HOLOGRAPHIC FORMULA FOR Q-CURVATURE
C. ROBIN GRAHAM AND ANDREAS JUHL
Introduction
In this paper we give a formula for Q-curvature in even-dimensional conformal
geometry. The Q-curvature was introduced by Tom Branson in [B] and has been
the subject of much research. There are now a number of characterizations of Q-
curvature; see for example [GZ], [FG1], [GP], [FH]. However, it has remained an open
problem to find an expression for Q-curvature which, for example, makes explicit the
relation to the Pfaffian in the conformally flat case.
Theorem 1. The Q-curvature of a metric g in even dimension n is given by
(0.1) 2ncn/2Q = nv
(n) +
n/2−1∑
(n− 2k)p∗2kv
(n−2k),
where cn/2 = (−1)
n/2 [2n(n/2)!(n/2− 1)!]
Here the v(2j) are the coefficients appearing in the asymptotic expansion of the
volume form of a Poincaré metric for g, the differential operators p2k are those which
appear in the expansion of a harmonic function for a Poincaré metric, and p∗2k denotes
the formal adjoint of p2k. These constructions are recalled in §1 below. We refer to
the papers cited above and the references therein for background about Q-curvature.
Each of the operators p∗2k for 1 ≤ k ≤ n/2− 1 can be factored as p
2k = δqk, where
δ denotes the divergence operator with respect to g and qk is a natural operator from
functions to 1-forms. So the second term on the right hand side is the divergence of
a natural 1-form. In particular, integrating (0.1) over a compact manifold recovers
the result of [GZ] that
(0.2) 2cn/2
Qdvg =
v(n)dvg.
This quantity is a global conformal invariant; the right hand side occurs as the coeffi-
cient of the log term in the renormalized volume expansion of a Poincaré metric (see
[G]).
The work of the first author was partially supported by NSF grant DMS-0505701. The work of
the second author was supported by SFB 647 “Raum-Zeit-Materie” of DFG.
http://arxiv.org/abs/0704.1673v1
2 C. ROBIN GRAHAM AND ANDREAS JUHL
As we also discuss in §1, if g is conformally flat then
v(n) = (−2)−n/2(n/2)!−1Pff ,
where Pff denotes the Pfaffian of g. So in the conformally flat case, Theorem 1 gives a
decomposition of the Q-curvature as a multiple of the Pfaffian and the divergence of a
natural 1-form. A general result in invariant theory ([BGP]) establishes the existence
of such a decomposition, but does not produce a specific realization.
We refer to (0.1) as a holographic formula because its ingredients come from the
Poincaré metric, involving geometry in n + 1 dimensions. Our proof is via the char-
acterization of Q-curvature presented in [FG1] in terms of Poincaré metrics; in some
sense Theorem 1 is the result of making explicit the characterization in [FG1]. How-
ever, passing from the construction in [FG1] to (0.1) involves a non-obvious appli-
cation of Green’s identity. The transformation law of Q-curvature under conformal
change, probably its most fundamental property, is not transparent from (0.1), but it
is from the characterization in [FG1]. In §2, we derive another identity involving the
p∗2kv
(n−2k) which is used in §3 and we discuss relations to the paper [CQY]. In §3, we
describe the relation between holographic formulae for Q-curvature and the theory of
conformally covariant families of differential operators of [J], and in particular explain
how this theory leads to the conjecture of a holographic formula for Q.
We are grateful to the organizing committee of the 2007 Winter School ’Geome-
try and Physics’ at Srńı, particularly to Vladimir Souc̆ek, for the invitation to this
gathering, which made possible the interaction leading to this paper.
We dedicate this paper to the memory of Tom Branson. His insights have led to
beautiful new mathematics and have greatly influenced our own respective work.
1. Derivation
Let g be a metric of signature (p, q) on a manifold M of even dimension n. In this
paper, by a Poincaré metric for (M, g) we will mean a metric g+ on M × (0, a) for
some a > 0 of the form
(1.1) g+ = x
−2(dx2 + gx),
where gx is a smooth 1-parameter family of metrics on M satisfying g0 = g, such
that g+ is asymptotically Einstein in the sense that Ric(g+) + ng+ = O(x
n−2) and
trg+(Ric(g+)+ng+) = O(x
n+2). Such a Poincaré metric always exists and gx is unique
up addition of a term of the form xnhx, where hx is a smooth 1-parameter family of
symmetric 2-tensors on M satisfying trg(h0) = 0 on M . The Taylor expansion of gx
is even through order n and the derivatives (∂x)
2kgx|x=0 for 1 ≤ k ≤ n/2− 1 and the
trace trg((∂x)
ngx|x=0) are determined inductively from the Einstein condition and are
given by polynomial formulae in terms of g, its inverse, and its curvature tensor and
covariant derivatives thereof. See [GH] for details.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 3
The first ingredient in our formula for Q-curvature consists of the coefficients in
the expansion of the volume form
(1.2) dvg+ = x
−n−1dvgxdx.
Because the expansion of gx has only even terms through order n, it follows that
dvgx =
det gx
det g
= (1 + v(2)x2 + · · ·+ v(n)xn + · · · )dvg,
(1.3)
where each of the v(2k) for 1 ≤ k ≤ n/2 is a smooth function on M expressible in
terms of the curvature tensor of g and its covariant derivatives. Set v(0) = 1.
The second ingredient in our formula is the family of differential operators which
appears in the expansion of a harmonic function for the metric g+. Given f ∈ C
∞(M),
one can solve formally the equation ∆g+u = O(x
n) for a smooth function u such that
u|x=0 = f , and such a u is uniquely determined modulo O(x
n). The Taylor expansion
of u is even through order n − 2 and these Taylor coefficients are given by natural
differential operators in the metric g applied to f which are obtained inductively by
solving the equation ∆g+u = O(x
n) order by order. See [GZ] for details. We write
the expansion of u in the form
(1.4) u = f + p2f x
2 + · · ·+ pn−2f x
n−2 +O(xn);
then p2k has order 2k and its principal part is (−1)
Γ(n/2− k)
22k k! Γ(n/2)
∆k. (Our convention
is ∆ = −∇i∇i.) Set p0f = f .
We remark that the volume coefficients v(2k) and the differential operators p2k also
arise in the context of an ambient metric associated to (M, [g]). If an ambient metric
is written in normal form relative to g, then the same v(2k) are coefficients in the
expansion of its volume form, and the same operators p2k appear in the expansion of
a harmonic function homogeneous of degree 0 with respect to the ambient metric.
Let g+ be a Poincaré metric for (M, g). In [FG1] it is shown that there is a unique
solution U mod O(xn) to
(1.5) ∆g+U = n +O(x
n+1 log x)
of the form
(1.6) U = log x+ A+Bxn log x+O(xn) ,
A,B ∈ C∞(M × [0, a)) , A|x=0 = 0 .
Also, A mod O(xn) is even in x and is formally determined by g, and
(1.7) B|x=0 = −2cn/2Q.
4 C. ROBIN GRAHAM AND ANDREAS JUHL
The proof of (1.7) presented in [FG1] used results from [GZ] about the scattering
matrix, so is restricted to positive definite signature. However, a purely formal proof
was also indicated in [FG1]. Thus (1.7) holds in general signature.
Proof of Theorem 1. Let g+ be a Poincaré metric for g and let U be a solution of
(1.5) as described above. Let f ∈ C∞(M) have compact support. Let u be a solution
of ∆g+u = O(x
n) with u|x=0 = f ; for definiteness we take u to be given by (1.4) with
the O(xn) term set equal to 0. Let 0 < ǫ < x0 with ǫ, x0 small.
