Conformal structure in Geometry, Analysis, and Physics

The American Institute of Mathematics

This is a hard–copy version of a web page available through http://www.aimath.org

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Table of Contents

A. A Primer on Q-Curvature . . . . . . . . . . . . . . . . . . . . . . . . 3 B. Origins, applications, and generalizations of the Q-curvature . . . . . . . . 11 C. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 D. Reference list . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter A: A Primer on Q-Curvature

by Michael Eastwood and Jan Slovák Disclaimer: These are rough notes only, aimed at setting the scene and promoting

discussion at the American Institute of Mathematics Research Conference Center Workshop ‘Conformal Structure in Geometry, Analysis, and Physics,’ 12^{th}–16^{th} August 2003. For simplicity, we have omitted all references. Curvature conventions are in an appendix. Conversations with Tom Branson and Rod Gover have been extremely useful.

Let M be an oriented even-dimensional Riemannian n-manifold. Branson’s Q-curvature is a canonically defined n-form on M. It is not conformally invariant but enjoys certain natural properties with respect to conformal transformations.

When n = 2, the Q-curvature is a multiple of the scalar curvature. With conventions as in the appendix Q = −\frac{1}{2}R. Under conformal rescaling of the metric, g_{ab} ↦ ĝ_{ab} = Ω^{2}g_{ab} we have

Q̂ = Q + Δ log Ω,

where Δ = ∇^{a}∇_{a} is the Laplacian.

When n = 4, the Q-curvature is given by

Q = \frac{1}{6}R^{2} − \frac{1}{2}R^{ab}R_{ab} − \frac{1}{6}ΔR.  (1)

Under conformal rescaling,

Q̂ = Q + P log Ω,

where P is the Paneitz operator

P f = ∇_{a} [∇^{a}∇^{b} + 2R^{ab} − \frac{2}{3}Rg^{ab}] ∇_{b}f.  (2)

For general even n, the Q-curvature transforms as follows:–

Q̂ = Q + P log Ω,  (3)

where P is a linear differential operator from functions to n-forms whose symbol is Δ^{n/2}. It follows from this transformation law that P is conformally invariant. To see this, suppose that

ĝ_{ab} = Ω^{2}g_{ab} and ˆ ĝ_{ab} = e^{2f} ĝ_{ab} = (e^{f}Ω)^{2}g_{ab}.

Then

ˆQ̂ = Q̂ + P̂ log e^{f} = Q + P log Ω + P̂f

but also

ˆQ̂ = Q + P log(e^{f}Ω) = Q + P f + P log Ω.

Therefore, P̂f = P f. With suitable normalisation, P is the celebrated Graham-Jenne-Mason-Sparling operator. Thus, Q may be regarded as more primitive than P and, therefore, is at least as mysterious.

Even when M is conformally flat, the existence of Q is quite subtle. We can reason as follows. When M is actually flat then Q must vanish. Therefore, in the conformally flat case, locally if we write g_{ab} = Ω^{2}η_{ab} where η_{ab} is flat, then (3) implies that

Q = Δ^{n/2} log Ω,  (4)

where Δ is the ordinary Laplacian in Euclidean space with η_{ab} as metric. An immediate problem is to verify that this purported construction of Q is well-defined. The problem is that there is some freedom in writing g_{ab} as proportional to a flat metric. If also g_{ab} = Ω̂^{2}η̂_{ab}, then we must show that

Δ^{n/2} log Ω = Δ̂^{n/2} log Ω̂.

This easily reduces to two facts:– fact 1:: Δ^{n/2} is conformally invariant on flat space. fact 2:: if g_{ab} is itself flat, then Δ^{n/2} log Ω = 0.

The second of these is clearly necessary in order that (37) be well-defined. For n = 2 it is immediate from (17). For n ≥ 4 it may be verified by direct calculation as follows. If g_{ab} and η_{ab} are both flat then

∇_{a}Υ_{b} = Υ_{a}Υ_{b} − \frac{1}{2}Υ^{c}Υ_{c}g_{ab},  (5)

where Υ_{a} = ∇_{a} log Ω. Therefore,

∇_{c}(Υ^{a}Υ_{a})^{k} = 2k(Υ^{a}Υ_{a})^{k−1}Υ^{a}∇_{c}Υ_{a} = k(Υ^{a}Υ_{a})^{k}Υ_{c}

and

∇_{b}∇_{c}(Υ^{a}Υ_{a})^{k} = k^{2}(Υ^{a}Υ_{a})^{k}Υ_{b}Υ_{c} + k(Υ^{a}Υ_{a})^{k}(Υ_{b}Υ_{c} − \frac{1}{2}Υ^{a}Υ_{a}g_{bc})

whence

Δ(Υ^{a}Υ_{a})^{k} = k(k + 1 − \frac{n}{2})(Υ^{a}Υ_{a})^{k+1}.  (6)

Taking the trace of (5) gives

Δ log Ω = ∇^{a}Υ_{a} = (1 − \frac{n}{2})Υ^{a}Υ_{a}

and now (6) gives, by induction,

Δ^{k+1} log Ω = k!(1 − \frac{n}{2})(2 − \frac{n}{2}) ⋯ (k + 1 − \frac{n}{2})(Υ^{a}Υ_{a})^{k+1}.

In particular, Δ^{n/2} log Ω = 0, as advertised.

That Δ^{n/2} is conformally invariant on flat space is well-known. It may also be verified directly by a rather similar calculation. For example, here is the calculation when n = 4. For general conformally related metrics ĝ_{ab} = Ω^{2}g_{ab} in dimension 4,

Δ̂^{2}f = Δ^{2}f + 2Υ^{a}Δ∇_{a}f − 2Υ^{a}∇_{a}Δf
+ 4(∇^{a}Υ^{b})∇_{a}∇_{b}f − 2(∇^{a}Υ_{a})Δf − 4Υ^{a}Υ^{b}∇_{a}∇_{b}f
+ 2(ΔΥ^{a})∇_{a}f − 4(∇^{a}Υ^{b})Υ_{a}∇_{b}f − 4(∇^{a}Υ_{a})Υ^{b}∇_{b}f.

If g_{ab} is flat then the third order terms cancel leaving

Δ̂^{2}f = Δ^{2}f + 4(∇^{a}Υ^{b})∇_{a}∇_{b}f − 2(∇^{a}Υ_{a})Δf − 4Υ^{a}Υ^{b}∇_{a}∇_{b}f
+ 2(ΔΥ^{a})∇_{a}f − 4(∇^{a}Υ^{b})Υ_{a}∇_{b}f − 4(∇^{a}Υ_{a})Υ^{b}∇_{b}f.

If ĝ_{ab} is also flat, then (5) implies

∇^{a}Υ^{b} = Υ^{a}Υ^{b} − \frac{1}{2}Υ^{c}Υ_{c}g^{ab} and ∇^{a}Υ_{a} = −Υ^{a}Υ_{a}

whence the second order terms cancel and the first order ones simplify:–

Δ̂^{2}f = Δ^{2}f + 2(ΔΥ^{b})∇_{b}f + 2Υ^{a}Υ_{a}Υ^{b}∇_{b}f.

But using (5) again,

ΔΥ^{b} = ∇_{a}(Υ^{a}Υ^{b} − \frac{1}{2}Υ^{c}Υ_{c}g^{ab})
= (∇^{a}Υ_{a})Υ^{b} + (∇^{a}Υ^{b})Υ_{a} − (∇^{b}Υ^{a})Υ_{a} = −Υ^{a}Υ_{a}Υ^{b}

and the first order terms also cancel leaving Δ̂^{2}f = Δ^{2}f, as advertised.

Conundrum: Deduce fact 2 from fact 1 or vice versa. Both are consequences of (5). Alternatively, construct a Lie algebraic proof of fact 2. There is a Lie algebraic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalised Verma modules for 𝔰𝔬(n + 1, 1).

What about a formula for Q, even in the conformally flat case? We have a recipe for Q, namely (37), but it is not a formula. We may proceed as follows.

If ĝ_{ab} = Ωg_{ab} and g_{ab} is flat, then (16) implies that

∇_{a}Υ_{b} = −P̂_{ab} + Υ_{a}Υ_{b} − \frac{1}{2}g_{ab}Υ^{c}Υ_{c}.  (7)

Taking the trace yields

Δ log Ω = ∇^{a}Υ_{a} = −P̂ − \frac{1}{2}(n − 2)Υ^{a}Υ_{a}.  (8)

This identity is also valid when n = 2: it is (17). Dropping the hat gives Q = −P = \frac{1}{2}R. This is the simplest of the desired formulae.

To proceed further we need two identities. If ϕ has conformal weight w, then as described in the appendix,

ˆ∇_{a}ϕ = ∇_{a}ϕ + wΥ_{a}ϕ,

which we rewrite as

∇_{a}ϕ = ˆ∇_{a}ϕ − wΥ_{a}ϕ.  (9)

Similarly, if ϕ_{a} has weight w, then

∇^{a}ϕ_{a} = ˆ∇^{a}ϕ_{a} − (n + w − 2)Υ^{a}ϕ_{a}  (10)

and, if ϕ_{ab} is symmetric and has weight w, then

∇^{a}ϕ_{ab} = ˆ∇^{a}ϕ_{ab} − (n + w − 2)Υ^{a}ϕ_{ab} + Υ_{b}ϕ^{a}_{a}.  (11)

The quantities in (8) have weight −2. Therefore, applying (9) gives

∇_{a}Δ log Ω = − ˆ∇_{a}P̂ − 2Υ_{a}P̂ − (n − 2)Υ^{b}∇_{a}Υ_{b}

wherein we may use (7) to replace ∇_{a}Υ_{b} to obtain

∇_{a}Δ log Ω = − ˆ∇_{a}P̂ − 2Υ_{a}P̂ + (n − 2)Υ^{b}P̂_{ab} − \frac{1}{2}(n − 2)Υ_{a}Υ^{b}Υ_{b}.

