of ω, i.e. of the linearized conformal transformation law of vk (g). The main ingredient is a formula for the conformal variation of the 1parameter family hr of metrics on M which arise when a given asymptotically

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hyperbolic metric g+ is written in the form (1.1). If one chooses a conformally related boundary metric b g = e2ω g, then up to a diffeomorphism of M × [0, ǫ), g+ can be written in the form (1.1) with a different 1-parameter family b hr satisfying b h0 = b g . It is possible to solve explicitly for the infinitesimal diffeomorphism in terms of ω and then for the infinitesimal conformal variation of hr . Letting δ denote infinitesimal conformal variation, the result when written in terms of gρ = hr defined as above is the following: (it is convenient to use also the notation gij (ρ) or simply gij for gρ ) (δg)ij = 2ω(1 − ρ∂ρ )gij + 2∇(i Yj)

(1.8) with (1.9)

i

Y (ρ) = −

Z

ρ

g ij (u) du ∂j ω,

Yj (ρ) = gij (ρ)Y i (ρ).

0

Here ∇i denotes the covariant derivative with respect to gρ with ρ fixed. An easy consequence of this is a formula for the infinitesimal conformal variation of the vk (g). Set 1/2 det gρ (1.10) v(ρ) = . det g0 Theorem 1.5. Let k ≥ 1 and k ≤ n/2 if n is even. The infinitesimal conformal variation of vk is given by: ij (1.11) δvk = −2kωvk + ∇i L(k) ∇j ω , where

(1.12)

Lij (k)

Z ρ k X 1 k 1 ij = − ∂ρ v(ρ) vk−l ∂ρl−1 g ij |ρ=0 . g (u) du =− k! l! ρ=0 0 l=1

In (1.11), ∇i denotes the covariant derivative with respect to the initial metric g = g0 . The infinitesimal transformation laws (1.8) and (1.11) clearly involve derivatives of ω of order at most 2. The second proof of Theorem 1.1 proceeds by arguing that if the infinitesimal conformal transformation law of a natural tensor involves at most m derivatives of ω for some m ≥ 0, then the same is true of the full transformation law. This is the content of Proposition 3.6. The Chang-Fang variational characterization of the equations vk (g) = c as EulerLagrange equations if n 6= 2k is an easy consequence of (1.11). The main point is that the second term on the right hand side of (1.11) is a divergence, which integrates to zero. Thus the only contribution to the Euler-Lagrange equation is the scaling contribution given by the first term, so that the Euler-Lagrange equation is vk (g) = c. The proof of Chang-Fang is also based on using the diffeomorphism

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invariance of g+ under conformal change of g to generate a divergence term; these amount to different versions of the same proof. But by working directly with metrics in the normal form (1.1), it is possible to give explicit formulae for the divergence terms, among other things making it clear that these terms depend on no more than second derivatives of ω. The approach to the linearization formulae used here is the same as in [ISTY], where the formulae (1.8) and (1.11) already appear. Formula (1.11) can be interpreted as identifying the linearization at ω = 0 of the second order fully nonlinear operator vk (e2ω g) with g fixed. In particular, the linearization is exhibited in divergence form modulo the zeroth order scaling term, 2 and its principal part is Lij (k) ∇ij ω. This may be useful in determining ellipticity of vk (e2ω g) = c. However, although all of these linearization formulae are explicit, they are written in terms of the coefficients in the ambient metric and renormalized volume expansions, and therefore are difficult to understand directly in terms of geometry of g. Additionally, they involve more and more derivatives of g as k increases. The results obtained in this paper support and extend the suggestion of ChangFang that the vk (g) are worthy of further study. However, significant algebraic complications remain and the geometric content of the equations vk (e2ω g) = c is unclear, particularly for large k. Perhaps it would be reasonable to try to extend directly the analytic theory of the σk equations to elliptic fully nonlinear equations allowing “lower order terms” with structure as in Theorem 1.4. 2. Extended Obstruction Tensors We begin this section by recalling the Poincar´e metric expansion and the definition of the renormalized volume coefficients. These are then reformulated in terms of the expansion of the ambient metric. After reviewing the theory of conformal curvature tensors from [FG2], we define the extended obstruction tensors as certain specific conformal curvature tensors. We establish the basic properties of the extended obstruction tensors. Then we prove Theorems 1.2 and 1.4. The section is concluded by giving some explicit formulae for small k. First recall the Poincar´e metric expansion and the definition of the renormalized volume coefficients vk . References for this material are [G], [GH], and [FG2]. Let g be a metric of signature (p, q) on a manifold M of dimension n ≥ 3. There are versions of most of the statements in dimension 2, but this case is anomalous and our main interest is in higher dimensions, so for simplicity we assume n ≥ 3. If n is odd, there is a smooth 1-parameter family hr , 0 ≤ r < 1, of metrics on M such that h0 = g and such that the metric g+ = r −2 (dr 2 + hr ) of signature (p + 1, q) on M × (0, 1) satisfies that Ric(g+ ) + ng+ vanishes to infinite order at r = 0. The Taylor expansion in r of hr at r = 0 can be chosen to be even in r, in which case it is uniquely determined.

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For n ≥ 4 even, the corresponding statement holds only to a finite order. We say that a tensor is O(r m) if all of its components relative to a frame smooth up to r = 0 are O(r m). We use lower case Latin indices to label objects on M. When n is even, hr can be chosen so that its Taylor expansion is even in r and such that Ric(g+ ) + ng+ = O(r n−2),

hij (Ric(g+ ) + ng+ )ij = O(r n ).

In the second equation, (Ric(g+ ) + ng+ )ij denotes the component with both indices in the M factor. In calculating the trace, hij can be taken to be either (h0 )ij = g ij or (hr )ij . These conditions uniquely determine hr mod O(r n ) and also trg hr mod O(r n+2). For n ≥ 4 even, there is a conformally invariant tracefree, divergence-free natural tensor Oij , the ambient obstruction tensor, which obstructs the existence of a formal power series solution for g+ to the next order. Oij depends on derivatives of g of order up to n. When n = 4, Oij is the classical Bach tensor. The passage from g to g+ is conformally invariant in the sense that if b g = e2ω g with ω ∈ C ∞ (M),and b hr denotes the expansion determined by b g , then the metrics hr are isometric by a diffeomorphism restricting to the g+ and gb+ = r −2 dr 2 + b

identity on M ×{0}, to infinite order if n is odd, and up to a term which is O(r n−2) and the trace of whose tangential component is O(r n ) if n is even. There are two special families of conformal structures in even dimensions for which the obstruction tensor vanishes and for which it is possible to uniquely determine the expansion of the Poincar´e metric to infinite order in a conformally invariant way. These are the locally conformally flat structures and the conformal classes containing an Einstein metric. In these cases, the normalized expansion can be written explicitly and terminates at order four: for all n ≥ 3, one has (2.1)

(hr )ij = gij − Pij r 2 + 14 Pik Pj k r 4

if g is Einstein or locally conformally flat. See [SS] and §7 of [FG2]. The volume form of g+ is 1/2 det hr −n−1 −n−1 dvg dr. dvhr dr = r dvg+ = r det h0 The renormalized volume coefficients are defined by the Taylor expansion: 1/2 ∞ X det hr (2.2) (−2)−k vk r 2k . ∼1+ det h0 k=1

Thus vk is uniquely determined by g = h0 for all k ≥ 1 if n is odd, and for 1 ≤ k ≤ n/2 if n is even. As will be discussed in more detail below, v1 and v2 are given by v1 = J, v2 = 21 J 2 − Pij P ij ,