Consider Green’s identity
(1.8)
ǫ<x<x0
(U∆g+u− u∆g+U) dvg+ =
(U∂νu− u∂νU) dσ,
where ν denotes the inward normal and dσ the induced volume element on the bound-
ary, relative to g+. Both sides have asymptotic expansions as ǫ→ 0; we calculate the
coefficient of log ǫ in these expansions.
Using the form of the expansion of U and the fact that ∆g+u = O(x
n), one sees
that the expansion of U∆g+u dvg+ has no x
−1 term, so
ǫ<x<x0
U∆g+u dvg+ has no
log ǫ term. Using (1.2), (1.3), (1.4), and (1.5), one finds that the log ǫ coefficient of
ǫ<x<x0
u∆g+U dvg+ is
(1.9) n
n/2−1∑
v(n−2k)p2kf dvg.
On the right hand side of (1.8),
is independent of ǫ, and
(U∂νu− u∂νU) dσ = ǫ
(U∂xu− u∂xU) dvgǫ.
A log ǫ term in the expansion of this quantity can arise only from the log x or xn log x
terms in the expansion of U . Substituting the expansions, one finds without difficulty
that the log ǫ coefficient is
n/2−1∑
2kv(n−2k)p2kf − nBf
 dvg.
Equating this to (1.9), using (1.7), and moving all derivatives off f gives the desired
identity. �
Since ∆g+1 = 0, it follows that p2k1 = 0 for 1 ≤ k ≤ n/2− 1. Thus these p2k have
no constant term, so p∗2k = δqk for some natural operator qk from functions to 1-forms,
where δ denotes the divergence with respect to the metric g. So in (0.1), the second
term on the right hand side is the divergence of a natural 1-form. As mentioned in
the introduction, integration gives (0.2). The proof of Theorem 1 presented above in
the special case u = 1 is precisely the proof of (0.2) presented in [FG1].
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 5
Theorem 1 provides an efficient way to calculate the Q curvature. Solving for the
beginning coefficients in the expansion of the Poincaré metric and then expanding its
volume form shows that the first few of the v(2k) are given by:
v(2) = −
v(4) =
(J2 − |P |2)
v(6) =
P ijBij + 3J |P |
2 − J3 − 2P ijPi
where
Pij =
Rij −
2(n− 1)
2(n− 1)
= P ii
Bij = Pij,k
k − Pik,j
k − P klWkijl
and Wijkl denotes the Weyl tensor. Similarly, one finds that the operators p2 and p4
are given by:
−2(n− 2)p2 = ∆
8(n− 2)(n− 4)p4 = ∆
2 + 2J∆+ 2(n− 2)P ij∇i∇j + (n− 2)J,
(1.10)
For n = 2, Theorem 1 states Q = −2v(2) = 1
R. For n = 4, substituting the above
into Theorem 1 gives:
Q = 2(J2 − |P |2) + ∆J,
and for n = 6:
Q = 8P ijBij + 16P
kPkj − 24J |P |
2 + 8J3
+∆2J + 4∆(J2) + 8(P ijJ,i),j − 4∆(|P |
In the formula for n = 6, the first line is (12c3)
−16v(6) and the second line is
(12c3)
4p∗2v
(4) + 2p∗4v
. Details of these calculations will appear in [J].
The expansion of the Poincaré metric g+ was identified explicitly in the case that
g is conformally flat in [SS]. (Since we are only interested in local considerations,
by conformally flat we mean locally conformally flat.) The two dimensional case is
somewhat anomalous in this regard, but the identification of Q curvature is trivial
when n = 2, so we assume n > 2 for this discussion. The conclusion of [SS] is that
if g is conformally flat and n > 2 (even or odd), then the expansion of the Poincaré
metric terminates at second order and
(1.11) (gx)ij = gij − Pijx
6 C. ROBIN GRAHAM AND ANDREAS JUHL
(The details of the computation are not given in [SS]. Details will appear in [FG2]
and [J].) This easily yields
Proposition 1. If g is conformally flat and n > 2, then
v(2k) =
(−2)−kσk(P ) 0 ≤ k ≤ n
0 n < k
where σk(P ) denotes the k-th elementary symmetric function of the eigenvalues of the
endomorphism Pi
Proof. Write g−1P for Pi
j. Then the σk(P ) are given by
det(I + g−1P t) =
σk(P )t
Equation (1.11) can be rewritten as g−1gx = (I−
g−1Px2)2. Taking the determinant
and comparing with (1.3) gives the result. �
We remark that for g conformally flat, gx given by (1.11) is uniquely determined to
all orders by the requirement that g+ be hyperbolic. So in this case the v
(2k) are
invariantly determined and given by Proposition 1 for all k ≥ 0 in all dimensions
n > 2.
Returning to the even-dimensional case, we define the Pfaffian of the metric g by
(1.12) 2n(n/2)! Pff = (−1)qµi1...inµj1...jnRi1i2j1j2 . . . Rin−1injn−1jn,
where µi1...in =
| det(g)| ǫi1...in is the volume form and ǫi1...in denotes the sign of the
permutation. For a conformally flat metric, one has Rijkl = 2(Pi[kgl]j−Pj[kgl]i). Using
this in (1.12) and simplifying gives
Pff = (n/2)! σn/2(P )
(see Proposition 8 of [V] for details). Combining with Proposition 1, we obtain for
conformally flat g:
v(n) = (−2)−n/2(n/2)!−1 Pff .
Hence in the conformally flat case, (0.1) specializes to
2Q = 2n/2(n/2− 1)! Pff +(ncn/2)
n/2−1∑
(n− 2k)p∗2kv
(n−2k),
and again the second term on the right hand side is a formal divergence.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 7
2. A Related Identity
In this section we derive another identity involving the p∗2kv
(n−2k). It is in gen-
eral impossible to choose the O(xn) term in (1.4) to make ∆g+u = O(x
n); in fact
x−n∆g+u|x=0 is independent of the O(x
n) term in (1.4) and is a conformally invariant
operator of order n applied to f , namely a multiple of the critical GJMS operator
Pn. Following [GZ], we consider the limiting behavior of the corresponding term in
the expansion of an eigenfunction for ∆g+ as the eigenvalue tends to 0.
Let g+ be a Poincaré metric as above. If 0 6= λ ∈ C is near 0, then for f ∈ C
∞(M),
one can solve formally the equation (∆g+ − λ(n − λ))uλ = O(x
n+λ+1) for uλ of the
(2.1) uλ = x
f + p2,λf x
2 + · · ·+ pn,λf x
n +O(xn+1)
where p2k,λ is a natural differential operator in the metric g of order 2k with principal
part (−1)k
Γ(n/2− k − λ)
22k k! Γ(n/2− λ)
∆k such that
Γ(n/2− λ)
Γ(n/2− k − λ)
p2k,λ is polynomial in λ. Set
p0,λf = f . The operators p2k,λ for k < n/2 extend analytically across λ = 0 and
p2k,0 = p2k for such k, where p2k are the operators appearing in (1.4). But pn,λ has a
simple pole at λ = 0 with residue a multiple of the critical GJMS operator Pn. Now
Pn is self-adjoint, so it follows that pn,λ − p
n,λ is regular at λ = 0. We denote its
value at λ = 0 by pn − p
n, a natural operator of order at most n − 2. Our identity
below involves the constant term (pn−p
n)1. Note that since Pn1 = 0, both pn,λ1 and
p∗n,λ1 are regular at λ = 0. We denote their values at λ = 0 by pn1 and p
n1; then
(pn − p
n)1 = pn1− p
n1. Moreover, (4.7), (4.13), (4.14) of [GZ] show that
(2.2) pn1 = −cn/2Q.