We may now apply ∇^{a}, using (9), (10), and (11) to replace ∇^{a} by ˆ∇^{a} on the right hand side and (7) to replace derivatives of Υ_{a}. We obtain an expression involving only complete contractions of P̂_{ab}, its hatted derivatives, and Υ_{a}:–

Δ^{2} log Ω = −Δ̂P̂ − (n − 2)P̂^{ab}P̂_{ab} + 2P̂^{2}
+ (n − 6)Υ^{a} ˆ∇_{a}P̂ + (n − 2)Υ^{a} ˆ∇^{b}P̂_{ab} + 2(n − 4)Υ^{a}Υ_{a}P̂
− (n − 2)(n − 4)Υ^{a}Υ^{b}P̂_{ab} + \frac{1}{4}(n − 2)(n − 4)Υ^{a}Υ_{a}Υ^{b}Υ_{b}.

Using the Bianchi identity ˆ∇^{b}P̂_{ab} = ˆ∇_{a}P̂, we may rewrite this as

Δ^{2} log Ω = −Δ̂P̂ − (n − 2)P̂^{ab}P̂_{ab} + 2P̂^{2}
+ 2(n − 4)Υ^{a} ˆ∇_{a}P̂ + 2(n − 4)Υ^{a}Υ_{a}P̂
− (n − 2)(n − 4)Υ^{a}Υ^{b}P̂_{ab} + \frac{1}{4}(n − 2)(n − 4)Υ^{a}Υ_{a}Υ^{b}Υ_{b}

(12) and, in particular, conclude that when n = 4,

Q = 2P^{2} − 2P^{ab}P_{ab} − ΔP.  (13)

Though it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression (1) in dimension 4.

Of course, we may continue in the vein, further differentiating (38) to obtain a formula for Δ^{k} log Ω expressed in terms of complete contractions of P̂_{ab}, its hatted derivatives, and Υ_{a}. With increasing k, this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naive derivation of Q fails for a general metric. Nevertheless, there are already some questions in the conformally flat case.

Conundrum: Find a formula for Q in the conformally flat case. Show that the procedure outlined above produces a formula for Q.

In fact, there is a tractor formula for the conformally flat Q. This is not the place to explain the tractor calculus but, for those who know it already:–

□ [ \begin{matrix}n − 2 \\ 0 \\ −P \end{matrix} ] = [ \begin{matrix}0 \\ 0 \\ Q \end{matrix} ]

where

□ = D_{A} ⋯ D_{B}(Δ − \frac{n − 2}{4(n − 1)}R) \underbrace{D^{B} ⋯ D^{A}}_{(n−4)/2} .

Unfortunately, this formula hides a lot of detail and does not seem to be of much immediate use. It is not valid in the curved case.

Recall that, like Q, the Pfaffian is an n-form canonically associated to a Riemannian metric on an oriented manifold in even dimensions. It is defined as a complete contraction of n/2 copies of the Riemann tensor with two copies of the volume form. For example, in dimension four it is

E = ϵ^{abpq}ϵ^{cdrs}R_{abcd}R_{pqrs},

where ϵ_{abcd} is the volume form normalised, for example, so that

ϵ^{abcd}ϵ_{abcd} = 4! = 24.

Therefore, in four dimensions,

E = 4R_{ab}^{ab}R_{cd}^{cd} − 16R_{ab}^{ac}R_{cd}^{bd} + 4R_{ab}^{cd}R_{cd}^{ab}
= 4R^{2} − 16R_{b}^{c}R_{c}^{b} + 4C_{abcd}C^{abcd} + 32P^{ab}P_{ab} + 16P^{2}
= 144P^{2} − 16(4P^{ab}P_{ab} + 8P^{2}) + 4C_{abcd}C^{abcd} + 32P^{ab}P_{ab} + 16P^{2}
= 32P^{2} − 32P^{ab}P_{ab} + 4C_{abcd}C^{abcd}.

The integral of the Pfaffian on a compact manifold is a multiple of the Euler characteristic. In dimension 4, for example,

∫_{M} E = 128π^{2} χ(M).

Notice the simple relationship between Q and E in dimension 4:–

Q = \frac{1}{16}E − \frac{1}{4}C^{abcd}C_{abcd} − ΔP.

Of course, it follows from (3) that ∫_{M} Q is a conformal invariant. Also, in the conformally flat case, it follows from a theorem of Branson, Gilkey, and Pohjanpelto that Q must be a multiple of the Pfaffian plus a divergence. However, the link between Q and the Pfaffian is extremely mysterious.

Conundrum: Find a direct link between Q and the Pfaffian in the conformally flat case. Prove directly that ∫_{M} Q is a topological invariant in this case.

Conundrum: Is it true that, on a general Riemannian manifold, Q may be written as a multiple of the Pfaffian plus a local conformal invariant plus a divergence?

Recall the conventions for Weyl structures as in the appendix. In particular, a metric in the conformal class determines a 1-form α_{a}. In fact, a Weyl structure may be regarded as a pair (g_{ab}, α_{a}) subject to equivalence under the simultaneous replacements

g_{ab} ↦ ĝ_{ab} = Ω^{2}g_{ab} and α_{a} ↦ α̂_{a} = α_{a} + Υ_{a} where Υ_{a} = ∇_{a}Ω.

A Riemannian structure induces a Weyl structure by taking the equivalence class with α_{a} = 0 but not all Weyl structures arise in this way. A Weyl structure gives rise to a conformal structure by discarding α_{a}. We may ask how Q-curvature is related to Weyl structures. From the transformation property (3), it follows that Q may be defined for a Weyl

structure as follows. Since Q is a Riemannian invariant, the differential operator P is necessarily of the form f ↦ S^{a}∇_{a}f for some Riemannian invariant linear differential operator from 1-forms to n-forms. Now, if [g_{ab}, α_{a}] is a Weyl structure, choose a representative metric g_{ab} and consider the n-form

Q − S^{a}α_{a},

where Q is the Riemannian Q-curvature associated to g_{ab} and α_{a} is the 1-form associated to g_{ab}. If ĝ_{ab} = Ω^{2}g_{ab}, then

Q̂ − Ŝ^{a}α̂_{a} = Q + S^{a}Υ_{a} − Ŝ^{a}α_{a} − Ŝ^{a}Υ_{a}
= Q + P log Ω − Ŝ^{a}α_{a} − P̂ log Ω
= Q − Ŝ^{a}α_{a}.

(14)

In dimension 4 we can proceed further as follows. From (2) we see that

S^{a}α_{a} = ∇_{b} [∇^{b}∇^{a} + 2R^{ab} − \frac{2}{3}Rg^{ab}] α_{a} = ∇_{b} [∇^{b}∇^{a} + 4P^{ab} − 2Pg^{ab}] α_{a}

and so we may calculate

Ŝ^{a}α_{a} = S^{a}α_{a} + 4∇^{a}(Υ^{b}∇_{[a}α_{b]}).

In combination with (14) we obtain

Q̂ − Ŝ^{a}α̂_{a} = Q − S^{a}α_{a} − 4∇^{a}(Υ^{b}∇_{[a}α_{b]}).

However,

ˆ∇^{a}(α̂^{b} ˆ∇_{[a}α̂_{b]}) = ∇^{a}(α̂^{b}∇_{[a}α_{b]}) = ∇^{a}(α^{b}∇_{[a}α_{b]}) + ∇^{a}(Υ^{b}∇_{[a}α_{b]})

and, therefore,

Q = Q − S^{a}α_{a} + 4∇^{a}(α^{b}∇_{[a}α_{b]})  (15)

is an invariant of the Weyl structure that agrees with Q when the Weyl structure arises from a Riemannian structure.

Conundrum: Can we find such a Q in general even dimensions? Presumably, this would restrict the choice of Riemannian Q.

Though Q given by (15) is an invariant of the Weyl structure, it is not manifestly so. Better is to rewrite it as follows. Using conventions from the appendix, we may write the Schouten tensor (18) of the Weyl structure in terms of the Schouten tensor of a representative metric g_{ab}:–

P_{ab} = P_{ab} + ∇_{a}α_{b} + α_{a}α_{b} − \frac{1}{2}α^{c}α_{c}g_{ab}.

In particular,

P = P + ∇^{a}α_{a} − α^{a}α_{a}
P^{ab}P_{ab} = P^{ab}P_{ab} + (∇^{a}α^{b})(∇_{a}α_{b}) + (α^{a}α_{a})^{2} + 2P^{ab}∇_{a}α_{b}
+ 2P^{ab}α_{a}α_{b} − Pα^{a}α_{a} + 2(∇^{a}α^{b})α_{a}α_{b} − (∇^{a}α_{a})α^{b}α_{b}
D^{a}D_{a}P = D^{a}(∇_{a}P + 2α_{a}P) = ∇^{a}(∇_{a}P + 2α_{a}P)
= ΔP + 2(∇^{a}α_{a})P + 2α_{a}∇^{a}P
= ΔP + Δ∇^{b}α_{b} − 2(Δα^{b})α_{b} − 2(∇^{a}α^{b})(∇_{a}α_{b})
+ 2(∇^{a}α_{a})P + 2(∇^{a}α_{a})∇^{b}α_{b} − 2(∇^{a}α_{a})α^{b}α_{b}
+ 2α_{a}∇^{a}P + 2α_{a}∇^{a}∇^{b}α_{b} − 4α_{a}(∇^{a}α^{b})α_{b}
P^{2} = P^{2} + (∇^{a}α_{a})^{2} + (α^{a}α_{a})^{2}
+ 2P∇^{a}α_{a} − 2Pα^{a}α_{a} − 2(∇^{a}α_{a})(α^{b}α_{b}).