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where J = R/2(n − 1) = Pi i . If g is Einstein or locally conformally flat, then the vk are determined by g for all k for both n even and odd. Proposition 1 of [GJ] uses (2.1) to show that vk (g) = σk (g −1 P ) for all k ≥ 1 if g is locally conformally flat. The same argument shows that this also holds if g is Einstein. In particular, vk (g) is constant for Einstein metrics. The evenness of the Poincar´e metric in r suggests to introduce r 2 as a new variable. Set ρ = − 21 r 2 and gρ = hr . Then the volume expansion (2.2) becomes (1.2). The 1-parameter family gρ can be characterized directly in terms of the expansion of the ambient metric e g associated to g, which is equivalent to the expansion of the Poincar´e metric g+ . Define e g , a metric of signature (p + 1, q + 1) on R+ × M × (−1/2, 0] ∋ (t, x, ρ), by (2.3)

g = 2t dt dρ + 2ρ dt2 + t2 gρ . e

The condition Ric(g+ )+ng+ = 0 is equivalent to Ric(e g ) = 0. Thus the expansion of gρ can be thought of as arising from formally solving Ric(e g ) = 0 to the appropriate order rather than Ric(g+ ) + ng+ = 0. An advantage of considering e g is that e g is smooth near ρ = 0, whereas g+ is singular at r = 0. As we will see, this makes it easier to pass objects constructed out of e g back to M. For each k ≥ 1 satisfying also k < n/2 if n is even, the Taylor coefficient ∂ρk gρ |ρ=0 is given by a polynomial natural tensor depending on the initial metric g. For n n/2 ij even, the trace g ∂ρ gij |ρ=0 at order n/2 is also a natural scalar invariant of g. It is possible to directly compute the beginning coefficients. For example, letting ′ = ∂ρ and suppressing the ρ in gρ , (3.6) and (3.18) of [FG2] show that one has at ρ = 0: (2.4)

gij′ = 2Pij ,

gij′′ =

2 Bij + 2Pi k Pkj , 4−n

where Bij = Pij ,k k − Pik ,j k − P kl Wkijl is the Bach tensor. Here Wijkl denotes the Weyl tensor. The last part of §6 of [FG2] considers a family of trace-free symmetric natural 2-tensors depending on a metric g. Here we call these the extended obstruction tensors. They have the feature that their transformation laws under conformal change are explicit and relatively simple. The first extended obstruction tensor is (1) (4 − n)−1 Bij , which we denote by Ωij . Its well-known transformation law under conformal change b g = e2ω g is: (2.5)

(1)

(1)

b = Ω − 2ω k C(ij)k + ω k ω l Wkijl , e2ω Ω ij ij

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where Cijk = Pij,k − Pik,j is the Cotton tensor. Equation (2.4) shows that gij′′ |ρ=0 (1) can be expressed in terms of Ωij by: (2.6)

1 ′′ g | 2 ij ρ=0

(1)

= Ωij + Pi k Pkj .

The definition and basic properties of the extended obstruction tensors are part of the theory of conformal curvature tensors developed in §6 of [FG2]. We summarize the relevant considerations and refer to [FG2] for details. Consider the curvature tensor and its covariant derivatives for an ambient metric e with components R eIJKL . Here capital (2.3). We denote its curvature tensor by R, Latin indices are used for objects on R+ × M × R ∋ (t, x, ρ), and we use ’0’ for the R+ factor (t component), lower case Latin indices for the M factor, and ’∞’ e rR e for the R factor (ρ component). For r ≥ 0, the r-th covariant derivative ∇ (r) e(r) , with components R e of the curvature tensor of ge will be denoted R IJKL,M1 ···Mr . (r) Sometimes the superscript is omitted when the list of indices makes clear the value of r. The conformal curvature tensors are tensors on M obtained from from the covariant derivatives of curvature of e g as follows. Choose an order r ≥ 0 of covariant differentiation. Divide the set of symbols IJKLM1 · · · Mr into three disjoint subsets labeled S0 , SM and S∞ . Set the indices in S0 equal to 0, those in S∞ equal to ∞, and let those in SM correspond to M in the decomposition R+ × M × R. eIJKL,M1···Mr at ρ = 0 and t = 1. This defines Evaluate the resulting component R e(r) a tensor on M, sometimes denoted by R S0 ,SM ,S∞ , whose rank is the cardinality of the set SM . In local coordinates, the indices in SM vary between 1 and n. e of e eIJK0 = 0, so The simplest case is r = 0. The curvature tensor R g satisfies R we must choose S0 = ∅ in order to get a nonzero component. Up to reordering the indices, there are only three possible nonzero choices (see (6.2) of [FG2]): (2.7)

eijkl |ρ=0,t=1 = Wijkl , R

e∞jkl |ρ=0, t=1 = Cjkl R

e∞ij∞ |ρ=0, t=1 = Bij . R 4−n

Thus the conformal curvature tensors which arise for r = 0 are precisely the Weyl, Cotton, and Bach tensors of g, except that when n = 4, the Bach tensor arises as the obstruction tensor rather than as a conformal curvature tensor. Since gρ is uniquely determined by g0 = g to infinite order for n odd, it follows e(r) that for n odd the conformal curvature tensors R S0 ,SM ,S∞ for all choices of r and S0 , SM , S∞ are defined and are polynomial natural tensors. However, when n is even, one must restrict the orders of differentiation to avoid the indeterminacy of e(r) gρ at order n/2. For n even, the tensor R S0 ,SM ,S∞ depends only on g and is a natural tensor so long as sM + 2s∞ ≤ n + 1, where sM , s∞ are the cardinalites of SM , S∞ , resp. If g is Einstein or locally conformally flat, then also for n even the

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conformal curvature tensors are defined for all choices of r and S0 , SM , S∞ . As will be seen below, they all vanish in the locally conformally flat case. Because e g changes by a diffeomorphism when g is changed conformally, the e rR e of ambient curvature transform tensorially under concovariant derivatives ∇ formal change of g. This leads to an explicit identification of the transformation laws of the conformal curvature tensors under conformal change. The following is Proposition 6.5 of [FG2]. Proposition 2.1. Let g and b g = e2ω g be conformally related metrics on M. Let IJKLM1 · · · Mr be a list of indices, s0 of which are 0, sM of which correspond to M, and s∞ of which are ∞. If n is even, assume that sM + 2s∞ ≤ n + 1. Then the conformal curvature tensors satisfy the conformal transformation law: be A Fr e (2.8) e2(s∞ −1)ω R IJKL,M1 ···Mr |ρb=0, b t=1 = RABCD,F1 ···Fr |ρ=0, t=1 p I · · · p Mr , where pA I is the matrix (2.9)

pA I

1 ωi − 12 ωk ω k −ω a . = 0 δ a i 0 0 1

eIJKL,M1···Mr |ρ=0, t=1 evaluated for the metric Here the conformal curvature tensor R be b and b t denote the coordinates g is denoted R b IJKL,M1···Mr |ρb=0, b t=1 . The variables ρ on R+ × M × R, thought of as a separate copy from the space for the unhatted metric. In (2.9), indices on ωi are raised using the initial metric g. In expanding the right hand side of (2.8), the leading term arises by replacing eIJKL,M1···Mr |ρ=0, t=1 . Because of the upper-triangular form each p by δ, giving R A of the matrix p I , the other terms on the right hand side all involve “earlier” conformal curvature tensors in the sense that each ’i’ can be replaced only by 0 and each ∞ only by an ’i’ or a 0. It is clear that the conformal transformation law of a conformal curvature tensor involves only other conformal curvature tensors and eIJK0 = 0, one first derivatives of ω. In case r = 0, using (2.7) and the fact that R sees that (2.8) reproduces the conformal invariance of the Weyl tensor and the usual conformal transformation laws of the Cotton and Bach tensors. Equation (2.8) can e rR e |ρ=0 defines a section of the (r + 4)-th tensor be interpreted as asserting that ∇ power of the cotractor bundle of the conformal manifold (M, [g]) with a particular conformal weight. It follows directly from the definition that the conformal curvature tensors all vanish if g is flat. Thus a consequence of (2.8) is that also they all vanish if g is locally conformally flat. By the infinite order invariance of the ambient metric for n even in the locally conformally flat case, this is true for all conformal curvature tensors in both even and odd dimensions. We now define the extended obstruction tensors.

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Definition 2.2. Let k ≥ 1. Suppose that n is odd or n is even and n > 2(k + 1). (k) Define the k-th extended obstruction tensor Ωij to be the conformal curvature tensor: (k) e∞ij∞, ∞...∞ |ρ=0, t=1 . Ωij = R | {z } k−1

(k)

According to the above discussion, Ωij is a polynomial natural tensor of the (1) initial metric g. For k = 1, (2.7) shows that Ωij is given by (1.5). It is clear that (k) Ωij is symmetric in ij, and it is also trace-free: (k)

Proposition 2.3. For each k ≥ 1 and in all dimensions n as above for which Ωij is defined, one has (k) g ij Ωij = 0.

Proof. This is a consequence of the Ricci-flatness of the ambient metric (to the appropriate order if n is even). First suppose that n is odd. Since Ric(e g ) = O(ρ∞ ), we have eKIJL,M1···Mr = 0 (2.10) geIJ R at ρ = 0 for all choices of KLM1 · · · Mr . Take ρ = 0 we have 0 0 g IJ = 0 t−2 g ij e t−1 0

all of KLM1 , · · · Mr to be ∞. At t−1 0 . 0

eKIJL in The terms in (2.10) with IJ = 0∞ or ∞0 vanish by skew-symmetry of R e∞ij∞,∞...∞ = 0 as desired. KI and JL. Thus (2.10) reduces to g ij R The same argument applies if n is even, so long as one checks that the order vanishing of Ric(e g ) is sufficient to under the restriction n > 2(k + 1). This is precisely the statement of Proposition 6.4 of [FG2].