It is evident that
pn1 dvg =
p∗n1 dvg. The next proposition expresses the differ-
ence pn1− p
n1 as a divergence.
Proposition 2.
(2.3) n (pn − p
n) 1 =
n/2−1∑
2k p∗2kv
(n−2k)
Proof. Take f ∈ C∞(M) to have compact support, let 0 6= λ be near 0, and define
uλ as in (2.1) with the O(x
n+1) term taken to be 0. Define wλ by the corresponding
expansion with f = 1:
wλ = x
1 + p2,λ1 x
2 + · · ·+ pn,λ1 x
As in the proof of Theorem 1, consider Green’s identity
(2.4)
ǫ<x<x0
(uλ∆g+wλ − wλ∆g+uλ)dvg+ = ǫ
(uλ∂xwλ − wλ∂xuλ) dvgǫ + cx0 ,
8 C. ROBIN GRAHAM AND ANDREAS JUHL
where cx0 is the constant (in ǫ) arising from the boundary integral over x = x0.
Consider the coefficient of ǫ2λ in the asymptotic expansion of both sides. The left
hand side equals
ǫ<x<x0
∆g+ − λ(n− λ)
wλ − wλ
∆g+ − λ(n− λ)
dvg+.
Now uλ
∆g+ − λ(n− λ)
wλ dvg+ and wλ
∆g+ − λ(n− λ)
uλ dvg+ are of the form
x2λψ dxdvg where ψ is smooth up to x = 0. It follows that the asymptotic expansion
of the left hand side of (2.4) has no ǫ2λ term. Consequently the coefficient of ǫn+2λ
must vanish in the asymptotic expansion of
(uλx∂xwλ − wλx∂xuλ) dvgǫ.
This is the same as the coefficient of ǫn in the expansion of
p2k,λf ǫ
(2k + λ)p2k,λ1 ǫ
p2k,λ1 ǫ
(2k + λ)p2k,λf ǫ
v(2k) ǫ2k
 dvg.
Evaluation of the ǫn coefficient gives
0≤k,l,m≤n/2
k+l+m=n/2
(2l − 2k)(p2k,λf)(p2l,λ1)v
(2m) dvg = 0,
and then moving the derivatives off f results in the pointwise identity
(2.5)
0≤k,l,m≤n/2
k+l+m=n/2
(2l − 2k) p∗2k,λ
(p2l,λ1)v
The limit as λ → 0 exists of all p2l,λ1 with 0 ≤ l ≤ n/2 and all p
2k,λ with 0 ≤ k ≤
n/2− 1. Since k = n/2 forces l = m = 0, the operator p∗n,λ occurs only applied to 1.
Thus we may let λ→ 0 in (2.5). Using p2l1 = 0 for 1 ≤ l ≤ n/2− 1 results in
npn1−
0≤k,m≤n/2
k+m=n/2
2k p∗2kv
(2m) = 0.
Separating the k = n/2 term in the sum gives (2.3). �
Proposition 2 may be combined with (0.1) and (2.2) to give other expressions for
Q-curvature. However, (0.1) seems the preferred form, as the other expressions all
involve some nontrivial linear combination of pn1 and p
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 9
We remark that the generalization of (2.5) obtained by replacing p2l,λ1 by p2l,λf
remains true for arbitrary f ∈ C∞(M). This follows by the same argument, taking
wλ to be given by the asymptotic expansion of the same form but with arbitrary
leading coefficient.
We conclude this section with some observations concerning relations to the paper
[CQY]:
(1) Recall that Theorem 1 was proven by consideration of the log ǫ term in (1.8),
generalizing the proof of (0.2) in [FG1] where u = 1. In [CQY], it was shown
that for a global conformally compact Einstein metric g+, consideration of the
constant term in
∆g+U dvg+ =
∂νU dσ
for U a global solution of ∆g+U = n gives a formula for the renormalized
volume V (g+, g) of g+ relative to a metric g in the conformal infinity of g+.
In our notation this formula reads
(2.6) V (g+, g) = −
(S(s)1) dvg +
2k ṗ∗2kv
(n−2k) dvg,
where ṗ2k =
|λ=0p2k,λ (which exists for k = n/2 when applied to 1) and
S(s) denotes the scattering operator relative to g. The operators ṗ2k arise in
this context because the coefficient of x2k in the expansion of U is ṗ2k1 for
1 ≤ k ≤ n/2−1, and the coefficient of xn involves ṗn1. Likewise, consideration
of the constant term in
u∆g+U dvg+ =
(u∂νU − U∂νu) dσ
for harmonic u gives an analogous formula for the finite part of
u dvg+ in
terms of boundary data.
(2) There is an analogue of Proposition 2 involving the ṗ∗2kv
(n−2k). Differentiating
(2.5) with respect to λ at λ = 0 and rearranging gives the identity
ṗ∗2kv
(n−2k) − (ṗ2k1)v
(n−2k)
(4l − 2k)p∗2k−2l
(ṗ2l1)v
(n−2k)
which expresses the left hand side as a divergence.
(3) In [CQY] it was also shown that under an infinitesimal conformal change, the
scattering term
S(g+, g) ≡
(S(s)1)dvg
10 C. ROBIN GRAHAM AND ANDREAS JUHL
satisfies
S(g+, e
2αΥg) = −2cn/2
ΥQdvg.
Comparing with
V (g+, e
2αΥg) =
Υv(n) dvg
(see [G]) and using (2.6) and Theorem 1, one deduces the curious conclusion
that the infinitesimal conformal variation of
2k ṗ∗2kv
(n−2k) dvg
n/2−1∑
(n− 2k)p∗2kv
(n−2k) dvg.
This statement involves the conformal variation only of local expressions. For
n = 2 this is the statement of conformal invariance of
Rdvg, while for
n = 4 it is the assertion that the infinitesimal conformal variation of
J2 dvg
Υ∆J dvg.
3. Q-curvature and families of conformally covariant differential
operators
In [J] one of the authors initiated a theory of one-parameter families of natural
conformally covariant local operators
(3.1) DN(X,M ; h;λ) : C
∞(X) → C∞(M), N ≥ 0
of orderN associated to a Riemannian manifold (X, h) and a hypersurface i :M → X ,
depending rationally on the parameter λ ∈ C. For such a family the conformal weights
which describe the covariance of the family are coupled to the family parameter in
the sense that
(3.2) e−(λ−N)ωDN(X,M ; ĥ;λ)e
λω = DN (X,M ; h;λ), ĥ = e
for all ω ∈ C∞(X) (near M).