Therefore, recalling the formula (39) for Q in dimension 4,

Q = 2P^{2} − 2P^{ab}P_{ab} − D^{a}D_{a}P
+ 4P^{ab}∇_{a}α_{b} + 4P^{ab}α_{a}α_{b} + Δ∇^{b}α_{b} − 2(Δα^{b})α_{b}
+ 2α_{a}∇^{a}P + 2α_{a}∇^{a}∇^{b}α_{b} − 2P∇^{a}α_{a} + 2Pα^{a}α_{a}

whence, from (15),

Q = 2P^{2} − 2P^{ab}P_{ab} − D^{a}D_{a}P
+ 4P^{ab}∇_{a}α_{b} + 4P^{ab}α_{a}α_{b} + Δ∇^{b}α_{b} − 2(Δα^{b})α_{b}
+ 2α_{a}∇^{a}P + 2α_{a}∇^{a}∇^{b}α_{b} − 2P∇^{a}α_{a} + 2Pα^{a}α_{a}
− ∇_{b} [∇^{b}∇^{a} + 4P^{ab} − 2Pg^{ab}] α_{a} + 4∇^{a}(α^{b}∇_{[a}α_{b]})
= 2P^{2} − 2P^{ab}P_{ab} − D^{a}D_{a}P
+ 4P^{ab}α_{a}α_{b} + 2α_{a}(∇^{a}∇^{b} − ∇^{b}∇^{a})α_{b} + 2Pα^{a}α_{a}
+ 2(∇^{a}α^{b})∇_{a}α_{b} − 2(∇^{a}α^{b})∇_{b}α_{a}
= 2P^{2} − 2P^{ab}P_{ab} − D^{a}D_{a}P + 2(∇^{a}α^{b})∇_{a}α_{b} − 2(∇^{a}α^{b})∇_{b}α_{a}.

However,

2(∇^{a}α^{b})∇_{a}α_{b} − 2(∇^{a}α^{b})∇_{b}α_{a} = 4(∇^{[a}α^{b]})∇_{[a}α_{b]} = 4P^{ab}P_{[ab]}

and so

Q = 2P^{2} − 2P^{ab}P_{ba} − D^{a}D_{a}P

a manifest invariant of the Weyl structure, as required.

Conundrum: Did we really need to go through this detailed calculation? What are the implications, if any, for the operator S : 1-forms → 4-forms?

Conundrum: Can we characterise the Riemannian Q by sufficiently many properties? Do Weyl structures help in this regard?

Tom Branson has suggested that, for two metrics g and ĝ = Ω^{2}g in the same conformal class on a compact manifold M, one should consider the quantity

H[ĝ, g] = ∫_{M}(log Ω)(Q̂ + Q).

That it is a cocycle,

H[ˆĝ, ĝ] + H[ĝ, g] = H[ˆĝ, g],

is easily seen to be equivalent to the GJMS operators P being self-adjoint.

Conundrum: Are there any deeper properties of Branson’s cocycle H[ĝ, g]?

One possible rôle for Q is in a curvature prescription problem:–

Conundrum: On a given manifold M, can one find a metric with specified Q?

One can also ask this question within a given conformal class or within the realm of conformally flat metrics though, of course, if M is compact, then ∫_{M} Q must be as specified by the conformal class and the topology of M. There is also the question of uniqueness:–

Conundrum: When does Q determine the metric up to constant rescaling within a given conformal class?

Since we know how Q changes under conformal rescaling (3), this question is equivalent to

Conundrum: When does the equation P f = 0 have only constant solutions?

On a compact manifold in two dimensions this is always true: harmonic functions are constant. In four dimensions, though there are conditions under which P f = 0 has only constant solutions, there are also counterexamples, even on conformally flat manifolds. The following counterexample is due to Michael Singer and the first author. Consider the metric in local coördinates

\frac{dx^{2} + dy^{2}}{(x^{2} + y^{2} + 1)^{2}} + \frac{ds^{2} + dt^{2}}{(s^{2} + t^{2} − 1)^{2}} .

It is easily verified that it is conformally flat, scalar flat, and has

R_{ab} = 4\frac{dx^{2} + dy^{2}}{(x^{2} + y^{2} + 1)^{2}} − 4\frac{ds^{2} + dt^{2}}{(s^{2} + t^{2} − 1)^{2}} .

From (2) we see that if f is a function of (x, y) alone, then P f = L(L + 8)f, where L is the Laplacian for the two-dimensional metric

\frac{dx^{2} + dy^{2}}{(x^{2} + y^{2} + 1)^{2}} .

More specifically, in these local coördinates

L = (x^{2} + y^{2} + 1)^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) .

It is easily verified that L + 8 annihilates the following functions:–

\frac{x}{x^{2} + y^{2} + 1},  \frac{y}{x^{2} + y^{2} + 1},  \frac{x^{2} + y^{2} − 1}{x^{2} + y^{2} + 1}.

In fact, (x, y) are stereographic coördinates on the sphere and these three functions extend to the sphere to span the spherical harmonics of minimal non-zero energy. On then other hand, the metric

\frac{ds^{2} + dt^{2}}{(s^{2} + t^{2} − 1)^{2}}

is the hyperbolic metric on the disc. We conclude that the Paneitz operator has at least a 4-dimensional kernel on S^{2} × H^{2}. The same conclusion applies to S^{2} × Σ where Σ is any Riemann surface of genus ≥ 2 equipped with constant curvature metric as a quotient of H^{2}. (In fact, the dimension in this case is exactly 4.)

APPENDIX: Curvature Conventions

Firstly, our conventions for conformal weight. A density f of conformal weight w may be identified as a function for any metric in the conformal class. At the risk of confusion, we shall also write this function as f. If however, our choice of metric g_{ab} is replaced by a conformally equivalent ĝ_{ab} = Ω^{2}g_{ab}, then the function f is replaced by f̂ = Ω^{w}f. Quantities that are not conformally invariant can still have a conformal weight with respect to constant rescalings. For example, the scalar curvature has weight −2 in this respect. Explicit conformal rescalings are generally suppressed.

The Riemann curvature is defined by

(∇_{a}∇_{b} − ∇_{b}∇_{a})ω_{c} = R_{abc}^{d}ω_{d}.

The Ricci and scalar curvatures are

R_{ac} = R_{abc}^{b} and R = R^{a}_{a},

respectively. The Schouten tensor is

P_{ab} = \frac{1}{n − 2} (R_{ab} − \frac{R}{2(n − 1)}g_{ab})

and transforms under conformal rescaling by

P̂_{ab} = P_{ab} − ∇_{a}Υ_{b} + Υ_{a}Υ_{b} − \frac{1}{2}g_{ab}Υ^{c}Υ_{c}.  (16)

In particular, if η̂_{ab} = Ω^{2}η_{ab} are two flat metrics, then

∇_{a}Υ_{b} = Υ_{a}Υ_{b} − \frac{1}{2}g_{ab}Υ^{c}Υ_{c},

a tensor version of the Riccati equation. When n = 2, the Schouten tensor itself is not defined but its trace is well-defined:–

P = \frac{1}{2}R P̂ = P − ∇^{a}Υ_{a} = P − Δ log Ω  (17)

and so, if η̂_{ab} = Ω^{2}η_{ab} are two flat metrics, then Δ log Ω = 0.

A Weyl structure is a conformal structure together with a choice of torsion-free connection D_{α} preserving the conformal structure. In other words, if we choose a metric g_{ab} in the conformal class, then

D_{a}g_{bc} = 2α_{a}g_{bc},

determining a smooth 1-form α_{a}. Conversely, α_{a} determines D_{a}:

D_{a}ϕ_{b} = ∇_{a}ϕ_{b} + α_{a}ϕ_{b} + α_{b}ϕ_{a} − α^{c}ϕ_{c}g_{ab},

where ∇_{a} is the Levi-Civita connection for the metric g_{ab}. Let W_{ab} denote the Ricci curvature of the connection D_{a}:–

(D_{a}D_{b} − D_{b}D_{a})V^{c} = W_{ab}^{c}_{d}V^{d} W_{ab} = W_{ca}^{c}_{b}.

We may compute these curvatures in terms of α_{a} and ∇_{a}, for a chosen metric in the conformal class:–

D_{a}D_{b}V^{c} = ∇_{a}(∇_{b}V^{c} − α_{b}V^{c} + α^{c}V_{b} − α_{d}V^{d}δ_{b}^{c})
+ α_{a}(∇_{b}V^{c} − α_{b}V^{c} + α^{c}V_{b} − α_{d}V^{d}δ_{b}^{c})
+ α_{b}(∇_{a}V^{c} − α_{a}V^{c} + α^{c}V_{a} − α_{d}V^{d}δ_{a}^{c})
+ α^{e}(∇_{e}V^{c} − α_{e}V^{c} + α^{c}V_{e} − α_{d}V^{d}δ_{e}^{c})g_{ab}
− α_{a}(∇_{b}V^{c} − α_{b}V^{c} + α^{c}V_{b} − α_{d}V^{d}δ_{b}^{c})
+ α^{c}(∇_{b}V_{a} − α_{b}V_{a} + α_{a}V_{b} − α_{d}V^{d}g_{ba})
− α_{e}(∇_{b}V^{e} − α_{b}V^{e} + α^{e}V_{b} − α_{d}V^{d}δ_{b}^{e})δ_{a}^{c}

so

(D_{a}D_{b} − D_{b}D_{a})V^{c} = (∇_{a}∇_{b} − ∇_{b}∇_{a})V^{c}
− (∇_{a}α_{b})V^{c} + (∇_{a}α^{c})V_{b} − (∇_{a}α_{d})V^{d}δ_{b}^{c}
+ (∇_{b}α_{a})V^{c} − (∇_{b}α^{c})V_{a} + (∇_{b}α_{d})V^{d}δ_{a}^{c}
+ α^{c}α_{a}V_{b} + α_{b}δ_{a}^{c}α_{e}V^{e} − α_{e}α^{e}δ_{a}^{c}V_{b}
− α^{c}α_{b}V_{a} − α_{a}δ_{b}^{c}α_{e}V^{e} + α_{e}α^{e}δ_{b}^{c}V_{a}

whence

\begin{aligned}W_{ab}^{c}_{d} & = & R_{ab}^{c}_{d} − 2δ^{c}_{d}∇_{[a}α_{b]} − 2g_{d[a}∇_{b]}α^{c} + 2δ_{[a}^{c}∇_{b]}α_{d} \\ & & + 2α^{c}α_{[a}g_{b]d} + 2δ_{[a}^{c}α_{b]}α_{d} − 2α_{e}α^{e}δ_{[a}^{c}g_{b]d} \end{aligned}

and

W_{ab} = R_{ab} + (n − 1)∇_{a}α_{b} − ∇_{b}α_{a} + g_{ab}∇^{c}α_{c} + (n − 2)α_{a}α_{b} − (n − 2)α^{c}α_{c}g_{ab}

whose trace is

W = R + 2(n − 1)∇^{c}α_{c} − (n − 1)(n − 2)α^{c}α_{c}.