Since for g locally conformally flat, all conformal curvature tensors are defined (k) and vanish whether n is even or odd, in particular it follows that Ωij is defined (k) and Ωij = 0 for all k for locally conformally flat g. This is also true if g is Einstein; see Proposition 7.6 of [FG2]. Note that general conformal curvature tensors do not vanish for Einstein metrics; for example the Weyl tensor is a conformal curvature (n/2−1) tensor. For n even and g Einstein, the vanishing of Ωij is actually the condition used to normalize the indeterminacy in the ambient metric; see Proposition 7.7 of [FG2]. Each obstruction tensor has divergence zero. But this property does not extend to the extended obstruction tensors. Already this fails for k = 1: the divergence of the Bach tensor is given by Bij, j = (n − 4)P jk Cjki .

RENORMALIZED VOLUME COEFFICIENTS

13

Next we define “higher Cotton tensors”, which will enter into the conformal transformation law of the extended obstruction tensors. Definition 2.4. Let k ≥ 1. Suppose that n is odd or n is even and n ≥ 2(k + 1). (k) Define the k-th higher Cotton tensor Cijl by: (k) e∞(ij)l, ∞...∞ + R e∞ij∞, ∞l∞...∞ + · · · + R e∞ij∞, ∞...∞l . e∞ij∞, l∞...∞ + R Cijl = 2R | {z } | {z } | {z } | {z } k−1

k−1

k−1

k−1

e components are evaluated at ρ = 0, t = 1. Here all R (k)

For k and n as in Definition 2.4, Cijl is a polynomial natural tensor of the initial metric g. As for the extended obstruction tensors, it is defined and vanishes for all k for g Einstein or locally conformally flat. Equation (2.7) shows that (1)

Cijl = 2C(ij)l . (1)

(1)

The tensors Cijl and Cijl are equivalent; Cijl can be recovered from Cijl by Cijl = 2 (1) C . 3 i[jl]

(k)

It is clear that Cijl is symmetric in ij, and it is also trace-free in these

indices: (k)

Proposition 2.5. For each k ≥ 1 and in all dimensions n as above for which Cijl is defined, one has (k) g ij Cijl = 0.

Proof. The proof is similar to that of Proposition 2.3. Again assume first that n is odd. Take K, L, and all but one of the Ms to be ∞ in (2.10) to deduce just as e∞ij∞, ∞...l...∞ = 0 for any location of the in the proof of Proposition 2.3 that g ij R | {z } k−1

index l after the comma. The same argument applied to the first term on the right hand side in Definition 2.4 shows that at ρ = 0 and t = 1 we have e∞ijl, ∞...∞ + R e∞0∞l, ∞...∞ = 0. g ij R | {z } | {z } k−1

k−1

Now (1) of Proposition 6.1 of [FG2] states that r X e e R RIJK0,M1···Mr = − cs ···Mr IJKMs ,M1 ···M s=1

e shows that R e∞0∞l, ∞...∞ = at t = 1. Applying this along with the symmetries of R | {z } k−1

e∞l∞0, ∞...∞ = 0, and the result follows. R | {z } k−1

Proposition 6.4 of [FG2] shows that the same argument applies if n is even and n ≥ 2(k + 1).

14

C. ROBIN GRAHAM (k)

We remark that g jlCijl = 0 for 1 ≤ k ≤ 3, but not for k = 4. Also, the (1)

(1)

(2)

symmetry C(ijl) = 0 satisfied by Cijl = 2C(ij)l does not hold for Cijl . A special case of Proposition 2.1 is the conformal tranformation law for the extended obstruction tensors: Proposition 2.6. Let k ≥ 1. Let n be odd or even with n > 2(k + 1). Under a conformal change b g = e2ω g, the conformally transformed extended obstruction tensor is given by: X ′e b (k) = Ω(k) + e2kω Ω RABCD,F1 ···Fk−1 |ρ=0, t=1 pA ∞ pB i pC j pD ∞ pF1 ∞ · · · pFk−1 ∞ , ij ij P′ where pA I is given by (2.9) and denotes the sum over all indices except for ABCDF1 · · · Fk−1 = ∞ij∞∞ · · · ∞.

Thus the conformal transformation law of the extended obstruction tensors is given explicitly in terms of conformal curvature tensors and first derivatives of the conformal factor. For k = 1, this reproduces (2.5). By the P upper-triangular A ′ form of p I , all of the conformal curvature tensors appearing in with nonzero coefficient are defined if n is even and n ≥ 2(k + 1). Next we identify the terms in the transformation law which are linear in ∇ω.

Proposition 2.7. Let k, n be as in Proposition 2.6. Under conformal change g = e2ω g, we have: b b (k) = Ω(k) − ω l C (k) + O(|∇ω|2). e2kω Ω ij ij ijl P′ Proof. For a term in in Proposition 2.6 to be linear in ∇ω, all p’s but one must 0 be δ, and p ∞ terms are excluded. If we suppress writing |ρ=0, t=1 , we obtain: (k)

(k)

b −Ω e2kω Ω ij ij = − ωl

elij∞, ∞...∞ + R e∞ijl, ∞...∞ + R e∞ij∞, l∞...∞ + · · · + R e∞ij∞, ∞...∞l R | {z } | {z } | {z } | {z } k−1

k−1

k−1

k−1

!

e∞0j∞, ∞...∞ + ωj R e∞i0∞, ∞...∞ + O(|∇ω|2) + ωi R | {z } | {z } k−1

=−ω

l

(k) Cijl

k−1

e∞i0∞, ∞...∞ + O(|∇ω|2). e∞0j∞, ∞...∞ + ωj R + ωi R | {z } | {z } k−1

k−1

e∞i0∞, ∞...∞ = 0 as in the proof of Proposition 2.5, and e∞0j∞, ∞...∞ = R However, R | {z } | {z } k−1

the result follows.

k−1

It is possible to view the dimension as a formal parameter and thus regard each of the extended obstruction tensors as a natural tensor depending rationally on n; see the discussion at the end of §6 of [FG2] (where, however, n is called d). The

RENORMALIZED VOLUME COEFFICIENTS

15

following result, which is Proposition 6.7 of [FG2], justifies the name “extended obstruction tensor”. (k)

Proposition 2.8. Viewed as a natural tensor rational in the dimension n, Ωij has a simple pole at n = 2(k + 1) with residue given by −1 (k) Resn=2(k+1) Ωij = (−1)k 2k−1 (k − 1)! Oij , where Oij denotes the obstruction tensor in dimension 2(k + 1).

As noted above, in the transformation law in Proposition 2.6, all of the conformal P′ curvature tensors appearing in with nonzero coefficient are regular at n = 2(k + 1). Therefore, formally taking the residue of this transformation law at n = 2(k + 1) recovers the conformal invariance of the obstruction tensor in dimension 2(k + 1). Likewise, for k > 1 we may consider the behavior as n → 2l with (k) 2 ≤ l ≤ k. It can be shown that Ωij and all the conformal curvature tensors appearing in its transformation law have at most simple poles at n = 2k. It is possible to justify the relation obtained by formally taking the residue at n = (k) 2k in the transformation law for Ωij ; this gives the conformal transformation (k) law of Resn=2k Ωij . In general, the order of the poles increases with k − l. For (3) example, Ωij has a double pole at n = 4, with leading coefficient a nonzero multiple of Bi k Bkj . In this case, consideration of the coefficient of (n − 4)−2 in the transformation law in Proposition 2.6 recovers the conformal invariance of Bi k Bkj in dimension 4. Now we turn to the proof of Theorem 1.2, which asserts that the Taylor coefficients in the ambient metric expansion can be written in terms of the Schouten tensor and the extended obstruction tensors by formulae universal in the dimension. Proof of Theorem 1.2. We prove by induction on k a stronger statement holding not only at ρ = 0. Consider a metric e g of the form (2.3), where now gρ is any smooth 1-parameter family of metrics on M, i.e. we make no assumption that e g is asymptotically Ricci-flat. For k ≥ 1, define (k)

e∞ij∞, ∞...∞ |t=1 , Λij = R | {z } k−1

a family of symmetric 2-tensors on M parametrized by ρ. We claim that for each (k) k ≥ 1, there is a linear combination Qij of partial contractions with respect to gρ−1 of gρ′ and the Λ(l) , 1 ≤ l ≤ k − 1, whose coefficients are independent of n, such that the identity (k) (2.11) ∂ρk gij = Qij g ′ , Λ(1) , . . . , Λ(k−1)