Two families are defined in [J]: one via a residue construction which has its origin
in an extension problem for automorphic functions of Kleinian groups through their
limit set ([J2], chapter 8), and the other via a tractor construction. Whereas the
tractor family depends on the choice of a metric h on X , the residue family depends
on the choice of an asymptotically hyperbolic metric h+ and a defining function x, to
which is associated the metric h = x2h+.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 11
Fix an asymptotically hyperbolic metric h+ on one side X+ of X in M and choose
a defining function x for M with x > 0 in X+. Set h = x
2h+. To an eigenfunction u
on X+ satisfying
∆h+u = µ(n− µ)u, Reµ = n/2, µ 6= n/2
is associated the family
〈Tu(ζ, x), ϕ〉 ≡
xζ uϕ dvh, ϕ ∈ C
c (X)
of distributions on X . The integral converges for Re ζ > −n/2− 1 and the existence
of a formal asymptotic expansion
xµ+jaj(µ) +
xn−µ+jbj(µ), x→ 0
with aj(µ), bj(µ) ∈ C
∞(M) implies the existence of a meromorphic continuation of
Tu(ζ, x) to C with simple poles in the ladders
−µ − 1− N0, −(n− µ)− 1− N0.
For N ∈ N0, its residue at ζ = −µ− 1−N has the form
a0δN(h;µ+N − n)(ϕ)dvi∗h,
where
δN (h;λ) : C
∞(X) → C∞(M)
is a family of differential operators of order N depending rationally on λ ∈ C. If
x̂ = eωx with ω ∈ C∞(X), then ĥ = e2ωh and it is easily checked that δN(h;λ)
satisfies (3.2). (The family δN (h;λ) should more correctly be regarded as determined
by x and h+, but we use this notation nonetheless.)
If g is a metric on M , then we can take h+ = g+ to be a Poincaré metric for g on
X+ =M×(0, a) and x to be the coordinate in the second factor, so that h = dx
2+gx.
Then (assuming N ≤ n if n is even), the family δN(h;λ) depends only on the initial
metric g. The residue can be evaluated explicitly and for even orders N = 2L one
obtains
(3.3) δ2L(h;µ+ 2L− n) =
(2L− 2k)!
p∗2l,µ ◦ v
(2k−2l)
◦ i∗∂2L−2kx ,
where the p2l,µ are the operators appearing in (2.1) and the coefficients v
(2j) are used
as multiplication operators. The corresponding residue family is defined by
(3.4) Dres2L (g;λ) = 2
Γ(−n/2 − λ+ 2L)
Γ(−n/2 − λ+ L)
δ2L(h;λ);
12 C. ROBIN GRAHAM AND ANDREAS JUHL
the normalizing factor makes Dres2L (g;λ) polynomial in λ. We are interested in the
critical case 2L = n for n even. Using
Res0(pn,λ) = −cn/2Pn
from [GZ], we see that
(3.5) Dresn (g; 0) = (−1)
n/2Pn(g)i
Direct evaluation from (3.3), (3.4) gives
Ḋresn (g; 0)1 = −(−1)
n/2c−1
p∗n1 +
n/2−1∑
p∗2kv
(n−2k)
where the dot refers to the derivative in λ.
Suppose now that g is transformed conformally: ĝ = e2Υg with Υ ∈ C∞(M). By
the construction of the normal form in §5 of [GL], the Poincaré metrics g+ and ĝ+ are
related by Φ∗ĝ+ = g+ for a diffeomorphism Φ which restricts to the identity onM and
for which the function Φ∗(x)/x restricts to eΥ. Using this the residue construction
easily implies
(3.6) e−(λ−n)ΥDresn (ĝ;λ) = D
n (g;λ) (Φ
∗(x)/x)
Applying (3.6) to the function 1, differentiating at λ = 0, and using (3.5) and Pn1 = 0
gives
enΥḊresn (ĝ; 0)1 = Ḋ
n (g; 0)1− (−1)
n/2Pn/2Υ.
This proves that the curvature quantity
−(−1)n/2Ḋresn (g; 0)1 = c
p∗n1 +
n/2−1∑
p∗2kv
(n−2k)
satisfies the same transformation law as the Q-curvature. It is natural to conjecture
that it equals the Q-curvature. Indeed, this follows from (0.1), (2.2), and (2.3):
p∗n1 +
n/2−1∑
p∗2kv
(n−2k) = pn1 +
p∗n1− pn1 +
n/2−1∑
2kp∗2kv
(n−2k)
n/2−1∑
p∗2kv
(n−2k) −
n/2−1∑
2kp∗2kv
(n−2k)
The first term is −cn/2Q by (2.2), the second term is 0 by (2.3), and the last term is
2cn/2Q by (0.1).
The relation Ḋresn (g; 0)1 = (−1)
n/2+1Q and (3.5) show that both the critical GJMS
operator Pn and the Q-curvature are contained in the one object D
n (g;λ). In that
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 13
respect, Dresn (g;λ) resembles the scattering operator in [GZ]. However, the family
Dresn (g;λ) is local and all operators in the family have order n.
References
[B] T. Branson, Sharp inequalities, the functional determinant, and the complementary series,
Trans. AMS 347 (1995), 3671-3742.
[BGP] T. Branson, P. Gilkey, and J. Pohjanpelto, Invariants of locally conformally flat manifolds,
Trans. AMS 347 (1995), 939–953.
[CQY] S.-Y. A. Chang, J. Qing, and P. Yang, On the renormalized volumes for conformally com-
pact Einstein manifolds, math.DG/0512376.
[FG1] C. Fefferman and C.R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett. 9
(2002), 139–151.
[FG2] C. Fefferman and C.R. Graham, The ambient metric, in preparation.
[FH] C. Fefferman and K. Hirachi, Ambient metric construction of Q-curvature in conformal
and CR geometries, Math. Res. Lett. 10 (2003), 819–832.
[GP] A.R. Gover and L.J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature
and tractor calculus, Comm. Math. Phys. 235 (2003), 339–378.
[G] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics,
Rend. Circ. Mat. Palermo, Ser. II, Suppl. 63 (2000), 31–42.
[GH] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Q-curvature, in
AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lec-
tures in Mathematics and Theoretical Physics 8 (2005), 59–71, math.DG/0405068.
[GJMS] C.R. Graham, R. Jenne, L.J. Mason, and G.A.J. Sparling Conformally invariant powers
of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557–565.
[GL] C.R. Graham and J. M. Lee, Einstein metrics with prescribed conformal infinity on the
ball, Adv. Math. 87 (1991), 186–225.
[GZ] C.R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math.
152 (2003), 89–118.
[J] A. Juhl, Families of conformally covariant differential operators, Q-curvature and holog-
raphy, book in preparation.
[J2] A. Juhl, Cohomological theory of dynamical zeta functions, Prog. Math. 194, Birkhäuser,
2001.
[SS] K. Skenderis and S. Solodukhin, Quantum effective action from the AdS/CFT correspon-
dence, Phys. Lett. B472 (2000), 316–322, hep-th/9910023.
[V] J. Viaclovsky, Conformal Geometry, Contact Geometry and the Calculus of Variations,
Duke Math. J. 101 (2000), 283–316.
Department of Mathematics, University of Washington, Box 354350, Seattle, WA
98195 USA
E-mail address : robin@math.washington.edu
Humboldt-Universität, Institut für Mathematik, Unter den Linden, 10099 Berlin
E-mail address : ajuhl@math.hu-berlin.de
http://arxiv.org/abs/math/0512376
http://arxiv.org/abs/math/0405068
http://arxiv.org/abs/hep-th/9910023
	Introduction
	1. Derivation
	2. A Related Identity
	3. Q-curvature and families of conformally covariant differential operators
	References

  This paper derives an explicit formula for Branson's Q-curvature in
even-dimensional conformal geometry. The ingredients in the formula come from
the Poincare metric in one higher dimension; hence the formula is called
holographic. When specialized to the conformally flat case, the holographic
formula expresses Q-curvature as a multiple of the Pfaffian and the divergence
of a natural one-form. The paper also outlines the relation between holographic
formulae for Q-curvature and a new theory of conformally covariant families of
differential operators due to the second author.