Therefore,

\frac{1}{n − 2} (W_{ab} − \frac{W}{2(n − 1)}g_{ab}) = P_{ab} + ∇_{a}α_{b} + α_{a}α_{b} − \frac{1}{2}α^{c}α_{c}g_{ab} + \frac{2}{n−2}∇_{[a}α_{b]}.

If two Weyl structures have the same underlying conformal structure, then we may, without loss of generality, represent them as (g_{ab}, α_{a}) and (g_{ab}, α_{a} − Υ_{a}) for the same metric g_{ab} and an arbitrary 1-form Υ_{a}. If we write hatted quantities to denote those computed with respect to (g_{ab}, α_{a} − Υ_{a}), then

for

P_{ab} = \frac{1}{n − 2} (W_{ab} − \frac{2}{n}W_{[ab]} − \frac{W}{2(n − 1)}g_{ab})  (18)

we have the convenient transformation law

P̂_{ab} = P_{ab} − D_{a}Υ_{b} + Υ_{a}Υ_{b} − \frac{1}{2}g_{ab}Υ^{c}Υ_{c}.  (19)

Chapter B: Origins, applications, and generalizations of the Q-curvature

by Tom Branson and Rod Gover

Curvature prescription

Everything below will take place in the setting of Riemannian manifolds (or Riemannian conformal manifolds) of even dimension n. Of course many statements will also be true for odd-dimensional manifolds and/or pseudo-Riemannian (conformal) manifolds, but our main intent is to make this blurb readable. There will be no reference list here, though there are plans to compile a separate reading list (of real papers) on the topic.

A touchstone in Differential Geometry is the Yamabe equation: for n > 2,

(Δ + \frac{n − 2}{4(n − 1)}K) u = \frac{n − 2}{4(n − 1)}K̂u^{\frac{n+2}{n−2}} .  (20)

This gives the conformal change law for the scalar curvature K. That is, if ω is a smooth function and

ĝ = e^{2ω}g, u := e^{\frac{n−2}{2} ω},

the K̂ given by (20) is the scalar curvature of ĝ.

Exercise 1. Show that (20) implies the conformal change law for the conformal Laplacian

Y := Δ + \frac{n − 2}{4(n − 1)}K,

namely

Ŷ = e^{− \frac{n+2}{2} ω}Y e^{\frac{n−2}{2} ω}.

Here, and in such formulas below, the function on the very right is to be viewed as a multiplication operator, so the relation really says that for all smooth functions f,

Ŷ f = e^{− \frac{n+2}{2} ω}Y (e^{\frac{n−2}{2} ω}f) .

When n = 2, the equation governing the conformal change of K is qualitatively different from a PDE standpoint:

Δω + \frac{1}{2}K = \frac{1}{2}K̂e^{2ω}.  (21)

Like the Yamabe equation, this is quasilinear, but in contrast to Yamabe, it has an exponential (as opposed to power) nonlinearity, and it has an inhomogeneity (the \frac{1}{2}K term). (21) is called the Gauss curvature prescription equation, the Gauss curvature in dimension 2 being \frac{1}{2}K.

There is a formal procedure of analytic continuation in dimension (which in fact can be made rigorous) that allows one to guess (or prove) (21) given (20). The Yamabe equation may be rewritten as

Δ (e^{\frac{n−2}{2} ω} − 1) + \frac{n − 2}{4(n − 1)}Ke^{\frac{n−2}{2} ω} = \frac{n − 2}{4(n − 1)}K̂e^{\frac{n+2}{2} ω}.

Note that we have slipped in an extra −Δ1 = 0 on the left. The advantage of this is that, as a power series in \frac{n−2}{2}, all terms in the equation begin at the first power. Dividing by \frac{n−2}{2} and then evaluating at n = 2, we get (21).

Exercise 2 (following C.R. Graham). Make the dimensional continuation argument rigorous by looking at stabilisations of the manifold M, i.e. the (n+p)-dimensional manifolds M × T^{p}, where T^{p} is the standard p-torus.

There is a generalisation of this whole picture to higher order, in which the role of the pair (Y, J) is played by a pair (P_{m}, Q_{m}) consisting of an operator and a local scalar invariant. The P_{m} are the celebrated Graham-Jenne-Mason-Sparling (GJMS) operators, which by construction have the following properties.

• P_{m} exists for m even and m − n ∉ 2ℤ^{+}.

• P_{m} = Δ^{m/2} + LOT. (Here and below, LOT=“lower order terms”.)

• P_{m} is formally self-adjoint.

• P_{m} is conformally invariant in the sense that

P̂_{m} = exp (−\frac{n + m}{2}ω) P_{m} exp (\frac{n − m}{2}ω) .

• P_{m} has a polynomial expression in ∇ and R in which all coefficients are rational in the dimension n.

• P_{m} has the form

δS_{m}d + \frac{n − m}{2}Q_{m},

where Q_{m} is a local scalar invariant, and S_{m} is an operator on 1-forms of the form

(dδ)^{m/2−1} + LOT or Δ^{m/2−1} + LOT.

In this last expression, d and δ are the usual de Rham operators and Δ is the form Laplacian δd + dδ. Note that P_{m} is unable to detect changes in the (dδ)^{m/2−1} term of the principal part of S_{m}, sandwiched as it is between a δ and a d.

With these properties, conditions are right to generalise the Yamabe equation to

(P_{m} + \frac{n − m}{2}Q_{m}) u = \frac{n − m}{2}Q̂_{m}u^{\frac{n+m}{n−m}} ,  m ∉ \{n, n + 2, n + 4, …\},  (22)

where

u := e^{\frac{n−m}{2} ω}.

Analytic continuation in dimension then yields the following analogue of the Gauss curvature prescription equation. If we denote P_{n} and Q_{n} simply by P and Q, then

P ω + Q = Q̂e^{nω}.  (23)

Though there is much more to be said about the Q-curvature, this is probably the central formula of the theory.

Exercise 3. Show that if we have a local invariant B satisfying a conformal change law like (23), Aω + B = B̂e^{nω}, with A a natural differential operator, then necessarily A is conformally invariant in the sense Â = e^{−nω}A.

The fact that P_{m} has an expression with rational dependence on the dimension is crucial to making the analytic continuation rigorous, whether one does it by stablilisation (generalising Exercise 2), or by an algebraic argument using ℝ(n)-linear combinations in a dimension-stable basis of invariants.

Very explicit formulas for P_{m} and Q_{m} are known up to m = 8. The m = 4 case, which was already being discussed in the early 1980’s, is particularly appealing as a source of intuition, since the formulas there are still quite manageable. Let r be the Ricci tensor and let

P := \frac{r − Jg}{n − 2} .

The Paneitz operator is

P_{4} := δS_{4}d + \frac{n − 4}{2}Q_{4},

where

Q_{4} := −2|P|^{2} + \frac{n}{2}J^{2} + ΔJ,

and

S_{4} := dδ + (n − 2)J − 4P · .

Here P· is the natural action of a symmetric 2-tensor on 1-forms.

The transition from m = 2 to m = 4 already points up the fact that the (P_{m}, Q_{m}) are not uniquely determined: if C is the Weyl conformal curvature tensor, we could add a suitable multiple of |C|^{2} to P_{4} without destroying any of its defining properties. (The coefficient should be rational in n, should have a zero at n = 4, and should not have poles that create new “bad” dimensions.)

The m = 4 situation is also already big enough to show that the study of Q is not just a disguised study of the conformal properties of the Pfaffian Pff. One of the salient properties of the Pfaffian is that it can be written as a polynomial in R, without any explicit occurrences of ∇. For example, in dimension 4,

32π^{2}Pff_{4} = |C|^{2} − 8|P|^{2} + 8J^{2}.

But the ΔJ term in Q_{4} is an absolutely essential aspect, and it generalises: see Exercise 6 below.

All the conformal change laws we’ve mentioned are good in odd dimensions, and that for P_{4} in dimension 3 shows that very strange nonlinearities can occur:

(δ(dδ + J − 4P·)d − \frac{1}{2}Q_{4}) u = −\frac{1}{2}Q̂_{4}u^{−7},

where u = e^{ω/2}.

Back to the general case, a celebrated property of Q is the conformal invariance of its integral on compact manifolds:

∫ Q dv_{g} is conformally invariant.

Indeed, since dvĝ = e^{nω}dv_{g},

∫ Q̂ dvĝ = ∫ (Q + P ω)dv_{g}.

But P ω is an exact divergence, by the δ ⋯ d property of P, and thus it integrates to 0, showing invariance. This has an immediate generalisation. If u is a smooth function,

∫ Q̂u dvĝ = ∫ (Q + P ω)u dv_{g}.

Since P is formally self-adjoint, we may move it over to u in the very last term under the integral. If it happens that u ∈ N (P), there is no contribution from this term. Thus

u ∈ N (P) ⊂ N (d) ⇒ ∫ Qu dv_{g} is conformally invariant.

Relativistic considerations

The Einstein equations are obtained by taking the Einstein-Hilbert action ∫ K dv in dimension 4, and taking the total metric variation. This means we take a compactly supported symmetric tensor h and a curve of metrics g_{ε} with (d/dε)|_{ε=0}g_{ε} = h, and compute that

(d/dε)|_{ε=0} (∫_{K} K dv) = ∫ ⟨h, r − \frac{1}{2}Kg⟩dv,

where K is any compact set containing supp(h). Here we may view ⟨·, ·⟩ as the pairing of a covariant with a contravariant symmetric tensor, or as the metric (g_{0}) pairing of two covariant tensors.