16

C. ROBIN GRAHAM

holds for all ρ. Since for e g asymptotically Ricci-flat, we have g ′ |ρ=0 = 2P and Λ(l) |ρ=0 = Ω(l) (for l < n/2 − 1 if n is even), the first statement of Theorem 1.2 follows upon setting ρ = 0. (1) Case k = 1 of (2.11) is trivial taking Qij = gij′ . For k = 2, we use an explicit e∞ij∞ of a metric (2.3). The Christoffel symbols of calculation of the component R g can be written explicitly; see (3.16) of [FG2]. From this it is straightforward to e calculate the curvature tensor of e g ; see (6.1) of [FG2]. One obtains in particular 1 1 kl ′ ′ ′′ e (2.12) R∞ij∞ |t=1 = g − g gik gjl . 2 ij 2 Thus 1 (1) ′ ′ (2.13) gij′′ = 2Λij + g kl gik gjl , 2 which is a relation of the form (2.11) for k = 2. We need a preliminary calculation before proceeding with the induction argument. The calculation of the covariant derivative in terms of Christoffel symbols gives e∞ij∞, ∞...∞ − Γ eA R e eA R e e∞ij∞, ∞...∞ = ∂ρ R −Γ R ∞...∞ ∞∞ Aij∞, ∞...∞ i∞ ∞Aj∞, | {z } | {z } | {z } | {z } k+1

k

k

k

eA R e eA R e −Γ −Γ ∞...∞ j∞ ∞iA∞, ∞...∞ ∞∞ ∞ijA, | | {z } {z } k

k

eA R e eA R e −Γ − ...− Γ . ∞∞ ∞ij∞, A...∞ ∞∞ ∞ij∞, ∞...A |{z} |{z} k

k

Now (3.16) of [FG2] shows that these Christoffel symbols are given by: eA = 0 for all A Γ ∞∞ and

e0 = 0, Γ i∞

el = 1 g lmg ′ , Γ i∞ im 2

e∞ = 0. Γ i∞

Therefore 1 lm ′ e e∞ij∞, ∞...∞ − 1 g lmg ′ R e∞ij∞, ∞...∞ = ∂ρ R e (2.14) R ∞...∞ − 2 g gjm R∞il∞, ∞...∞ . im ∞lj∞, | | {z } 2 {z } | {z } | {z } k+1

k

k

k

The ρ derivative commutes with restriction to t = 1, so this can be written in (k) terms of the Λij as (2.15)

(k)

(k+1)

∂ρ Λij = Λij

(k)

′ + g lm gm(i Λj)l .

Now we prove that there is an identity of the form (2.11) by induction on k ≥ 2. Suppose that (2.11) holds for k. Differentiate this relation with respect to ρ. (k) Each of the summands in Qij is a product of factors g −1 , g ′ , and the Λ(l) for 1 ≤ l ≤ k − 1. The derivative of any such factor is again a sum of products of

RENORMALIZED VOLUME COEFFICIENTS

17

the same form, except that also Λ(k) can appear. In fact, (g −1 )′ = −g −1 g ′ g −1, g ′′ is given by (2.13), and the derivative of a Λ(l) by (2.15). Therefore the Leibnitz rule gives a relation of the form (2.11) for k + 1. This completes the induction and thus also the proof of the first statement of Theorem 1.2. It is easily seen by induction starting with (2.13) and using (2.15) that for k ≥ 2, (k) Qij has the form (k) (k) (k−1) (2.16) Qij = 2Λij + Qij g ′, Λ(1) , . . . , Λ(k−2) , (k)

where Qij is a linear combination of partial contractions of the indicated tensors. Thus (k) (k) (k−1) Gij = 2Ωij + G ij P, Ω(1) , . . . , Ω(k−2) (k)

for some G ij . It follows that

(2.17)

(k)

g ij ∂ρk gij |ρ=0 = g ij G ij

P, Ω(1) , . . . , Ω(k−2)

for all k ≥ 2 if n is odd and for 2 ≤ k ≤ n/2 − 1 if n is even. However, this (n/2−1) reasoning does not apply for k = n/2 if n is even, since Ωij is not defined. Nonetheless we claim that this is true also for k = n/2, so that (k)

T (k) = g ij G ij

(2.18)

in Theorem 1.2. To see this, the discussion following (3.16) of [FG2] shows that n/2 e∞∞ = O(ρn/2−1 ). Now for n even, g ij ∂ρ gij |ρ=0 is determined by the condition R e∞∞, ∞...∞ = −e e∞IJ∞, ∞...∞ = −t−2 g ij R e∞ij∞, ∞...∞ . R g IJ R | {z } | {z } | {z } n/2−2

n/2−2

n/2−2

e∞ij∞, ∞...∞ |ρ=0 = 0 if R e∞∞ = O(ρn/2−1 ). Hence g ij ∂ρn/2 gij |ρ=0 is Therefore g ij R | {z } n/2−2

(n/2−1)

determined by requiring g ij Λij |ρ=0 = 0. So setting k = n/2 in (2.16), taking the trace, and restricting to ρ = 0 proves the second statement of Theorem 1.2 with T (k) given by (2.18). Equations (2.4) and (2.6) show that (1)

Gij = 2Pij ,

(2)

(1)

Gij = 2Ωij + 2Pi k Pkj . (k)

Thus T (2) = 2P ij Pij . Formulae for Gij for k = 3, 4, 5 are given in (2.22). Proof of Corollary 1.3. Taylor expanding the square root of the determinant in (1.2) shows that vk can be written as a linear combination of complete contractions of the Taylor coefficients ∂ρl gij |ρ=0 for 1 ≤ l ≤ k − 1 and also g ij ∂ρk gij |ρ=0 . (See the end of this section for more details.) Equation (1.6) shows that ∂ρl gij |ρ=0 for

18

C. ROBIN GRAHAM (s)

1 ≤ l ≤ k − 1 involves only the Ωij with s ≤ k − 2, and (2.17) shows that this is also the case for g ij ∂ρk gij |ρ=0 .

Theorem 1.1 implies that for a fixed background metric g, the equation vk (e2ω g) = c is second order in the unknown ω, even though for k ≥ 2, vk (g) depends on derivatives of g of order up to 2k −2. It is possible to say more about the form of vk (e2ω g) (k) as a function of ω with g fixed. First we show that Gij and Vk have a weighted homogeneity with respect their arguments. Consider a constant rescaling b g = s2 g with 0 < s ∈ R. The ambient metrics (2.3) are related by the diffeomorphism b t = ts−1 , x b = x, ρb = ρs2 , (k) (k) with gbρb = s2 gρ . It follows that ∂ρbk gbij |ρb=0 = s2−2k ∂ρk gij |ρ=0 . Thus if Gbij denotes Gij evaluated for the metric b g , then (k) (k) (2.19) Gb = s2−2k G . ij

ij

(k)

Suppose a term appears in Gij whose homogeneity degrees with respect to P , Pk−1 Ω(1) , . . . , Ω(k−1) are d0 , d1 , . . . , dk−1, resp., and let d = l=0 dl denote the total degree. Such a term necessarily involves d − 1 contractions with respect to g −1 . b (l) = s−2l Ω(l) , By Proposition 2.6, the extended obstruction tensors transform by Ω −1 −2 −1 and of course Pb = P and b g = s g . Thus (2.19) gives −2(d − 1) +

or k−1 X

(2.20)

k−1 X l=1

(−2l)dl = 2 − 2k,

(l + 1)dl = k.

l=0

This same relation holds for terms appearing in Vk since vbk = s−2k vk and Vk involves one more contraction because it is a scalar. Of course, dk−1 = 0 for Vk . Proof of Theorem 1.4. Write vk (g) = Vk P, Ω(1) , . . . , Ω(k−2) as a linear combination of complete contractions of P and the Ω(l) as in Corollary 1.3. The contractions which occur all satisfy (2.20) with dk−1 = 0. Collect the contractions according to their homogeneity degree m(= d0 ) in P : write vk (g) =

k X

m=0

Vk,m P, Ω(1) , . . . , Ω(k−2) ,

where Vk,m is the sum of the contractions which are homogeneous of degree m in P . Observe first that Vk,k−1 = 0 since there are no solutions to (2.20) with d0 = k − 1. Next, note that Vk,k depends only on P since d0 = k in (2.20) forces dl = 0 for l > 0. Also, vk (g) = Vk,k (P ) if g is conformally flat, since in this case all Ω(l) = 0.