Introduction
In this paper we give a formula for Q-curvature in even-dimensional conformal
geometry. The Q-curvature was introduced by Tom Branson in [B] and has been
the subject of much research. There are now a number of characterizations of Q-
curvature; see for example [GZ], [FG1], [GP], [FH]. However, it has remained an open
problem to find an expression for Q-curvature which, for example, makes explicit the
relation to the Pfaffian in the conformally flat case.
Theorem 1. The Q-curvature of a metric g in even dimension n is given by
(0.1) 2ncn/2Q = nv
(n) +
n/2−1∑
(n− 2k)p∗2kv
(n−2k),
where cn/2 = (−1)
n/2 [2n(n/2)!(n/2− 1)!]
Here the v(2j) are the coefficients appearing in the asymptotic expansion of the
volume form of a Poincaré metric for g, the differential operators p2k are those which
appear in the expansion of a harmonic function for a Poincaré metric, and p∗2k denotes
the formal adjoint of p2k. These constructions are recalled in §1 below. We refer to
the papers cited above and the references therein for background about Q-curvature.
Each of the operators p∗2k for 1 ≤ k ≤ n/2− 1 can be factored as p
2k = δqk, where
δ denotes the divergence operator with respect to g and qk is a natural operator from
functions to 1-forms. So the second term on the right hand side is the divergence of
a natural 1-form. In particular, integrating (0.1) over a compact manifold recovers
the result of [GZ] that
(0.2) 2cn/2
Qdvg =
v(n)dvg.
This quantity is a global conformal invariant; the right hand side occurs as the coeffi-
cient of the log term in the renormalized volume expansion of a Poincaré metric (see
[G]).
The work of the first author was partially supported by NSF grant DMS-0505701. The work of
the second author was supported by SFB 647 “Raum-Zeit-Materie” of DFG.
http://arxiv.org/abs/0704.1673v1
2 C. ROBIN GRAHAM AND ANDREAS JUHL
As we also discuss in §1, if g is conformally flat then
v(n) = (−2)−n/2(n/2)!−1Pff ,
where Pff denotes the Pfaffian of g. So in the conformally flat case, Theorem 1 gives a
decomposition of the Q-curvature as a multiple of the Pfaffian and the divergence of a
natural 1-form. A general result in invariant theory ([BGP]) establishes the existence
of such a decomposition, but does not produce a specific realization.
We refer to (0.1) as a holographic formula because its ingredients come from the
Poincaré metric, involving geometry in n + 1 dimensions. Our proof is via the char-
acterization of Q-curvature presented in [FG1] in terms of Poincaré metrics; in some
sense Theorem 1 is the result of making explicit the characterization in [FG1]. How-
ever, passing from the construction in [FG1] to (0.1) involves a non-obvious appli-
cation of Green’s identity. The transformation law of Q-curvature under conformal
change, probably its most fundamental property, is not transparent from (0.1), but it
is from the characterization in [FG1]. In §2, we derive another identity involving the
p∗2kv
(n−2k) which is used in §3 and we discuss relations to the paper [CQY]. In §3, we
describe the relation between holographic formulae for Q-curvature and the theory of
conformally covariant families of differential operators of [J], and in particular explain
how this theory leads to the conjecture of a holographic formula for Q.
We are grateful to the organizing committee of the 2007 Winter School ’Geome-
try and Physics’ at Srńı, particularly to Vladimir Souc̆ek, for the invitation to this
gathering, which made possible the interaction leading to this paper.
We dedicate this paper to the memory of Tom Branson. His insights have led to
beautiful new mathematics and have greatly influenced our own respective work.
1. Derivation
Let g be a metric of signature (p, q) on a manifold M of even dimension n. In this
paper, by a Poincaré metric for (M, g) we will mean a metric g+ on M × (0, a) for
some a > 0 of the form
(1.1) g+ = x
−2(dx2 + gx),
where gx is a smooth 1-parameter family of metrics on M satisfying g0 = g, such
that g+ is asymptotically Einstein in the sense that Ric(g+) + ng+ = O(x
n−2) and
trg+(Ric(g+)+ng+) = O(x
n+2). Such a Poincaré metric always exists and gx is unique
up addition of a term of the form xnhx, where hx is a smooth 1-parameter family of
symmetric 2-tensors on M satisfying trg(h0) = 0 on M . The Taylor expansion of gx
is even through order n and the derivatives (∂x)
2kgx|x=0 for 1 ≤ k ≤ n/2− 1 and the
trace trg((∂x)
ngx|x=0) are determined inductively from the Einstein condition and are
given by polynomial formulae in terms of g, its inverse, and its curvature tensor and
covariant derivatives thereof. See [GH] for details.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 3
The first ingredient in our formula for Q-curvature consists of the coefficients in
the expansion of the volume form
(1.2) dvg+ = x
−n−1dvgxdx.
Because the expansion of gx has only even terms through order n, it follows that
dvgx =
det gx
det g
= (1 + v(2)x2 + · · ·+ v(n)xn + · · · )dvg,
(1.3)
where each of the v(2k) for 1 ≤ k ≤ n/2 is a smooth function on M expressible in
terms of the curvature tensor of g and its covariant derivatives. Set v(0) = 1.
The second ingredient in our formula is the family of differential operators which
appears in the expansion of a harmonic function for the metric g+. Given f ∈ C
∞(M),
one can solve formally the equation ∆g+u = O(x
n) for a smooth function u such that
u|x=0 = f , and such a u is uniquely determined modulo O(x
n). The Taylor expansion
of u is even through order n − 2 and these Taylor coefficients are given by natural
differential operators in the metric g applied to f which are obtained inductively by
solving the equation ∆g+u = O(x
n) order by order. See [GZ] for details. We write
the expansion of u in the form
(1.4) u = f + p2f x
2 + · · ·+ pn−2f x
n−2 +O(xn);
then p2k has order 2k and its principal part is (−1)
Γ(n/2− k)
22k k! Γ(n/2)
∆k. (Our convention
is ∆ = −∇i∇i.) Set p0f = f .
We remark that the volume coefficients v(2k) and the differential operators p2k also
arise in the context of an ambient metric associated to (M, [g]). If an ambient metric
is written in normal form relative to g, then the same v(2k) are coefficients in the
expansion of its volume form, and the same operators p2k appear in the expansion of
a harmonic function homogeneous of degree 0 with respect to the ambient metric.
Let g+ be a Poincaré metric for (M, g). In [FG1] it is shown that there is a unique
solution U mod O(xn) to
(1.5) ∆g+U = n +O(x
n+1 log x)
of the form
(1.6) U = log x+ A+Bxn log x+O(xn) ,
A,B ∈ C∞(M × [0, a)) , A|x=0 = 0 .
Also, A mod O(xn) is even in x and is formally determined by g, and
(1.7) B|x=0 = −2cn/2Q.
4 C. ROBIN GRAHAM AND ANDREAS JUHL
The proof of (1.7) presented in [FG1] used results from [GZ] about the scattering
matrix, so is restricted to positive definite signature. However, a purely formal proof
was also indicated in [FG1]. Thus (1.7) holds in general signature.