Weyl relativity is one proposal for replacing the Einstein-Hilbert action with an action that is invariant under multiplication of the metric by a positive constant: under the variation above,

(d/dε)|_{ε=0} ∫_{K} |C|^{2}dv = ∫ ⟨h, B⟩dv,

where B is the Bach tensor, a trace-free symmetric 2-tensor with the property (when viewed as a contravariant tensor) that under the usual conformal variation,

B̂ = e^{−2ω}B.

One aspect of B is that the nonlinear differential operator

g ↦ B_{g},  (24)

carrying a metric to its Bach tensor, is fourth-order quasilinear. Its linearisation is an interesting fourth-order conformally invariant linear differential operator.

In attempting to generalise this to higher dimensions, it’s clear that ∫ Pff won’t help – its total metric variation is 0, since it’s a topological invariant. Choices like ∫ |C|^{n/2} for n = 8, 12, … are uninteresting because the linearisations of the analogues of the operators (24) have order lower than one might hope for – less than n. In fact these linearisations will even vanish when we vary at a conformally flat metric.

Coming to the rescue of the situation is Q:

(d/dε)|_{ε=0} ∫_{K} Q dv = ∫ ⟨h, A⟩dv,

where A is the Fefferman-Graham tensor, a symmetric contravariant 2-tensor with the conformal invariance law Â = e^{(2−n)ω}A. The linearisation D of the map g ↦ A_{g} has order n. The operator D is conformally invariant in the sense that D̂ = e^{−(n+2)ω}De^{−2ω} when acting on trace-free symmetric contravariant 2-tensors.

Exercise 4. Show that if ∫ S dv is conformally invariant, then its total metric variation tensor C is conformally invariant. Show that if T is any conformally invariant tensor, then the linearisation of the map g ↦ T_{g} is conformally invariant on trace-free perturbations of g (and 0 on pure trace perturbations).

Quantum considerations

Let A be a natural differential operator with positive definite leading symbol, and suppose A is a positive power of a conformally invariant operator. For example, A could be one of the GJMS operators, or it could be the square of the Dirac operator. Then in dimensions 2,4,6, and conjecturally in higher even dimensions,

− log \frac{det Â}{det A} = α \{\frac{1}{2} ∫ ωP ω dv + ∫ ωQ dv\} + ∫ (\overline{F dv} − F dv) + H,  (25)

where α is a constant, F is a local scalar invariant, and H is a term depending on the null space of A. In particular, if the conformally invariant condition N (A) = 0 is satisfied, then H = 0. The determinant involved is the zeta-regularised functional determinant of a positively elliptic operator.

Such formulas are the finite variational formulas corresponding to Polyakov formulas, which are infinitesimal variational formulas for the determinant; these take the form

(− log det A)^{•} = ∫ ω(L + h)  (26)

where L is a local invariant and h depends on the null space of A, and the superscripted bullet denotes the variation (d/dε)|_{ε=0}. Here the curve along which we vary can be any curve of conformal metrics with g^{•} = 2ωg; it is particularly convenient to consider curves g_{ε} = e^{2εω}g_{0} for given g_{0} and ω. As part of the package producing the Polyakov formula, one gets the conformal index property, that

∫ L dv is conformally invariant.

In fact, getting from (26) to (40) may be viewed as a process of finding conformal primitives. We say that a functional F on the conformal class [g_{0}] is a conformal primitive for a local invariant L if (d/dε)|_{ε=0}F = ∫ ωL. Of course this should happen at all possible choices of background metric g_{0}, and all directions of variation ω. This can be said in a more invariant way, following a suggestion of Mike Eastwood. Putting the “running” metric g and the background metric on the same footing in a two-metric functional F(ĝ, g) on the conformal class, we require of a conformal primitive that it be

• alternating and in the ĝ and g arguments;

• cocyclic in the sense that

F(ˆĝ, g) = F(ˆĝ, ĝ) + F(ĝ, g);  (27)

• having variation L in the sense above when varied in ĝ for fixed g.

That the ∫ (\overline{F dv} − F dv) term in (40) should be this way is obvious; for the term involving P and Q, it is a subtle point.

Some local invariants have other local invariants as conformal primitives. For example, since

J^{•} = −2ωJ + Δω,

we have

(J^{n/2})^{•} = −nωJ + \frac{n}{2}J^{n/2−1}Δω,

so

(∫ J^{n/2}dv)^{•} = \frac{n}{2} ∫ ωΔ(J^{n/2−1}).

This makes 2J^{n/2}/n a conformal primitive for Δ(J^{n/2−1}). Q is an example of a local invariant that does not have such a local conformal primitive.

In order to handle these objects more cleanly, let’s view P and Q as being densityvalued objects P and Q, so that a “weight term” involving the conformal factor does not appear explicitly. In other words, replace P f by Pf = P f dv_{g}, and Q by Q = Q dv_{g}. (For readers unfamiliar with densities, not much is lost conceptually in assuming our manifold is oriented and talking about n-forms instead of scalar densities.) Then

Q̂ = Q + Pω, ∫ Q̂ = ∫ Q.  (28)

Let’s also take all the local invariants we consider to be density valued. A local primitive for L will be a functional F with F^{•} = ∫ ωL. The right side of (40) (ignoring the term H, which may be shown to have a conformal primitive, or which may be eliminated by redefining the determinant) goes over to

α \{\frac{1}{2} ∫ ωPω + ∫ ωQ\} + ∫ (F̂ − F) .  (29)

Recall that when we wrote this, we were thinking about a background metric g_{0} and a perturbed metric g_{ω}. But there is an interesting way of rewriting the first term, as

\frac{1}{2}α ∫ ω(Q_{ω} + Q_{0}).  (30)

To eliminate the appearance of a choice (of g_{0}) being made, we define c(ĝ, g) := ω for two conformally related metrics ĝ = e^{2ω}g. The two-metric functional c is alternating and cocyclic as above. From this viewpoint, (30) is (α times) a two-point functional

G(ĝ, g) := \frac{1}{2} ∫ c(ĝ, g)(Qĝ + Q_{g}),

and clearly

G(ĝ, g) = −G(g, ĝ).

The functional (29) is

αG(ĝ, g) + ∫ (Fĝ − F_{g}).  (31)

Since the log-determinant functional will obviously satisfy the cocycle condition (27), and since the second functional in (31) satisfies such a condition, we expect G to behave similarly. One way to see that this expectation is fulfilled is to use the conformal primitive property: for fixed g and ĝ, with G for F above, the two sides of (27) have the same conformal variation (of ˆ ĝ), and the same value at ˆ ĝ = g.

Exercise 5. Show that if g_{0}, g_{ω} = e^{2ω}g_{0}, g_{ζ} = e^{2ω}g_{0}, and g_{η} = e^{2η}g_{0} are 4 conformally related metrics, then

∫ ω(Q_{ζ} − Q_{η}) + ∫ ζ(Q_{η} − Q_{ω}) + ∫ η(Q_{ω} − Q_{ζ}) = 0.

The following conjecture would be enough to prove the conjecture mentioned at the beginning of the section on the form of the determinant quotient.

Conjecture 1. If S is a natural n-form and ∫ S is conformally invariant, then

S = const · Q + L + G,

where L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.

The point of separating these 3 kinds of terms is that ∫ ωL will have a very banal conformal primitive, namely itself, or (to write it in a way that makes the properties of a conformal primitive more apparent),

\frac{1}{2} ∫ c(ĝ, g)(Lĝ + L_{g}).

Q has an interesting conformal primitive, as discussed above. Q is not uniquely defined, but the The Q in the statement of the conjecture could be anyone’s favorite version of Q. In fact, Q is well-defined up to addition of an L.

Then there are the following related conjectures: Conjecture 2. Any S as above may be written

const · Q + L + V,

where V is an exact divergence.

Conjecture 3. Any S as above may be written

const · Pff + L + V.

There are at least 2 filtrations of the local invariants of this type that should be relevant. First, any invariant can be written as a sum of monomial expressions in R and ∇ with k_{∇} + 2k_{R} = n, where k_{∇} (resp. k_{R}) is the number of occurrences of ∇ (resp. R) in the monomial. If an invariant T can be written with k_{∇} ≤ p for each monomial term, let’s say T ∈ P_{p}. Then

P_{0} ⊂ P_{2} ⊂ ⋯ ⊂ P_{n−2}, P_{odd} = 0.

Pff is in the most elite space, P_{0}. The exact divergences inject into P_{2}/P_{0}.

Exercise 6. Use the conformal change law for Q to show that the class of Q in P_{n−2}/P_{n−4} is nontrivial, and agrees with the class of (Δ^{(n−2)/2}J)dv_{g}. This establishes that Pff and Q are “at opposite ends” of the P-filtration.

The other filtration is by the degree of the conformal change law. If

T̂ = T + T_{1}(dω) + ⋯ + T_{n}(dω),

with T_{i} of homogeneity i with respect to scalar multiples of ω, then say T ∈ L_{p} if T_{q} = 0 for q > p. Then local conformal invariants are in L_{0}, and Q is in L_{1}. Pff on the other hand does not look so great in this filtration.

Other routes to Q and its variants

There is an alternative definition of Q which avoids dimensional continuation. We write E for the space of smooth functions, E^{1} for space of smooth 1-forms and define the special section

I^{g} := ( \begin{matrix}2 − n \\ 0 \\ J \end{matrix} )

of the direct sum bundle E ⊕ E^{1} ⊕ E. Let us first set the dimension to be 4, simply present the some results and then explain how this works. Then we get

□I^{g} = ( \begin{matrix}0 \\ 0 \\ Q_{4} \end{matrix} ) ,

where □ is the coupled conformal Laplacian operator. More precisely □ = −∇^{a}∇_{a} + (n − 2)K/(4n − 4), which appears to be the usual formula for the conformal Laplacian (cf. Y above), but now ∇ is a connection which couples the usual metric connection with the connection

∇_{a} ( \begin{matrix}σ \\ μ \\ τ \end{matrix} ) = ( \begin{matrix}∇σ − μ \\ ∇μ + gτ + Pσ \\ ∇τ − μ_{−}^{|} P \end{matrix} ) .

on the sum bundle T := E ⊕E^{1} ⊕E. This bundle T is called the standard tractor bundle and this connection is usually termed the (normal conformal) tractor connection. It is equivalent to a principal bundle structure known as the (normal conformal) Cartan connection. For those who know about Cartan connections we can say that the tractor bundle and connection is an associated bundle and connection for the Cartan bundle. We have used a metric g to express these objects in terms of a Riemannian structure but in fact the bundle and connection are conformally invariant and so descend to well defined structures on a conformal manifold. In fact, to be more accurate, the decomposition of the standard tractor bundle T is really T = E[1] ⊕ E^{1} ⊗ E[1] ⊕ E[−1] where E[w] indicates the space of conformal densities of weight w. The field I^{g} is a section of T ⊗ E[−1]. In this section and the next we are allowing tensors and tractor fields to be density valued, to simplify the notation, but partially suppressing the details of weights involved.