RENORMALIZED VOLUME COEFFICIENTS

19

Since vk (g) = σk (g −1P ) for g conformally flat, it follows that Vk,k (P ) = σk (g −1 P ) for general g because any symmetric 2-tensor Pij at a point arises as the Schouten tensor of some conformally flat metric. Thus −1

vk (g) = σk (g P ) +

k−2 X

m=0

Vk,m P, Ω(1) , . . . , Ω(k−2) ,

where Vk,m P, Ω(1) , . . . , Ω(k−2) is homogeneous of degree m in P . Evaluating at b g gives vk (b g ) = σk (b g Pb) + −1

k−2 X

m=0

bk,m , V

bk,m denotes Vk,m P, Ω(1) , . . . , Ω(k−2) evaluated for the metric b where V g , i.e. P and (l) (l) b , and the contractions are taken with respect to b the Ω are replaced by Pb, Ω g. −1 b −2kω −1 b Now σk (b g P) = e σk (g P ). If we take into account the scaling of vk and of the Ω(l) as in the proof of (2.20), it follows that bk,m = e−2kω Vk,m Pb, e2ω Ω b (1) , . . . , e2(k−2)ω Ω b (k−2) , V b (1) , . . . , e2(k−2)ω Ω b (k−2) denotes the sum where on the right hand side, Vk,m Pb, e2ω Ω b (l) of the contractions with respect to g of the indicated tensors. Each of the e2lω Ω is given by Proposition 2.6, so is a polynomial in ∇ω with coefficients depending on g. So if we set b (1) , . . . , e2(k−2)ω Ω b (k−2) , rk,m(x, ∇ω, Pb) = Vk,m Pb, e2ω Ω

b (1) , . . . , e2(k−2)ω Ω b (k−2) where the x, ∇ω arguments in rk,m correspond to the e2ω Ω arguments in Vk,m and the Pb arguments correspond on both sides, then rk,m is a polynomial in (∇ω, Pb) homogeneous of degree m in Pb, with coefficients depending on g. It remains only to bound its degree in ∇ω. b (l) given by Proposition 2.6. Set k0k = 0, kik = 1 Consider the expression of e2lω Ω ij for 1 ≤ i ≤ n, k∞k = 2, and kAB · · · Ck = kAk + kBk + · · · + kCk. Now pA I = 0 if kAk > kIk and pA I is homogeneous of degree kIk − kAk in ∇ω for kIk ≥ kAk. So the term eABCD,F1 ···F |ρ=0, t=1 pA ∞ pB i pC j pD ∞ pF1 ∞ · · · pFl−1 ∞ R l−1

in Proposition 2.6 is of degree ≤ 2l + 4 − kABCDF1 · · · Fl−1 k in ∇ω. The coneABCD,F1 ···F |ρ=0, t=1 = 0 if formal curvature tensors have the property that R l−1 kABCDF1 · · · Fl−1 k ≤ 3. This is because in this case at most three of the inb (l) dices ABCDF1 · · · Fl−1 are nonzero; see Proposition 6.1 of [FG2]. Thus e2lω Ω ij

20

C. ROBIN GRAHAM

is of degree ≤ 2l in ∇ω. If d1 , · · · , dk−2 denote the homogeneity degrees with re(1) (k−2) spect to Ωij , · · · , Ωij , resp., of a contraction appearing in Vk,m , it follows that rk,m (x, ∇ω, Pb) has degree in ∇ω at most k−2 X l=1

2ldl = 2(k − m −

k−2 X

dl ).

l=1

Since d0 = m < k, (2.20) shows that dl > 0 for at least one l ≥ 1, giving the upper bound 2k − 2m − 2 for the degree of rk,m , as claimed. Clearly, for a specific Pk−2 contraction this argument gives a possibly better bound depending on l=1 dl .

It is possible to derive by hand formulae for some of the extended obstruction tensors and expressions for ambient metric coefficients and renormalized volume coefficients in terms of them. Consider first the extended obstruction tensors. We (1) have already seen that Ωij is given by (1.5). Formulae for higher extended obstruction tensors in terms of the Taylor coefficients ∂ρk gρ |ρ=0 of the ambient metric may be derived inductively starting with (2.12) and using (2.14). For instance, (2.12) together with (2.14) for k = 0 give: k e e∞ij∞,∞ = 1 g ′′′ − 1 g ′′ g ′ k + 1 g ′kl g ′ g ′ − g ′ R R ik jl k(i j)∞∞ . 2 ij 2 k(i j) 4

g ′ and g ′′ at ρ = 0 are given by (2.4) and g ′′′ |ρ=0 in (3.18) of [FG2]. Substituting these gives (2)

(n − 4)(n − 6)Ωij = Bij,k k − 2Wkijl B kl − 4Pk k Bij

+ (n − 4) 4P klC(ij)k,l − 2C k i l Cljk + Ci kl Cjkl + 2P k k,l C(ij) l − 2Wkijl P k m P ml .

Carrying out the algorithm by hand to derive the formulae for a few more extended obstruction tensors in terms of the ∂ρk gij |ρ=0 is manageable; this uses only the form (2.3) of the ambient metric. But deriving formulae for ∂ρk gij |ρ=0 in terms of the curvature of the base metric for k ≥ 4 by solving the Einstein equation is more lengthy. A similar calculation gives the second Cotton tensor (2) e∞(ij)l,∞ + R e∞ij∞,l Cijl = 2R . ρ=0,t=1

The Bianchi identity allows this to be rewritten as (2) e e e (2.21) Cijl = 3R∞ij∞,l − R∞li∞,j − R∞lj∞,i

ρ=0,t=1

.

The covariant derivative can be evaluated using (2.7) and the formulae for the Christoffel symbols of e g given by (3.16) of [FG2] to obtain e∞ij∞,l |ρ=0,t=1 = Bij,l − 2Pl m C(ij)m . R 4−n

RENORMALIZED VOLUME COEFFICIENTS

21

(2)

Substituting this into (2.21) gives the desired formula for Cijl . The proof of Theorem 1.2 gives the algorithm to make explicit the formulae (1.6) for the ambient metric coefficients in terms of the Schouten tensor and the extended obstruction tensors. This involves the same ingredients as in the derivation of the formulae for the extended obstruction tensors discussed above; it is just a matter of which set of quantities one is solving for inductively in terms of which others. Again, these relations depend only on the form (2.3) of the ambient metric and not on the values of its Taylor coefficients obtained by solving the Einstein equation for e g. (k) Set gij = ∂ρk gij |ρ=0 . We have already seen that 1 (1) g 2 ij 1 (2) g 2 ij

= Pij

(1)

= Ωij + Pi k Pjk .

Carrying out the algorithm of the proof of Theorem 1.2, one obtains:

(2.22)

1 (3) g 2 ij

=Ωij + 4P k (i Ωj)k

(2)

(1)

1 (4) g 2 ij

=Ωij + 6P k (i Ωj)k + 4Ω(1)k i Ωjk + 4P k i P l j Ωkl

(3)

(2)

1 (5) g 2 ij

=Ωij + 8P k (i Ωj)k + 14 Ω(2)k (i Ωj)k + 10P k i P l j Ωkl + 16 P k (i Ωj) Ωkl .

(4)

(3)

(1)

(1)

(1)

(2)

(1)l

(1)

The algorithm also easily gives the leading terms: for k ≥ 3 one has 1 (k) g 2 ij

(k−1)

= Ωij

(k−2)

+ 2(k − 1)P l (i Ωj)l

(k)

+ Jij ,

(k)

where Jij is a linear combination of contractions of the tensors P, Ω(1) , . . . , Ω(k−3) satisfying (2.20). Finally, the renormalized volume coefficients can be calculated from the ambient metric coefficients by expanding the volume form. Set D = det gρ / det g0 . Then D ′ = Dg ij gij′ . Successive differentiation of this relation gives formulae for ∂ρk D/D in terms of g −1 and derivatives of g. For example, ′ ′ D ′′ = Dg ij gij′′ − Dg ik g jl gkl gij + D(g ij gij′ )2 .

The Taylor coefficients of D are then obtained by evaluating these relations at ρ = 0 and substituting the above formulae for the Taylor coefficients of gρ . Composing √ with the Taylor expansion of x about x = 1 gives the vk according to (1.2). It is straightforward but tedious to carry this out. The result for the first few vk is: v1 =σ1 v2 =σ2 (2.23)

v3 =σ3 − 31 tr P Ω(1)

v4 =σ4 + 31 tr P 2 Ω(1) − 31 (tr P ) tr P Ω(1) −

1 12

tr P Ω(2) −

1 12

tr Ω(1)

2

.