Proof of Theorem 1. Let g+ be a Poincaré metric for g and let U be a solution of
(1.5) as described above. Let f ∈ C∞(M) have compact support. Let u be a solution
of ∆g+u = O(x
n) with u|x=0 = f ; for definiteness we take u to be given by (1.4) with
the O(xn) term set equal to 0. Let 0 < ǫ < x0 with ǫ, x0 small.
Consider Green’s identity
(1.8)
ǫ<x<x0
(U∆g+u− u∆g+U) dvg+ =
(U∂νu− u∂νU) dσ,
where ν denotes the inward normal and dσ the induced volume element on the bound-
ary, relative to g+. Both sides have asymptotic expansions as ǫ→ 0; we calculate the
coefficient of log ǫ in these expansions.
Using the form of the expansion of U and the fact that ∆g+u = O(x
n), one sees
that the expansion of U∆g+u dvg+ has no x
−1 term, so
ǫ<x<x0
U∆g+u dvg+ has no
log ǫ term. Using (1.2), (1.3), (1.4), and (1.5), one finds that the log ǫ coefficient of
ǫ<x<x0
u∆g+U dvg+ is
(1.9) n
n/2−1∑
v(n−2k)p2kf dvg.
On the right hand side of (1.8),
is independent of ǫ, and
(U∂νu− u∂νU) dσ = ǫ
(U∂xu− u∂xU) dvgǫ.
A log ǫ term in the expansion of this quantity can arise only from the log x or xn log x
terms in the expansion of U . Substituting the expansions, one finds without difficulty
that the log ǫ coefficient is
n/2−1∑
2kv(n−2k)p2kf − nBf
 dvg.
Equating this to (1.9), using (1.7), and moving all derivatives off f gives the desired
identity. �
Since ∆g+1 = 0, it follows that p2k1 = 0 for 1 ≤ k ≤ n/2− 1. Thus these p2k have
no constant term, so p∗2k = δqk for some natural operator qk from functions to 1-forms,
where δ denotes the divergence with respect to the metric g. So in (0.1), the second
term on the right hand side is the divergence of a natural 1-form. As mentioned in
the introduction, integration gives (0.2). The proof of Theorem 1 presented above in
the special case u = 1 is precisely the proof of (0.2) presented in [FG1].
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 5
Theorem 1 provides an efficient way to calculate the Q curvature. Solving for the
beginning coefficients in the expansion of the Poincaré metric and then expanding its
volume form shows that the first few of the v(2k) are given by:
v(2) = −
v(4) =
(J2 − |P |2)
v(6) =
P ijBij + 3J |P |
2 − J3 − 2P ijPi
where
Pij =
Rij −
2(n− 1)
2(n− 1)
= P ii
Bij = Pij,k
k − Pik,j
k − P klWkijl
and Wijkl denotes the Weyl tensor. Similarly, one finds that the operators p2 and p4
are given by:
−2(n− 2)p2 = ∆
8(n− 2)(n− 4)p4 = ∆
2 + 2J∆+ 2(n− 2)P ij∇i∇j + (n− 2)J,
(1.10)
For n = 2, Theorem 1 states Q = −2v(2) = 1
R. For n = 4, substituting the above
into Theorem 1 gives:
Q = 2(J2 − |P |2) + ∆J,
and for n = 6:
Q = 8P ijBij + 16P
kPkj − 24J |P |
2 + 8J3
+∆2J + 4∆(J2) + 8(P ijJ,i),j − 4∆(|P |
In the formula for n = 6, the first line is (12c3)
−16v(6) and the second line is
(12c3)
4p∗2v
(4) + 2p∗4v
. Details of these calculations will appear in [J].
The expansion of the Poincaré metric g+ was identified explicitly in the case that
g is conformally flat in [SS]. (Since we are only interested in local considerations,
by conformally flat we mean locally conformally flat.) The two dimensional case is
somewhat anomalous in this regard, but the identification of Q curvature is trivial
when n = 2, so we assume n > 2 for this discussion. The conclusion of [SS] is that
if g is conformally flat and n > 2 (even or odd), then the expansion of the Poincaré
metric terminates at second order and
(1.11) (gx)ij = gij − Pijx
6 C. ROBIN GRAHAM AND ANDREAS JUHL
(The details of the computation are not given in [SS]. Details will appear in [FG2]
and [J].) This easily yields
Proposition 1. If g is conformally flat and n > 2, then
v(2k) =
(−2)−kσk(P ) 0 ≤ k ≤ n
0 n < k
where σk(P ) denotes the k-th elementary symmetric function of the eigenvalues of the
endomorphism Pi
Proof. Write g−1P for Pi
j. Then the σk(P ) are given by
det(I + g−1P t) =
σk(P )t
Equation (1.11) can be rewritten as g−1gx = (I−
g−1Px2)2. Taking the determinant
and comparing with (1.3) gives the result. �
We remark that for g conformally flat, gx given by (1.11) is uniquely determined to
all orders by the requirement that g+ be hyperbolic. So in this case the v
(2k) are
invariantly determined and given by Proposition 1 for all k ≥ 0 in all dimensions
n > 2.
Returning to the even-dimensional case, we define the Pfaffian of the metric g by
(1.12) 2n(n/2)! Pff = (−1)qµi1...inµj1...jnRi1i2j1j2 . . . Rin−1injn−1jn,
where µi1...in =
| det(g)| ǫi1...in is the volume form and ǫi1...in denotes the sign of the
permutation. For a conformally flat metric, one has Rijkl = 2(Pi[kgl]j−Pj[kgl]i). Using
this in (1.12) and simplifying gives
Pff = (n/2)! σn/2(P )
(see Proposition 8 of [V] for details). Combining with Proposition 1, we obtain for
conformally flat g:
v(n) = (−2)−n/2(n/2)!−1 Pff .
Hence in the conformally flat case, (0.1) specializes to
2Q = 2n/2(n/2− 1)! Pff +(ncn/2)
n/2−1∑
(n− 2k)p∗2kv
(n−2k),
and again the second term on the right hand side is a formal divergence.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 7
2. A Related Identity
In this section we derive another identity involving the p∗2kv
(n−2k). It is in gen-
eral impossible to choose the O(xn) term in (1.4) to make ∆g+u = O(x
n); in fact
x−n∆g+u|x=0 is independent of the O(x
n) term in (1.4) and is a conformally invariant
operator of order n applied to f , namely a multiple of the critical GJMS operator
Pn. Following [GZ], we consider the limiting behavior of the corresponding term in
the expansion of an eigenfunction for ∆g+ as the eigenvalue tends to 0.
Let g+ be a Poincaré metric as above. If 0 6= λ ∈ C is near 0, then for f ∈ C
∞(M),
one can solve formally the equation (∆g+ − λ(n − λ))uλ = O(x
n+λ+1) for uλ of the
(2.1) uλ = x
f + p2,λf x
2 + · · ·+ pn,λf x
n +O(xn+1)
where p2k,λ is a natural differential operator in the metric g of order 2k with principal
part (−1)k
Γ(n/2− k − λ)
22k k! Γ(n/2− λ)
∆k such that
Γ(n/2− λ)
Γ(n/2− k − λ)
p2k,λ is polynomial in λ. Set
p0,λf = f . The operators p2k,λ for k < n/2 extend analytically across λ = 0 and
p2k,0 = p2k for such k, where p2k are the operators appearing in (1.4). But pn,λ has a
simple pole at λ = 0 with residue a multiple of the critical GJMS operator Pn. Now
Pn is self-adjoint, so it follows that pn,λ − p
n,λ is regular at λ = 0. We denote its
value at λ = 0 by pn − p
n, a natural operator of order at most n − 2. Our identity
below involves the constant term (pn−p
n)1. Note that since Pn1 = 0, both pn,λ1 and
p∗n,λ1 are regular at λ = 0. We denote their values at λ = 0 by pn1 and p
n1; then
(pn − p
n)1 = pn1− p
n1. Moreover, (4.7), (4.13), (4.14) of [GZ] show that
(2.2) pn1 = −cn/2Q.