This construction generalises. In each even dimension n there is a conformally invariant differential operator □_{n−2} so that for any metric g we have

□_{n−2}I^{g} = ( \begin{matrix}0 \\ 0 \\ Q_{n} \end{matrix} ) .  (32)

Here I^{g} is as above, while □_{n−2} has the form Δ^{n/2−1} + LOT. The tractor field I^{g} is not conformally invariant, but it does have an interesting conformal transformation. If ĝ is a metric related to g conformally according to ĝ = e^{2ω}g (ω a smooth function) then

Iĝ = I^{g} + Dω,  (33)

where D is a well known second order conformally invariant linear differential operator known as the tractor D operator. From this and (41) it follows that the Q-curvature Q̂_{n}, for ĝ, differs from Q_{n} by a linear conformally invariant operator acting on ω. In fact it follows easily from the definition of □_{n−2} that

□_{n−2}Dω = ( \begin{matrix}0 \\ 0 \\ P_{n}ω \end{matrix} )

where P_{n} is the GJMS operator of order n. So we have recovered the now famous property Q̂_{n} = Q_{n} + P_{n}ω (cf. (28)).

As a final comment on the above story we should clarify the origins of the tractor field I^{g} defined and used above. For those who are familiar with tractors a more enlightening alternative definition is

I_{A}^{g} := −\frac{1}{n}D^{A}σ^{−1}D_{AB}σ.

Here D^{A} is the tractor D operator and D_{AB} is the so-called fundamental D operator. 0 ≠ σ ∈ ΓE[1] is the conformal scale corresponding to the metric g. The point is that these operators are both conformally invariant and under g ↦ ĝ = e^{2ω}g we have σ ↦ e^{ω}σ. Since D_{AB} satisfies a Leibniz rule and σ^{−1}D_{AB}σ is a “logarithmic derivative” the conformal transformation law of I^{g} is no surprise.

The main results above are derived via the ambient metric construction of Fefferman and Graham. Explaining this construction would be a significant detour at this point. Suffice to say that this construction geometrically associates to an n-dimensional conformal manifold M an (n + 2)-dimensional pseudo-Riemannian manifold M̃. The GJMS operators P_{2ℓ} arise from powers of the Laplacian Δ̃^{ℓ}, of M̃, acting on suitably homogeneous functions. The operators □_{2ℓ} arise in a similar way from Δ̃^{ℓ} on appropriately homogeneous sections of the tangent bundle T M̃. Such homogeneous sections correspond to tractor fields on the conformal manifold M. The results above are given by an easy calculation on the ambient manifold. Thus we can take (41) as a definition of the Q-curvature; it is simply the natural scalar field that turns up on the right hand side.

While this definition avoids dimensional continuation, there is still the issue of getting a formula for Q_{n}. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives. This solves the problem for small n. For example

Q_{6} = 8P_{ij|k}P^{i}^{j}_{|}^{k} + 16P_{ij}P^{i}^{j}_{|}_{k}^{k} − 32P_{ij}P^{i}_{k}P^{jk} − 16P_{ij}P^{ij}J+
8J^{3} − 8J_{|}_{k}^{k}J + J_{|}_{j}^{j}_{k}^{k} + 16P_{ij}P_{kl}C^{ikjl}.

While such formulae shed some light on the nature of the Q-curvature it would clearly be ideal to give a general formula or simple inductive formula. From the angle discussed here the missing information is a general formula for the operators □_{n−2}.

Problem 1: Give general formulae or inductive formulae for the operators □_{2ℓ}.

This seems to be a difficult problem. In another direction there is another exercise to which we already have some answers. One of the features of the Q-curvature is that it “transforms by a linear operator” within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law

N ĝ = N^{g} + Lω,  (34)

L being some universal linear differential operator. (Here ω has the usual meaning; ĝ = e^{2ω}g.) Problem 2: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.)

From the transformation law for I^{g} above, we can evidently manufacture solutions to this problem. We have observed already that I^{g} is a section of the bundle T [−1] := T ⊗E[−1]. If P is any scalar (or rather density) valued natural conformally invariant differential operator which acts on T [−1] then P can act on I^{g}, and P I^{g} has a conformal transformation of the the form (43). Using the calculus naturally associated to tractor bundles (or equally effectively, using the ambient metric) it is in fact a simple matter to write down examples, and the possibilities increase with dimension. This is most interesting when the resulting scalar field gives a possible modification to the original Q-curvature. For those familiar with densities this means that P should take values in densities of weight −n; this is the weight at which densities that can be integrated on a conformal manifold. For example, in any dimension we may take P to be ι(D)|C|^{2} where ι(D) indicates a contracted action of the tractor D operator and the square of the Weyl curvature is here viewed as a multiplication operator. In dimension 6 this takes values in E[−6] and we have

ι(D)|C|^{2}I^{g} = 4Δ|C|^{2}

and

ι(D)|C|^{2}Iĝ = 4Δ|C|^{2} − 16δ|C|^{2}dω.

Note that in this example the conformally invariant “L-operator” δ|C|^{2}d is formally selfadjoint. So for any constant α, Q_{6} + αι(D)|C|^{2}I^{g} is another scalar field with almost the same properties as Q_{6}. It is not so closely related to the GJMS operator P_{6}, but it is related instead to a modification of P_{6} by δ|C|^{2}d. It is clear that solutions to Problem 2 have a role to play in the problem of characterising the Q-curvature and the GJMS operators.

A generalisation: maps like Q

So far we have viewed the Q-curvature as a natural scalar field. It turns out that if instead we view it as an operator then it fits naturally into a bigger picture. To simplify matters suppose we are working with a compact, oriented, but not necessarily connected, manifold of even dimension n. We fix n and so omit n in the notation for Q. We can view Q as a multiplication operator from the closed 0-forms C^{0} (i.e. the locally constant functions) into the space of n-forms E^{n} (which we identify with E[−n] via the conformal Hodge ⋆). With the observations above we have the following properties:

A. Q : C^{0} → E^{n} is not conformally invariant but Q̂ = Q + P_{n}ω, where P_{n} is a formally self-adjoint operator from 0-forms to n-forms. P_{n} has the form dMd which implies the next properties.

B. Q : C^{0} → H^{n}(M) is conformally invariant and non-trivial in general.

C. If c ∈ C^{0} and u ∈ N (P_{n}) then ∫ uQc is conformally invariant.

D. (See the discussion immediately below.) In each choice of metric Q : E^{0} → E[−n] is

formally self-adjoint. E. (See the discussion immediately below.) Q1 is the Q-curvature.

The last properties are trivial since the operator is multiplication by a scalar field and by definition Q1 = Q. However we should note that we can add to Q any differential operator that annihilates constants, and properties 1–3 will be unaffected. Property 4 is suggesting that if we do that, then we should insist that the result is formally self-adjoint.

The idea now is to look for analogous operators on other forms. We write C^{k} for the space of closed k-forms. Consider the operator

M^{g} = dδ + 2J − 4P♯ : E^{n/2−1} → E^{n/2−1}[−2].  (35)

(Note that by the conformal Hodge star E^{n/2−1}[−2] ≅ E^{n/2+1}, so we can also view this as an operator into (n/2 + 1)-forms.) Then:

Exercise 7. On C^{n/2−1} we have

M ĝ = M^{g} + βδdω,

where β is some nonzero constant, ĝ = e^{2ω}g, and in the display ω is viewed as a multiplication operator.

Note that the conformal variation term δd is the Maxwell operator and is formally self-adjoint. So M^{g} satisfies the analogue of property 1 above. The analogue of property 3 is an immediate consequence, i.e., ∫ ⟨u, Mc⟩ is conformally invariant where now c is a closed (n/2−1)-form and u ∈ N (δd) (so in fact by compactness u, c are both closed). Next observe, by inspection, that M^{g} is formally self-adjoint. So we have analogues for 1,3,4. There is also a bonus property, which is clear from the transformation law displayed:

δM^{g}d : E^{n/2−2} → E^{n/2−2}[−4]

is a non-trivial conformally invariant operator. In dimension 4 this is the Paneitz operator.

So finally we need an analogue for property 2. It is clear that M^{g} is conformally invariant as a map C^{n/2−1} → E^{n/2−1}[−2]/R(δ), so this is an analogue. But we can do more. There is no reason to suppose the image is co-closed. On the other hand note that δM^{g} is conformally invariant on C^{n/2−1} and so we have the following:

\begin{matrix}M^{g} : H^{n/2−1} → H^{n/2+1}(M) with \\ H^{n/2−1} := N (δM : C^{n/2−1} → E^{n/2−2}[−4]) \end{matrix}  (36)

is conformally invariant. The space H^{n/2−1} may be viewed as the space of “conformal harmonics”. Evidently dim(H^{n/2−1}) is not always the same as the Betti number b^{n/2−1}, but the elliptic coercivity of the pair (d, δM) gives it a good chance of returning the Betti number off a set of conformal structures that is somehow small. One should also check that the map (36) is non-trivial.

Fact: Let M = S^{p} × S^{q}, where p = n/2 − 1, q = n/2 + 1, with the standard Riemannian structure. Then ϕ ∈ H^{p} if and only if ϕ is harmonic. Furthermore, the map (36) is nontrivial.