22

C. ROBIN GRAHAM

Here we have omitted the argument g −1 P of the σk . Also omitted are the g −1 factors raising the indices in the trace terms. These σk are given by: σ1 =J σ2 = 21 J 2 − tr P 2 σ3 = 61 2 tr P 3 − 3J tr P 2 + J 3 1 σ4 = 24 −6 tr P 4 + 8J tr P 3 + 3(tr P 2 )2 − 6J 2 tr P 2 + J 4 , where J = tr P = R/2(n − 1).

3. Linearization Let X be a manifold-with-boundary and set ∂X = M. If [g] is a conformal class of metrics of signature (p, q) on M, recall that a metric g+ of signature (p + 1, q) on X ◦ is said to be conformally compact with conformal infinity (M, [g]) if u2 g+ extends smoothly to X with u2 g+ |M nondegenerate and u2 g+ |T M ∈ [g], where u is a defining function for M. The function |du|2u2g+ M is independent of the choice of u; g+ is said to be asymptotically hyperbolic if |du|2u2g+ M = 1. Let g+ be asymptotically hyperbolic and let g be a choice of metric in the conformal class on M. Then there is an open neighborhood of M (= M × {0}) in M × [0, ∞) on which there is a unique diffeomorphism ϕ to a neighborhood of M in X such that ϕ|M is the identity, and such that ϕ∗ g+ takes the form ϕ∗ g+ = r −2 dr 2 + h(r) ,

where h(r) (denoted hr previously) is a 1-parameter family of metrics on M of signature (p, q) satisfying h(0) = g. Here r denotes the variable in [0, ∞). See §5 of [GL]. Suppose we choose a conformally related metric b g = e2ω g, where ω ∈ C ∞ (M). Then b g induces another diffeomorphism ϕ b from a neighborhood of M in M ×[0, ∞) 2 ∗ −2 dr + b h(r) , where b h(r) is to a neighborhood of M in X such that ϕ b g+ = r a 1-parameter family of metrics on M, satisfying b h(0) = b g , uniquely determined by g+ , g, and ω. Consider the infinitesimal dependence of b h(r) on ω. For each t t, denote by b h (r) the 1-parameter family of metrics obtained from the conformal representative b g t = e2tω g. Let δ = ∂t |t=0 denote the operation of taking the infinitesimal conformal variation. For example, δh(r) = ∂tb ht (r)|t=0 . We sometimes suppress writing the argument for h(r) and δh(r); the r dependence of h and δh is to be understood. Theorem 3.1. Under infinitesimal conformal change of g, h(r) transforms by: (3.1)

(δh)ij = ω(2 − r∂r )hij + 2∇(i Xj) ,

RENORMALIZED VOLUME COEFFICIENTS

23

where X i is the r-dependent family of vector fields on M given by Z r i (3.2) X (r) = shij (s) ds ∂j ω. 0

i

Here Xj (r) = hij (r)X (r), and ∇i denotes the covariant derivative on M with respect to h(r) with r fixed. Note in (3.2) that ∂j ω is independent of r. An immediate consequence of Theorem 3.1 is the fact that for each r, the transformation rule for infinitesimal conformal change of h(r) involves at most second derivatives of ω. Proof. For each t, we have a diffeomorphism ϕt such that ϕt |M is the identity and ht (r) . So ϕ∗t g+ = r −2 dr 2 + b ϕ−1 0

◦ ϕt

∗

dr 2 + h(r) r2

=

dr 2 + b ht (r) . r2

Differentiate with respect to t at t = 0 to deduce that there is a vector field X near M in M × [0, ∞) such that X|M = 0 and 2 δh dr + h = 2, LX 2 r r where L denotes the Lie derivative. Expanding the left hand side and then multiplying by r 2 gives (3.3) − 2r −1 X(r) dr 2 + h + LX (dr 2 ) + LX h = δh. Now write X = X 0 ∂r + X i ∂i . Then X(r) = X 0 LX (dr 2) = 2dX 0 dr = 2∂r X 0 dr 2 + 2∂i X 0 dxi dr LX h = 2∇(i Xj) + X 0 ∂r hij dxi dxj + 2hij ∂r X j drdxi ,

where Xj = hjk X k and ∇i is the covariant derivative on M with respect to h(r) with r fixed. Substitute these into (3.3) and then equate the coefficients of dr 2 , drdxi and dxi dxj on the two sides of (3.3). One obtains (3.4)

−2r −1 X 0 + 2∂r X 0 = 0 2∂i X 0 + 2hij ∂r X j = 0

−2r −1 X 0 hij + 2∇(i Xj) + X 0 ∂r hij = δhij . The first equation shows that X 0 = cr where c is independent of r, i.e. c is just a function of x ∈ M. Substitute this into the last equation and evaluate at r = 0.

24

C. ROBIN GRAHAM

Recalling that X = 0 at r = 0 and δh = 2ωh = 2ωg at r = 0, one obtains c = −ω. So now we know X 0 = −ωr. Substitute this into the second equation to obtain ∂r X i = rhij ∂j ω.

Now integrate in r to solve for X i ; ∂j ω is a constant with respect to the integration. Using the initial condition X i = 0 at r = 0, one obtains Z r i X = shij (s) ds ∂j ω. 0

Substituting X 0 = −ωr into the third line of (3.4) gives (3.1).

It is useful to introduce the new variable ρ = − 12 r 2 as in §2 in the infinitesimal transformation law (3.1). Set g(ρ) = h(r) (denoted gρ previously) and Y i (ρ) = X i (r). Then (3.1), (3.2) become (1.8), (1.9). Consider now the case where g+ = r −2 (dr 2 + h(r)) is a Poincar´e metric whose Taylor expansion (to the appropriate order for n even) is determined along M by the choice of an initial metric g via the Einstein equation Ric(g+ ) = −ng+ . The Taylor coefficients of g(ρ) are the natural tensors studied in §2, so the Taylor expansion of (1.8) gives the infinitesimal transformation laws of these tensors. For example, (1.9) shows that ∂ρ Y i |ρ=0 = −g ij ∂j ω, so differentiating (1.8) once at ρ = 0 and recalling (2.4) recovers the infinitesimal transformation law δPij = −ωij of the Schouten tensor. In general, in (1.8) the term 2ω(1 − ρ∂ρ )gij encodes the scaling of each coefficient and the term 2∇(i Yj) carries the dependence on derivatives of ω. It follows that the infinitesimal transformation laws of all these natural tensors (subject to the usual truncation for n even) involves at most second derivatives of ω. An easy consequence of Theorem 3.1 is a similar formula for the infinitesimal transformation laws of the renormalized volume coefficients. First suppose that g+ is asymptotically hyperbolic with conformal infinity (M, [g]) but not necessarily asymptotically Einstein, as in the setting of Theorem 3.1. Define v(ρ) by (1.10). Proposition 3.2. Under infinitesimal conformal change of g(0), v(ρ) transforms by: (ρ)

(3.5)

δv = −2ωρ∂ρ v + v∇i Y i (0)

= −2ωρ∂ρ v + ∇i (ρ)

where Y i is given by (1.9) and ∇i with ρ fixed.

vY i ,

is the covariant derivative with respect to g(ρ)

Proof. Under a conformal tranformation, δ det g(0) = 2nω det g(0), so δv = δ log v = 21 g ij δgij − 2nω . v

RENORMALIZED VOLUME COEFFICIENTS

25

Substitution of (1.8) gives δv ρ∂ρ v (ρ) (ρ) = −ωg ij ρ∂ρ gij + g ij ∇i Yj = −2ω + ∇i Y i . v v (ρ)

(0)

This gives the first line of (3.5). The second line follows since v∇i Y i = ∇i (vY i ) for any vector field Y i . Proof of Theorem 1.5. Take g+ in Proposition 3.2 to be an asymptotically Einstein metric whose Taylor expansion is determined by g = g(0). By (1.2), the coefficient of ρk in δv is δvk . So taking Taylor coefficients in (3.5) and recalling (1.9) gives (k) (1.11) with Lij given by the first equality of (1.12). The second equality of (1.12) follows upon expanding via the Leibnitz rule. The fact that the second term on the right hand side of (3.5) is a divergence implies that it drops out when considering the infinitesimal conformal change of the volume of M relative to the metrics g(ρ). Suppose M is compact and set Z V (ρ) ≡ Volg(ρ) (M) = v(ρ) dvg(0) . M

Integration of (3.5) gives for each ρ: Z Z δv dvg(0) = −2 ωρ∂ρ v dvg(0) . (3.6) δV = M

M

Taking Taylor coefficients in (3.6) (or integrating (1.11) over M), it follows that: Proposition 3.3. Suppose k ≥ 1 with k ≤ n/2 if n is even, and suppose M is compact. Then Z Z δvk dvg = −2k vk ω dvg . M

M

Proposition 3.3 is the main ingredient in the proof of the result of Chang-Fang [CF]. Consider the functionals Z Fk (g) = vk (g) dvg M

as g varies over a conformal class. Fk is defined for all k ≥ 1 if k is odd and for 1 ≤ k ≤ n/2 if n is even. It was shown in [G] that if n is even, then Fn/2 (g) is conformally invariant, i.e. is constant on each conformal class. The Chang-Fang theorem gives the constrained Euler-Lagrange equation for the other values of k:

Theorem 3.4. Suppose k ≥ 1 and k < n/2 if n is even. The Euler-Lagrange equation for Fk (g) as g varies over a conformal class, subject to the constraint Volg (M) = 1, is vk (g) = c.