It is evident that
pn1 dvg =
p∗n1 dvg. The next proposition expresses the differ-
ence pn1− p
n1 as a divergence.
Proposition 2.
(2.3) n (pn − p
n) 1 =
n/2−1∑
2k p∗2kv
(n−2k)
Proof. Take f ∈ C∞(M) to have compact support, let 0 6= λ be near 0, and define
uλ as in (2.1) with the O(x
n+1) term taken to be 0. Define wλ by the corresponding
expansion with f = 1:
wλ = x
1 + p2,λ1 x
2 + · · ·+ pn,λ1 x
As in the proof of Theorem 1, consider Green’s identity
(2.4)
ǫ<x<x0
(uλ∆g+wλ − wλ∆g+uλ)dvg+ = ǫ
(uλ∂xwλ − wλ∂xuλ) dvgǫ + cx0 ,
8 C. ROBIN GRAHAM AND ANDREAS JUHL
where cx0 is the constant (in ǫ) arising from the boundary integral over x = x0.
Consider the coefficient of ǫ2λ in the asymptotic expansion of both sides. The left
hand side equals
ǫ<x<x0
∆g+ − λ(n− λ)
wλ − wλ
∆g+ − λ(n− λ)
dvg+.
Now uλ
∆g+ − λ(n− λ)
wλ dvg+ and wλ
∆g+ − λ(n− λ)
uλ dvg+ are of the form
x2λψ dxdvg where ψ is smooth up to x = 0. It follows that the asymptotic expansion
of the left hand side of (2.4) has no ǫ2λ term. Consequently the coefficient of ǫn+2λ
must vanish in the asymptotic expansion of
(uλx∂xwλ − wλx∂xuλ) dvgǫ.
This is the same as the coefficient of ǫn in the expansion of
p2k,λf ǫ
(2k + λ)p2k,λ1 ǫ
p2k,λ1 ǫ
(2k + λ)p2k,λf ǫ
v(2k) ǫ2k
 dvg.
Evaluation of the ǫn coefficient gives
0≤k,l,m≤n/2
k+l+m=n/2
(2l − 2k)(p2k,λf)(p2l,λ1)v
(2m) dvg = 0,
and then moving the derivatives off f results in the pointwise identity
(2.5)
0≤k,l,m≤n/2
k+l+m=n/2
(2l − 2k) p∗2k,λ
(p2l,λ1)v
The limit as λ → 0 exists of all p2l,λ1 with 0 ≤ l ≤ n/2 and all p
2k,λ with 0 ≤ k ≤
n/2− 1. Since k = n/2 forces l = m = 0, the operator p∗n,λ occurs only applied to 1.
Thus we may let λ→ 0 in (2.5). Using p2l1 = 0 for 1 ≤ l ≤ n/2− 1 results in
npn1−
0≤k,m≤n/2
k+m=n/2
2k p∗2kv
(2m) = 0.
Separating the k = n/2 term in the sum gives (2.3). �
Proposition 2 may be combined with (0.1) and (2.2) to give other expressions for
Q-curvature. However, (0.1) seems the preferred form, as the other expressions all
involve some nontrivial linear combination of pn1 and p
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 9
We remark that the generalization of (2.5) obtained by replacing p2l,λ1 by p2l,λf
remains true for arbitrary f ∈ C∞(M). This follows by the same argument, taking
wλ to be given by the asymptotic expansion of the same form but with arbitrary
leading coefficient.
We conclude this section with some observations concerning relations to the paper
[CQY]:
(1) Recall that Theorem 1 was proven by consideration of the log ǫ term in (1.8),
generalizing the proof of (0.2) in [FG1] where u = 1. In [CQY], it was shown
that for a global conformally compact Einstein metric g+, consideration of the
constant term in
∆g+U dvg+ =
∂νU dσ
for U a global solution of ∆g+U = n gives a formula for the renormalized
volume V (g+, g) of g+ relative to a metric g in the conformal infinity of g+.
In our notation this formula reads
(2.6) V (g+, g) = −
(S(s)1) dvg +
2k ṗ∗2kv
(n−2k) dvg,
where ṗ2k =
|λ=0p2k,λ (which exists for k = n/2 when applied to 1) and
S(s) denotes the scattering operator relative to g. The operators ṗ2k arise in
this context because the coefficient of x2k in the expansion of U is ṗ2k1 for
1 ≤ k ≤ n/2−1, and the coefficient of xn involves ṗn1. Likewise, consideration
of the constant term in
u∆g+U dvg+ =
(u∂νU − U∂νu) dσ
for harmonic u gives an analogous formula for the finite part of
u dvg+ in
terms of boundary data.
(2) There is an analogue of Proposition 2 involving the ṗ∗2kv
(n−2k). Differentiating
(2.5) with respect to λ at λ = 0 and rearranging gives the identity
ṗ∗2kv
(n−2k) − (ṗ2k1)v
(n−2k)
(4l − 2k)p∗2k−2l
(ṗ2l1)v
(n−2k)
which expresses the left hand side as a divergence.
(3) In [CQY] it was also shown that under an infinitesimal conformal change, the
scattering term
S(g+, g) ≡
(S(s)1)dvg
10 C. ROBIN GRAHAM AND ANDREAS JUHL
satisfies
S(g+, e
2αΥg) = −2cn/2
ΥQdvg.
Comparing with
V (g+, e
2αΥg) =
Υv(n) dvg
(see [G]) and using (2.6) and Theorem 1, one deduces the curious conclusion
that the infinitesimal conformal variation of
2k ṗ∗2kv
(n−2k) dvg
n/2−1∑
(n− 2k)p∗2kv
(n−2k) dvg.
This statement involves the conformal variation only of local expressions. For
n = 2 this is the statement of conformal invariance of
Rdvg, while for
n = 4 it is the assertion that the infinitesimal conformal variation of
J2 dvg
Υ∆J dvg.
3. Q-curvature and families of conformally covariant differential
operators
In [J] one of the authors initiated a theory of one-parameter families of natural
conformally covariant local operators
(3.1) DN(X,M ; h;λ) : C
∞(X) → C∞(M), N ≥ 0
of orderN associated to a Riemannian manifold (X, h) and a hypersurface i :M → X ,
depending rationally on the parameter λ ∈ C. For such a family the conformal weights
which describe the covariance of the family are coupled to the family parameter in
the sense that
(3.2) e−(λ−N)ωDN(X,M ; ĥ;λ)e
λω = DN (X,M ; h;λ), ĥ = e
for all ω ∈ C∞(X) (near M).
Two families are defined in [J]: one via a residue construction which has its origin
in an extension problem for automorphic functions of Kleinian groups through their
limit set ([J2], chapter 8), and the other via a tractor construction. Whereas the
tractor family depends on the choice of a metric h on X , the residue family depends
on the choice of an asymptotically hyperbolic metric h+ and a defining function x, to
which is associated the metric h = x2h+.