In some recent work the authors have used the ambient metric, and its relationship to tractors, to show that the above construction generalises along the following lines: There are operators M_{k}^{g} : E^{k} → E^{n−k} (k ≤ n/2 − 1), given by a uniform construction, with the following properties:

A. M_{k}^{g} : C^{k} → E^{n−k} has the conformal transformation law M ĝ ; k = M_{k}^{g} + L_{k}ω, where L_{k} is a formally self-adjoint operator from k-forms to (n − k)-forms, and is a constant multiple of dM_{k+1}^{g}d.

B. H^{k} := N (dM_{k}^{g} : C^{k} → E^{n−k+1}) is a conformally invariant subspace of C^{k} and M_{k} : H^{k} → H^{n−k}(M) is conformally invariant. There are conformal manifolds on which M_{k} is non-trivial.

C. If c ∈ C^{k} and u ∈ N (L_{k}) then

∫ ⟨u, M_{k}^{g}c⟩

is conformally invariant. D. For each choice of metric g, M_{k}^{g} : E^{k} → E^{n−k} is formally self-adjoint. E. M_{0}^{g}1 is the Q-curvature.

From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator E^{n/2−1} → E^{n/2−1}[−2]), and the explicit formula (44) it is clear M_{n/2−1}^{g} is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the M_{k} generally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M_{k}^{g} give a genuine generalisation of the Q-curvature to an operator on closed forms.

Problems k: There are analogues for the operators M_{k}^{g} of most of the conundrums and problems for the Q-curvature.

Chapter C: Open problems Conformal Structure in Geometry, Analysis, and Physics August 12 to 16, 2003 at the American Institute of Mathematics, Palo Alto, California

I. Problems suggested by the participants Thomas Branson. Anti-conformal perturbations. Problem 1a: Given any functional of the metric that is well understood conformally, is there information that can arise going across conformal classes? If the functional is the integral of a local invariant we can obtain information by computing its anti-conformal variation. If the functional is a nonlocal spectral invariant, like the functional determinant, then it is even a challenge to compute the anti-conformal deformation. Problem 1b: How to obtain the information that arises going across conformal classes? Problem 1c: Study variational problems arising from conformally invariant problems. Michael Eastwood. Problem 2: Find an explicit relation between Q and Pff(R) in the conformally flat case. Problem 3: Is there a global ambient metric construction? Problem 4: Can we explicitly write Q in dimension 6 uniquely as constant times Pff(R) plus local conformally invariant plus divergence? Answer to problem 4: Robin Graham reports the answer to be YES. Alice Chang. General problems in conformal geometry: Problem 5a: How to decide which curvature invariants have a conformal primitive? For example on manifold M, we have Δ(J^{n/2−1}) has \frac{2}{n}J^{n/2} as a conformal primitive, i.e.

(∫ \frac{2}{n}J^{n/2})^{•} (ω) = Δ(J^{n/2−1})

for all smooth function ω on M, see “Origins, applications and generalizations of the Qcurvature” by T. Branson and R. Gover. Available through http://www.aimath.org. Problem 5b: What characterizes such curvature invariants? A related problem is posed by T. Branson: On M^{n}, Q curvature is a local invariant (of density weight −n) which does not have a conformal primitive. The local invariants that have conformal primitives form a vector subspace, say L′, of the space of local invariants L. Thus the quotient space L/L′ is the space which measures “how many things” do not have a conformal primitive. There are also local conformal invariants, L′′ say.

Problem 6: Is L/(L′ + L′′) one-dimensional and generated by the class of Q?

Problem 7: On M^{4}, Gursky (“The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE.” Comm. Math. Phys., 207(1):131-143, 1999.) proved that if Sc > 0, and if ∫ Q > 0, then the Paneitz operator P_{4} is positive with its kernel consisting of constants. The original proof given by Gursky depends on estimates of solution of some non-linear PDE. Can one also see this fact from the construction method of the general GJMS operators?

Claude LeBrun. Problem 8: Explicitly expess the Gauss-Bonnet integrand as a sum of σ_{n/2}(P) plus terms involving the Weyl curvature, and then use this to explicitly understand relationships between Q and topology.

Problem 9: Given a compact manifold of even dimension > 2, show that there exists a sequence of metrics such that ∫ Q → +∞.

Robin Graham Problem 10: If n ≥ 4 is even, is there a nonzero scalar conformal invariant of weight −n which is expressible as a linear combination of complete contractions of the tensors ∇^{l}P, l ≥ 0?

If the answer to this question is no, then the Q-curvature defined via the ambient metric construction is uniquely determined by its transformation law in terms of the GJMS operator P_{n} and the fact that it can be written just in terms of P and its derivatives. The answer is no if n = 4. It is worth pointing out that there are scalar conformal invariants of more negative weight which can be so expressed: the norm squared of the Bach tensor is of this form if n = 4, as is the norm squared of the ambient obstruction tensor in higher even dimensions.

Problem 11: If n ≥ 4 is even, is the GJMS operator P_{n} the only natural differential operator with principal part Δ^{n/2} whose coefficients can be expressed purely in terms of the tensors ∇^{l}P, l ≥ 0, and which is conformally invariant from E(0) to E(−n)?

If the answer is yes, then this gives a characterization of the GJMS operator P_{n}. Combined with a negative answer to Problem 11, this would provide a unique specification of Q.

Rod Gover Alice Chang and Jie Qing have an order 3 operator P_{3}, on 3-manifolds (boundary of a 4-dimensional manifold, or embedded in a 4-dimensional manifold). There is a version of Q_{3} associated to this P_{3}.

Problem 12: What sort of information is encoded by Q_{3} and/or ∫ Q_{3}?

Helga Baum On a spin manifold (M, g) with spin bundle S, we have two conformally covariant operators. The Dirac operator D_{g} and the twistor operator P_{g}. If ∇^{S} represents the spin connection then,

∇^{S} : Γ(S) → Γ(T^{∗}M ⊗ S) ≅ Γ(S) ⊕ Γ(T_{w})

and we define D_{g} = pr_{1}∇^{S} and P_{g} = pr_{2}∇^{S}, with pr_{i} the projection on the i-th factor. Let h(g) be the dimension of ker(D_{g}) (harmonic spinors) and let t(g) be the dimension of ker(P_{g}) (twistor spinors/conformal Killing spinors). Both numbers are conformal invariants. In case of Riemannian conformal structures these invariants are rather well studied. In the Lorentzian case much less is known.

Problem 13: Find all Lorentzian conformal structures (M, [g]) with t(g) > 0 or h(g) > 0.

Problem 14: How t(g) and h(g) relate to other conformal invariants?

Problem 15: Relate t(g) to the holonomy of conformal Cartan connections.

Problem 16: Relate h(g) to the dynamic of null geodesics.

Problem 17: Describe conformally flat Lorentzian manifolds with h(g) > 0 or t(g) > 0.

II. Problems extracted from the document “A Primer on Q-curvature” by M. Eastwood and J. Slovàck. ^{1}

In the conformally flat case, locally by setting g_{ab} = Ω^{2}η_{ab} where η_{ab} is flat, then

Q = Δ^{n/2} log Ω,  (37)

where Δ is the ordinary Laplacian in Euclidean space with η_{ab} as metric. For this construction of Q to be well-defined it is necessary that, if also g_{ab} = Ω̂^{2}η̂_{ab}, then

Δ^{n/2} log Ω = Δ̂^{n/2} log Ω̂.

This reduces to two facts:– fact 1:: Δ^{n/2} is conformally invariant on flat space. fact 2:: if g_{ab} is itself flat, then Δ^{n/2} log Ω = 0.

The second of these is necessary in order that (37) be well-defined. There is a Lie algebraic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalized Verma modules for 𝔰𝔬(n + 1, 1).

Problem 18: Deduce fact 2 from fact 1 or vice versa. Alternatively, construct a Lie algebraic proof of fact 2.

About a formula for Q, Eastwood and Slováck have deduce:

Δ^{2} log Ω = −Δ̂P̂ − (n − 2)P̂^{ab}P̂_{ab} + 2P̂^{2}
+ 2(n − 4)Υ^{a} ˆ∇_{a}P̂ + 2(n − 4)Υ^{a}Υ_{a}P̂
− (n − 2)(n − 4)Υ^{a}Υ^{b}P̂_{ab} + \frac{1}{4}(n − 2)(n − 4)Υ^{a}Υ_{a}Υ^{b}Υ_{b}.

(38)

1This section is the recompilation of the conundra in that document. Refer to the original for more details.

Though it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression in dimension 4,

Q = 2P^{2} − 2P^{ab}P_{ab} − ΔP.  (39)

It is possible, by further differentiating (38), to obtain a formula for Δ^{k} log Ω expressed in terms of complete contractions of P̂_{ab}, its hatted derivatives, and Υ_{a}. With increasing k, this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naive derivation of Q fails for a general metric.

Problem 19: Find a formula for Q in the conformally flat case. Show that the procedure outlined by Eastwood and Slováck produces a formula for Q.

In the conformally flat case, it follows from a theorem of Branson, Gilkey, and Pohjanpelto that Q must be a multiple of the Pfaffian plus a divergence.

Problem 20: Find a direct link between Q and the Pfaffian in the conformally flat case. Prove directly that ∫_{M} Q is a topological invariant in this case.

Problem 21: Is it true that, on a general Riemannian manifold, Q may be written as a multiple of the Pfaffian plus a local conformal invariant plus a divergence?

See Problem 4 for the 6 dimensional case. Also, T. Branson has appointed that if it is true that any local invariant L of density weight −n has the form

constant_{L}Pff + divergence_{L} + (local conformal invariant)_{L}

where L signals the dependence on L then, in this decomposition for Q, we have constant_{Q} ≠ 0. In fact we know constant_{Q} exactly, since we know (the constant values of) Q and Pff on the sphere.