26

C. ROBIN GRAHAM

Proof. The constrained Euler-Lagrange equation for Fk is obtained by requiring that Fk − λ Vol(M) vanishes to first order in ω under a conformal change b g = e2ω g, where λ is a Lagrange multiplier. This is therefore the condition (3.7)

δ (Fk − λ Vol(M)) (g) = 0 for all ω.

Proposition 3.3 together with the fact that δ dvg = nω dvg give Z Z δFk (g) = (δvk dvg + vk δ dvg ) = (n − 2k) vk ω dvg . M

M

If n = 2k we recover the conformal invariance of Fn/2 . Otherwise (3.7) becomes Z Z (n − 2k) vk ω dvg − nλ ω dvg = 0 for all ω, M

which gives vk (g) = nλ/(n − 2k) = c.

M

Thus if we fix a background metric g in the conformal class, then the critical points of Fk (e2ω g) as a function of ω are those ω for which vk (e2ω g) = c. So we recover the second order fully nonlinear operator whose structure was studied in §2. Its linearization at ω = 0 is of course just δvk , so Theorem 1.5 gives: Proposition 3.5. Suppose k ≥ 1 with k ≤ n/2 if n is even. Let Pk (ω) denote the linearization at ω = 0 of the operator ω → vk (e2ω g) with g fixed. Then Pk (ω) = ∇i Lij ∇ ω − 2kvk ω, (k) j with Lij (k) given by (1.12).

In considering (1.12), recall that gij (ρ) is the series determined by g = g(0) upon formally solving the Einstein equation Ric(e g ) = 0. Hence identification of l−1 ij ∂ρ g |ρ=0 requires knowledge of the coefficients ∂ρm gij |ρ=0 in the ambient metric expansion as well as calculation of the Taylor coefficients of the inverse in terms of these. In any case, it is clear from Theorem 1.2 and Corollary 1.3 that each Lij (k) (with k ≤ n/2 for n even) is a natural tensor which can be written as a linear combination of contractions of the Schouten tensor and the extended obstruction tensors with coefficients independent of the dimension. For small k, it is possible to calculate Lij (k) from (1.12) using (2.22) and (2.23). Alternately, one can simply linearize the explicit expressions (2.23). In the latter approach, one uses from Proposition 3.5 that Pk is determined once one knows its principal part Lij (k) . Thus it suffices to calculate the principal part of the linearization from (2.23). So in linearizing (2.23), one can ignore contributions from the Ω(l) and simply apply the Leibnitz rule to Pij and use δPij = −ωij . Recall that the linearization of σk can be expressed in terms of its polarization. If Ai j (t) is a 1-parameter family of endomorphisms of a vector space, then the

RENORMALIZED VOLUME COEFFICIENTS

relation

27

σk (A)· = tr T(k−1) (A)A˙

(3.8)

defines an endomorphism-valued polynomial T(k−1) (A) homogeneous of degree k−1 in A. Let σ k denote the symmetric k-linear form obtained by complete polarization of σk , i.e. σ k (A1 , . . . , Ak ) is linear in each Al , symmetric, and satisfies σ k (A, . . . , A) = σk (A). Then the Leibnitz rule shows that tr T(k−1) (A)B = kσ k (A, . . . , A, B). In the sequel, our vector space will be equipped with a non-degenerate quadratic form which we use to raise and lower indices, and Aij will be symmetric. So we will usually write (3.8) in the form ij σk (A)· = T(k−1) (A)A˙ ij .

The homogeneity of σk together with δPij = −ωij give ij (3.9) δ σk (g −1P ) = −T(k−1) (g −1 P )ωij − 2kσk (g −1 P )ω,

ij so that the principal part of the linearization of σk (g −1 P ) is −T(k−1) (g −1P ). In ij and we write Ω(l) the following, we suppress writing the argument g −1 P of T(k−1) (l) instead of Ω . Either directly linearizing (2.23) or calculating from (1.12), one obtains: ij Lij (1) = −g

ij Lij (2) = −T(1)

ij 1 ij Lij (3) = −T(2) + 3 Ω(1)

j)k

ij (i kl ij k ij 2 1 1 Lij (4) = −T(3) − 3 Pk Ω(1) + 3 Pkl Ω(1) g + 3 Pk Ω(1) +

1 ij Ω . 12 (2)

Recall from the discussion in §2 that if g is locally conformally flat, then the vk are defined for all k also for n even, and vk (g) = σk (g −1 P ) for all k in all dimensions. The invariance of the ambient metric holds to all orders in all dimensions and this was the fundamental ingredient used in the proof of Theorem 3.1. Thus all the arguments and results of this section apply without the restriction k ≤ n/2 for n even if g is locally conformally flat. In particular, this gives another argument for the variational character of the σk in this case. The relations asserted by Theorem 3.1, Proposition 3.2 and Theorem 1.5 are not obvious for locally conformally flat metrics. Equation (2.1) can be written (3.10)

gij (ρ) = gij (0) + 2Pij ρ + Pik P k j ρ2 .

28

C. ROBIN GRAHAM

Let us set γ = g(0) and A = γ −1 P . Then (3.10) can be written g(ρ) = γ(I + ρA)2 . Therefore Z ρ Z ρ −1 (3.11) g (u) du = (I + uA)−2 du γ −1 = ρ(I + ρA)−1 γ −1 . 0

0

Hence

g(ρ)

Z

ρ

g −1 (u) du = ργ(I + ρA) γ −1 .

0

Comparing with (1.9) gives (3.12)

Yj = −ρ δj k + ρPj k ωk .

Taking the conformal variation in (3.10) yields

(δg)ij = 2ωγij − 2ωij ρ + −2ωPik P k j − 2ωk(i Pj) k ρ2 .

The fact that this agrees with the right hand side of (1.8) can be verified using (3.12) and the relation between the Levi-Civita connections of g(ρ) and γ derived in Lemma 7.3 of [FG2]. Similarly, it can be verified directly that if g is locally conformally flat, then ij −1 (1.12) reduces to Lij (k) = −T(k−1) (g P ) for k ≤ n and to 0 for k > n, and that (1.11) reduces to (3.9). The identification of the Lij (k) is clearly equivalent to showing that Z ρ n X ij ij v(ρ) T(k−1) (g −1 P )ρk . g (u) du = 0

k=1

Now v(ρ) = det(I + ρA), so (3.11) shows that this can be rewritten as det(I + ρA)(I + ρA)

−1

=

n−1 X

T(k) (A)ρk .

k=0

It is standard and can be seen in a variety of ways that this is a reformulation of ij the definition of the T(k) (A). Thus one concludes that Lij = −T(k−1) (g −1 P ). (k) ij Finally, (1.11) reduces to (3.9) since ∇i T(k−1) (g −1P ) = 0. The fact that ij ∇i T(k−1) (g −1 P ) = 0 for locally conformally flat metrics follows from the vanishing of the Cotton tensor and is essentially equivalent to the variational characterization of the σk . See [V] or [BG]. Theorem 3.1 can be used as the basis for another proof of Theorem 1.1, the fact that under conformal change, the ambient metric coefficients ∂ρk gij |ρ=0 and the vk depend on at most second derivatives of ω. As observed above, the fact that this is true under infinitesimal conformal change is immediate from Theorem 3.1. Thus Theorem 1.1 follows if we can prove that the full conformal transformation law depends on at most 2 derivatives of the conformal factor, assuming that this is the

RENORMALIZED VOLUME COEFFICIENTS

29

case for the infinitesimal transformation law. We formulate a general result along these lines. Consider a polynomial natural tensor T (g) depending on a Riemannian metric in dimension n ≥ 2, of contravariant rank a and covariant rank b. T (g) may be expressed by evaluating a linear combination of partial contractions of covariant indices against contravariant indices of g, g −1, and the covariant derivatives ∇r R, r ≥ 0, of the curvature tensor R of g. Each such partial contraction can be written in the form (3.13) pcontr ∇r1 R ⊗ · · · ⊗ ∇rM R ⊗ g ⊗ · · · ⊗ g ⊗ g −1 ⊗ · · · ⊗ g −1 . Our convention is that the curvature tensor R has contravariant rank 0 and covariant rank 4. We say that T has homogeneity h ∈ R if T (e2ω g) = ehω T (g),

ω ∈ R.