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 11
Fix an asymptotically hyperbolic metric h+ on one side X+ of X in M and choose
a defining function x for M with x > 0 in X+. Set h = x
2h+. To an eigenfunction u
on X+ satisfying
∆h+u = µ(n− µ)u, Reµ = n/2, µ 6= n/2
is associated the family
〈Tu(ζ, x), ϕ〉 ≡
xζ uϕ dvh, ϕ ∈ C
c (X)
of distributions on X . The integral converges for Re ζ > −n/2− 1 and the existence
of a formal asymptotic expansion
xµ+jaj(µ) +
xn−µ+jbj(µ), x→ 0
with aj(µ), bj(µ) ∈ C
∞(M) implies the existence of a meromorphic continuation of
Tu(ζ, x) to C with simple poles in the ladders
−µ − 1− N0, −(n− µ)− 1− N0.
For N ∈ N0, its residue at ζ = −µ− 1−N has the form
a0δN(h;µ+N − n)(ϕ)dvi∗h,
where
δN (h;λ) : C
∞(X) → C∞(M)
is a family of differential operators of order N depending rationally on λ ∈ C. If
x̂ = eωx with ω ∈ C∞(X), then ĥ = e2ωh and it is easily checked that δN(h;λ)
satisfies (3.2). (The family δN (h;λ) should more correctly be regarded as determined
by x and h+, but we use this notation nonetheless.)
If g is a metric on M , then we can take h+ = g+ to be a Poincaré metric for g on
X+ =M×(0, a) and x to be the coordinate in the second factor, so that h = dx
2+gx.
Then (assuming N ≤ n if n is even), the family δN(h;λ) depends only on the initial
metric g. The residue can be evaluated explicitly and for even orders N = 2L one
obtains
(3.3) δ2L(h;µ+ 2L− n) =
(2L− 2k)!
p∗2l,µ ◦ v
(2k−2l)
◦ i∗∂2L−2kx ,
where the p2l,µ are the operators appearing in (2.1) and the coefficients v
(2j) are used
as multiplication operators. The corresponding residue family is defined by
(3.4) Dres2L (g;λ) = 2
Γ(−n/2 − λ+ 2L)
Γ(−n/2 − λ+ L)
δ2L(h;λ);
12 C. ROBIN GRAHAM AND ANDREAS JUHL
the normalizing factor makes Dres2L (g;λ) polynomial in λ. We are interested in the
critical case 2L = n for n even. Using
Res0(pn,λ) = −cn/2Pn
from [GZ], we see that
(3.5) Dresn (g; 0) = (−1)
n/2Pn(g)i
Direct evaluation from (3.3), (3.4) gives
Ḋresn (g; 0)1 = −(−1)
n/2c−1
p∗n1 +
n/2−1∑
p∗2kv
(n−2k)
where the dot refers to the derivative in λ.
Suppose now that g is transformed conformally: ĝ = e2Υg with Υ ∈ C∞(M). By
the construction of the normal form in §5 of [GL], the Poincaré metrics g+ and ĝ+ are
related by Φ∗ĝ+ = g+ for a diffeomorphism Φ which restricts to the identity onM and
for which the function Φ∗(x)/x restricts to eΥ. Using this the residue construction
easily implies
(3.6) e−(λ−n)ΥDresn (ĝ;λ) = D
n (g;λ) (Φ
∗(x)/x)
Applying (3.6) to the function 1, differentiating at λ = 0, and using (3.5) and Pn1 = 0
gives
enΥḊresn (ĝ; 0)1 = Ḋ
n (g; 0)1− (−1)
n/2Pn/2Υ.
This proves that the curvature quantity
−(−1)n/2Ḋresn (g; 0)1 = c
p∗n1 +
n/2−1∑
p∗2kv
(n−2k)
satisfies the same transformation law as the Q-curvature. It is natural to conjecture
that it equals the Q-curvature. Indeed, this follows from (0.1), (2.2), and (2.3):
p∗n1 +
n/2−1∑
p∗2kv
(n−2k) = pn1 +
p∗n1− pn1 +
n/2−1∑
2kp∗2kv
(n−2k)
n/2−1∑
p∗2kv
(n−2k) −
n/2−1∑
2kp∗2kv
(n−2k)
The first term is −cn/2Q by (2.2), the second term is 0 by (2.3), and the last term is
2cn/2Q by (0.1).
The relation Ḋresn (g; 0)1 = (−1)
n/2+1Q and (3.5) show that both the critical GJMS
operator Pn and the Q-curvature are contained in the one object D
n (g;λ). In that
HOLOGRAPHIC FORMULA FOR Q-CURVATURE 13
respect, Dresn (g;λ) resembles the scattering operator in [GZ]. However, the family
Dresn (g;λ) is local and all operators in the family have order n.
References
[B] T. Branson, Sharp inequalities, the functional determinant, and the complementary series,
Trans. AMS 347 (1995), 3671-3742.
[BGP] T. Branson, P. Gilkey, and J. Pohjanpelto, Invariants of locally conformally flat manifolds,
Trans. AMS 347 (1995), 939–953.
[CQY] S.-Y. A. Chang, J. Qing, and P. Yang, On the renormalized volumes for conformally com-
pact Einstein manifolds, math.DG/0512376.
[FG1] C. Fefferman and C.R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett. 9
(2002), 139–151.
[FG2] C. Fefferman and C.R. Graham, The ambient metric, in preparation.
[FH] C. Fefferman and K. Hirachi, Ambient metric construction of Q-curvature in conformal
and CR geometries, Math. Res. Lett. 10 (2003), 819–832.
[GP] A.R. Gover and L.J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature
and tractor calculus, Comm. Math. Phys. 235 (2003), 339–378.
[G] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics,
Rend. Circ. Mat. Palermo, Ser. II, Suppl. 63 (2000), 31–42.
[GH] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Q-curvature, in
AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lec-
tures in Mathematics and Theoretical Physics 8 (2005), 59–71, math.DG/0405068.
[GJMS] C.R. Graham, R. Jenne, L.J. Mason, and G.A.J. Sparling Conformally invariant powers
of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557–565.
[GL] C.R. Graham and J. M. Lee, Einstein metrics with prescribed conformal infinity on the
ball, Adv. Math. 87 (1991), 186–225.
[GZ] C.R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math.
152 (2003), 89–118.
[J] A. Juhl, Families of conformally covariant differential operators, Q-curvature and holog-
raphy, book in preparation.
[J2] A. Juhl, Cohomological theory of dynamical zeta functions, Prog. Math. 194, Birkhäuser,
2001.
[SS] K. Skenderis and S. Solodukhin, Quantum effective action from the AdS/CFT correspon-
dence, Phys. Lett. B472 (2000), 316–322, hep-th/9910023.
[V] J. Viaclovsky, Conformal Geometry, Contact Geometry and the Calculus of Variations,
Duke Math. J. 101 (2000), 283–316.
Department of Mathematics, University of Washington, Box 354350, Seattle, WA
98195 USA
E-mail address : robin@math.washington.edu
Humboldt-Universität, Institut für Mathematik, Unter den Linden, 10099 Berlin
E-mail address : ajuhl@math.hu-berlin.de
http://arxiv.org/abs/math/0512376
http://arxiv.org/abs/math/0405068
http://arxiv.org/abs/hep-th/9910023
	Introduction
	1. Derivation
	2. A Related Identity
	3. Q-curvature and families of conformally covariant differential operators
	References