How is Q-curvature related to Weyl structures? Q may be defined for a Weyl structure as follows. Since Q is a Riemannian invariant, the differential operator P is necessarily of the form f ↦ S^{a}∇_{a}f for some Riemannian invariant linear differential operator from 1-forms to n-forms. Now, if [g_{ab}, α_{a}] is a Weyl structure, choose a representative metric g_{ab} and consider the n-form

Q − S^{a}α_{a},

where Q is the Riemannian Q-curvature associated to g_{ab} and α_{a} is the 1-form associated to g_{ab}. If ĝ_{ab} = Ω^{2}g_{ab}, then

Q̂ − Ŝ^{a}α̂_{a} = Q − Ŝ^{a}α_{a}.

In dimension 4, Eastwood and Slováck have appointed that

Q − S^{a}α_{a} + 4∇^{a}(α^{b}∇_{[a}α_{b]})

is an invariant of the Weyl structure that agrees with Q when the Weyl structure arises from a Riemannian structure.

Problem 22: Can we find such a Q in general even dimensions? Presumably, this would restrict the choice of Riemannian Q.

Though Q is an invariant of the Weyl structure, it is not manifestly so. With a detailed calculation, Eastwood and Slov‘ack have shown that in dimension 4:

Q = 2P^{2} − 2P^{ab}P_{ba} − D^{a}D_{a}P

a manifest invariant of the Weyl structure. Problem 23: Did we really need to go through that detailed calculation? What are the implications, if any, for the operator S : 1-forms → 4-forms? Problem 24 a: Can we characterise the Riemannian Q by sufficiently many properties? Problem 24 b: Do Weyl structures help in this regard? Tom Branson has suggested that, for two metrics g and ĝ = Ω^{2}g in the same conformal class on a compact manifold M, one should consider the quantity

H[ĝ, g] = ∫_{M}(log Ω)(Q̂ + Q).

That it is a cocycle,

H[ˆĝ, ĝ] + H[ĝ, g] = H[ˆĝ, g],

is easily seen to be equivalent to the GJMS operators P being self-adjoint. Problem 25: Are there any deeper properties of Branson’s cocycle H[ĝ, g]? One possible rôle for Q is in a curvature prescription problem: Problem 26: On a given manifold M, can one find a metric with specified Q? One can also ask this question within a given conformal class or within the realm of conformally flat metrics though, of course, if M is compact, then ∫_{M} Q must be as specified by the conformal class and the topology of M. Problem 27: When does Q determine the metric up to constant rescaling within a given conformal class? Since we know how Q changes under conformal rescaling:

Q̂ = Q + P log Ω,

where P is a linear differential operator from functions to n-forms whose symbol is Δ^{n/2} this question is equivalent to Problem 28: When does the equation P f = 0 have only constant solutions? On a compact manifold in two dimensions this is always true: harmonic functions are constant. In four dimensions, though there are conditions under which P f = 0 has only constant solutions, there are also counterexamples, even on conformally flat manifolds.

III. Problems extracted from the document “Origins, applications, and generalizations of the Q-curvature” by T. Branson and R. Gover. ^{2}

Let A be a natural differential operator with positive definite leading symbol, and suppose A is a positive power of a conformally invariant operator. For example, A could be one of the GJMS operators, or it could be the square of the Dirac operator. Then in dimensions 2,4,6,

− log \frac{det Â}{det A} = α \{\frac{1}{2} ∫ ωP ω dv + ∫ ωQ dv\} + ∫ (\overline{F dv} − F dv) + H,  (40)

where α is a constant, F is a local scalar invariant, and H is a term depending on the null space of A. In particular, if the conformally invariant condition N (A) = 0 is satisfied, then H = 0. The determinant involved is the zeta-regularized functional determinant of a positively elliptic operator.

Problem 29: ^{3} Is (40) true in higher even dimensions?

The following conjecture would be enough to answer the previous problem.

Problem 30: If S is a natural n-form and ∫ S is conformally invariant, then

S = const · Q + L + G,

where L is a local conformal invariant and G has a local conformal primitive. That is, there is a local invariant F for which the conformal variation of ∫ F is ∫ ωG.

Problem 31: Is it possible to write any S, as in Problem 30, in the form

const · Q + L + V,

where V is an exact divergence?

Problem 32: Is it possible to write any S, as in Problem 30, in the form

const · Pff + L + V?

Other routes to Q. There is an alternative definition of Q which avoids dimensional continuation. Let E be the space of smooth functions, let E^{1} be space of smooth 1-forms and define the special section

I^{g} := ( \begin{matrix}2 − n \\ 0 \\ J \end{matrix} )

of the direct sum bundle E ⊕ E^{1} ⊕ E. In dimension 4:

□I^{g} = ( \begin{matrix}0 \\ 0 \\ Q_{4} \end{matrix} ) ,

2This section is the recompilation of the problems and conjectures in that document. We strongly suggest its lecture for a better understanding of the following problems.

3Problems 29 – 32 are actually conjectures that T. Branson and R Gover address in their document.

where

□ = −∇^{a}∇_{a} + (n − 2)K/(4n − 4),

which appears to be the usual formula for the conformal Laplacian, but now ∇ is a connection which couples the usual metric connection with the connection

∇_{a} ( \begin{matrix}σ \\ μ \\ τ \end{matrix} ) = ( \begin{matrix}∇σ − μ \\ ∇μ + gτ + V σ \\ ∇τ − μ_{−}^{|} P \end{matrix} )

on the sum bundle T := E ⊕E^{1}⊕E. In any even dimension n, there is a conformally invariant differential operator □_{n−2} so that for any metric g :

□_{n−2}I^{g} = ( \begin{matrix}0 \\ 0 \\ Q_{n} \end{matrix} ) .  (41)

Here I^{g} is as above, while □_{n−2} has the form Δ^{n/2−1} + lot (with lot = “lower order terms”). If ĝ is a metric related to g conformally according to ĝ = e^{2ω}g (ω a smooth function) then

Iĝ = I^{g} + Dω,  (42)

where D is a well known second order conformally invariant linear differential operator (the tractor D operator). From this and (41) it follows that the Q-curvature Q̂_{n}, for ĝ, differs from Q_{n} by a linear conformally invariant operator acting on ω. In fact

□_{n−2}Dω = ( \begin{matrix}0 \\ 0 \\ P_{n}ω \end{matrix} )

where P_{n} is the GJMS operator of order n, recovering the property Q̂_{n} = Q_{n} + P_{n}ω.

While this definition avoids dimensional continuation, there is still the issue of getting a formula for Q_{n}. There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives, solving the problem for small n.

Problem 33: Give general formulae or inductive formulae for the operators □_{2ℓ}.

In another direction there is another exercise to which already are some answers. One of the features of the Q-curvature is that it “transforms by a linear operator” within a conformal class. More precisely, it is an example of a natural Riemannian tensor-density field with a transformation law

N ĝ = N^{g} + Lω,  (43)

L being some universal linear differential operator. (Here ω has the usual meaning; ĝ = e^{2ω}g.)

Problem 34: Construct other natural tensor-densities which transform according to (43). (Note that any solution yields a conformally invariant natural operator L.)

Solutions to Problem 34 have a role to play in the problem of characterizing the Q-curvature and the GJMS operators.

Generalizations of Q. In a compact, oriented, but not necessarily connected, manifold of even dimension n, Q can be seen as a multiplication operator from the closed 0-forms C^{0} (i.e. the locally constant functions) into the space of n-forms E^{n} (identified with E[−n] via the conformal Hodge ⋆). This operator has the following properties:

A. Q : C^{0} → E^{n} is not conformally invariant but Q̂ = Q + P_{n}ω, where P_{n} is a formally self-adjoint operator from 0-forms to n-forms. P_{n} has the form dMd which implies the next properties.

B. Q : C^{0} → H^{n}(M) is conformally invariant and non-trivial in general.

C. If c ∈ C^{0} and u ∈ N (P_{n}) then ∫ uQc is conformally invariant. D. In each choice of metric Q : E^{0} → E[−n] is formally self-adjoint. E. Q1 is the Q-curvature.

The idea now is to look for analogous operators on other forms. T. Branson and R. Gover (see math.DG/0309085) have used the ambient metric, and its relationship to tractors, to show that the previous generalizes along the following lines: There are operators M_{k}^{g} : E^{k} → E^{n−k} (k ≤ n/2 − 1), given by a uniform construction, with the following properties:

A. M_{k}^{g} : C^{k} → E^{n−k} has the conformal transformation law M ĝ ; k = M_{k}^{g} + L_{k}ω, where L_{k} is a formally self-adjoint operator from k-forms to (n − k)-forms, and is a constant multiple of dM_{k+1}^{g}d. Here C^{k} is the space of closed k-forms,

B. H^{k} := N (dM_{k}^{g} : C^{k} → E^{n−k+1}) is a conformally invariant subspace of C^{k} and M_{k} : H^{k} → H^{n−k}(M) is conformally invariant. There are conformal manifolds on which M_{k} is non-trivial.

C. If c ∈ C^{k} and u ∈ N (L_{k}) then

∫ ⟨u, M_{k}^{g}c⟩

is conformally invariant. D. For each choice of metric g, M_{k}^{g} : E^{k} → E^{n−k} is formally self-adjoint. E. M_{0}^{g}1 is the Q-curvature.

From the uniqueness of the Maxwell operator at leading order (as a conformally invariant operator E^{n/2−1} → E^{n/2−1}[−2]), and the explicit formula

M^{g} = dδ + 2J − 4P♯ : E^{n/2−1} → E^{n/2−1}[−2],  (44)

it is clear M_{n/2−1}^{g} is not the difference between any conformally invariant differential operator and a divergence (even as an operator on closed forms). A similar argument applies to the M_{k} generally. Thus, from the point of view that the Q-curvature is a non-conformally invariant object that in a deep sense cannot be made conformally invariant, but one which nevertheless determines a global conformal invariant, the operators M_{k}^{g} give a genuine generalization of the Q-curvature to an operator on closed forms.

Problem 35: There are analogues for the operators M_{k}^{g} of most of the Problems in Sections II. and III. for the Q-curvature.

Chapter D: Reference list

A list of references related to the topic of the workshop is available at http://www.und.nodak.edu/instruct/lapeters/bib