We assume throughout that T has a well-defined homogeneity; this is no loss of generality since a general natural tensor is the sum of its homogeneous parts in this sense and all of our considerations respect homogeneity. If the contraction (3.13) has P factors of g, Q factors of g −1, and involves C contractions of a covariant index against a contravariant index, then the contravariant rank a, covariant rank b, and homogeneity h of the resulting tensor are given by a = 2Q − C b = 4M +

M X i=1

ri + 2P − C

h = 2(M + P − Q). In particular, the quantity L ≡ 2M +

X

ri = b − a − h

is determined just by the rank and homogeneity of T . L is called the level of T ; it is the total number of derivatives of g occuring in T . Clearly L ≥ 0. If T has homogeneity h, the full conformal variation ∆T (g, ω) of T is defined to be ∆T (g, ω) ≡ e−hω T (e2ω g) − T (g)

for smooth ω. Then ∆T (g, ω) is a natural tensor depending on g and the scalar function ω. It can be obtained by evaluating a linear combination of partial contractions of g, g −1 , the ∇r R for r ≥ 0, and the covariant derivatives ∇l ω, l ≥ 1, of ω with respect to the Levi-Civita connection of g, each contraction of which contains at least one of the ∇l ω. In this discussion we use a slightly modified

30

C. ROBIN GRAHAM

infinitesimal conformal variation operator δ by subtracting the scaling term from δ. If T is a natural tensor of homogeneity h, define d −htω 2tω δT (g, ω) ≡ e T (e g) = δT − hωT. dt t=0 Then δT (g, ω) is a natural tensor depending on g and ω; it is obtained from ∆T (g, ω) by keeping only the terms which are linear in the derivatives of ω. It is evident that when viewed as a function of g, ∆T (g, ω) and δT (g, ω) also have homogeneity h: ∆T (e2Υ g, ω) = ehΥ ∆T (g, ω),

δT (e2Υ g, ω) = ehΥ δT (g, ω),

We may consider the infinitesimal conformal variation of δT in g: d −htΥ 2 2tΥ δ T (g, ω, Υ) ≡ e δT (e g, ω) . dt t=0 The equality of second mixed partials implies that (3.14)

Υ ∈ R.

δ 2 T (g, ω, Υ) = δ 2 T (g, Υ, ω).

Let U(g, ω) be a polynomial natural tensor depending on g and ω (for example U = δT or U = ∆T ). For m ≥ 0, we will say that U involves at most m derivatives of ω if it can be obtained by evaluating a linear combination of partial contractions in which only the tensors ∇l ω, 1 ≤ l ≤ m appear, together with g, g −1, and the ∇r R for r ≥ 0. Proposition 3.6. Let m ≥ 0. If δT involves at most m derivatives of ω, then the same is true for ∆T . The case m = 0 is the well-known statement that if a natural tensor is infinitesimally conformally invariant, then it is conformally invariant. The proof in this case is simpler than in the case m > 0. Clearly δT involves at most m derivatives of ω if and only if δT involves at most m derivatives of ω. Proof. The proof is by induction on the level L = b − a − h of T . First consider the case L = 0. If aPcontraction (3.13) appears in an expression for T , then the relation L = 2M + ri forces M = 0. Thus T is a linear combination of partial contractions only of g and g −1 . Such a T is conformally invariant, so the desired conclusion is automatic. Assume now that the result is true for natural tensors whose level L satisfies L ≤ N for some N ≥ 0. Suppose T (g) is a natural tensor of some homogeneity h and level N + 1, for which δT (g, ω) involves at most m derivatives of ω. We can write (3.15)

i1 ···im i1 i1 i2 δT (g, ω) = ωi1 S(1) (g), (g) + ωi1 i2 S(2) (g) + · · · + ωi1 ···im S(m)

where for 1 ≤ l ≤ m, S(l) (g) is a natural tensor of homogeneity h whose covariant rank equals that of T and whose contravariant rank is l more than that of T . Here

RENORMALIZED VOLUME COEFFICIENTS

31

ωi1 ···il denotes the components of ∇l ω. Since the level of T is N + 1, it follows that the level of S(l) is N + 1 − l ≤ N. Since for each l, the skew-symmetrization of ∇l ω in any two indices can be expressed by the Ricci identity in terms of the tensors ∇j ω with 1 ≤ j ≤ l − 2 and the ∇r R, it follows inductively that S(l) (g) can be taken to be symmetric in the indices i1 . . . il . Under this condition the S(l) (g) are uniquely determined. We claim that each of the δS(l) (g, ω) involves at most m derivatives of ω. To see this, take the infinitesimal conformal variation of (3.15) in g with respect to a conformal change b gt = e2tΥ g. The infinitesimal conformal variation of the right hand side may be calculated via the Leibnitz rule. Each of the terms ∇l ω has a variation corresponding to the change of the connection. It is clear that δ(∇l ω) involves at most l derivatives of ω and Υ. Thus it follows that δ 2 T (g, ω, Υ) i1 i1 i2 i1 ···im = ωi1 δS(1) (g, Υ) + U(g, ω, Υ), (g, Υ) + ωi1 i2 δS(2) (g, Υ) + · · · + ωi1 ···im δS(m)

where U(g, ω, Υ) is a natural tensor depending on g, ω, and Υ which involves at most m derivatives of ω and Υ. Using (3.14), we obtain i1 i1 i2 i1 ···im e ω, Υ) ωi1 δS(1) (g, Υ) + U(g, (g, Υ) + ωi1 i2 δS(2) (g, Υ) + · · · + ωi1 ···im δS(m)

i1 i1 i2 i1 ···im (g, ω), (g, ω) + Υi1 i2 δS(2) (g, ω) + · · · + Υi1 ···im δS(m) = Υi1 δS(1)

e involves at most m derivatives of ω and Υ. Since the left hand where again U side involves at most m derivatives of ω, the same is true of the right hand side. Therefore this must also hold for the coefficient of each of the Υi1 ···il . Hence each of the δS(l) (g, ω) involves at most m derivatives of ω as claimed. Thus the induction hypothesis applies to each of the S(l) (g), and we deduce that for 1 ≤ l ≤ m, ∆S(l) (g, ω) involves at most m derivatives of ω. Next, recall that ∆T can be recovered by integrating δT . To see this, note first that d −h(t+s)ω d −htω e T (e2tω g) = e T (e2(t+s)ω g) dt ds s=0 −hsω 2sω 2tω −htω d e T (e e g) = e−htω δT (e2tω g, ω). =e s=0 ds Thus Z 1 d −htω −hω 2ω e T (e2tω g) dt ∆T (g, ω) = e T (e g) − T (g) = 0 dt (3.16) Z 1 = e−htω δT (e2tω g, ω)dt. 0

2tω

Apply (3.15) to evaluate δT (e g, ω). The occurrences of ∇l ω, 1 ≤ l ≤ m on the right hand side of (3.15) now have to be evaluated using the Levi-Civita connection

32

C. ROBIN GRAHAM

of e2tω g. It is clear that for fixed t, each such evaluation gives rise to a natural tensor depending on g and ω which involves at most m derivatives of ω. Likewise, for each t we have e−htω S(l) (e2tω g) = S(l) (g) + ∆S(l) (g, tω), and the right hand side is a family parametrized by t of natural tensors depending on g and ω which involves at most m derivatives of ω. Substituting into (3.16) and integrating in t, it follows that ∆T (g, ω) involves at most m derivatives of ω. This completes the induction step. References [BG] [CF] [FG1] [FG2] [G]

[GH]

[GJ] [GL] [ISTY] [SS] [V]

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