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 $\hbox{Pff}(R)$ 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 $\hbox{Pff}(R)$ 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 $\Delta(J^{n/2-1})$ has $\frac{2}{n} J^{n/2}$ as a conformal primitive, i.e.

$$\left(\int \frac{2}{n} J^{n/2} \right)^\bullet(\omega) = \Delta(J^{n/2-1})$$

for all smooth function $\omega$ on $M,$ see ``Origins, applications and generalizations of the Q-curvature'' 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 $\mathcal{L}.$ Thus the quotient space $\mathcal{L}/\mathcal{L}'$ is the space which measures ``how many things'' do not have a conformal primitive. There are also local conformal invariants, $\mathcal{L}''$ say.

Problem 6: Is $\mathcal{L}/(\mathcal{L}'+\mathcal{L}'')$ 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 $\hbox{Sc}>0,$ and if $\int Q >0$, 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 $\sigma_{n/2}({\mbox{\sf P}})$ 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 $\int Q \to +\infty$.

Robin Graham

Problem 10: If $n \geq 4$ 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 $\nabla^l {\mbox{\sf P}}$, $l \geq 0$?

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 ${\mbox{\sf P}}$ 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 $n \geq 4$ is even, is the GJMS operator $P_n$ the only natural differential operator with principal part $\Delta ^{n/2}$ whose coefficients can be expressed purely in terms of the tensors $\nabla^l {\mbox{\sf P}}$, $l \geq 0$, and which is conformally invariant from ${\mathcal{E}}(0)$ to ${\mathcal{E}}(-n)$?

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 $\int Q_3?$

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 $\nabla^S$ represents the spin connection then,

$$\nabla^S : \Gamma (S) \to \Gamma (T^*M \otimes S) \cong \Gamma (S) \oplus \Gamma (T_w)$$

and we define $D_g=\hbox{pr}_1 \nabla^S$ and $P_g=\hbox{pr}_2 \nabla^S,$ with $\hbox{pr}_i$ 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 $g_{ab}=\Omega^2\eta_{ab}$ where $\eta_{ab}$ is flat, then

$$Q=\Delta^{n/2}\log\Omega,$$ (1)

where $\Delta$ is the ordinary Laplacian in Euclidean space with $\eta_{ab}$ as metric. For this construction of $Q$ to be well-defined it is necessary that, if also  $g_{ab}=\widehat\Omega^2\widehat\eta_{ab}$, then

$$\Delta^{n/2}\log\Omega=\widehat\Delta^{n/2}\log\widehat\Omega.$$

This reduces to two facts:-

    fact 1:

$\Delta ^{n/2}$ is conformally invariant on flat space.

    fact 2:

if $g_{ab}$ is itself flat, then $\Delta^{n/2}\log\Omega=0$.

The second of these is necessary in order that () 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 ${\mathfrak{so}}(n+1,1)$.

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:

$$\raisebox{-10pt}{\makebox[0pt]{$\begin{array}{rcl}\Delta^2\l... ...2)(n-4)\Upsilon^a\Upsilon_a\Upsilon^b\Upsilon_b. \end{array}$}}$$ (2)

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,

$$Q=2{\mbox{\sf P}}^2-2{\mbox{\sf P}}^{ab}{\mbox{\sf P}}_{ab}-\Delta{\mbox{\sf P}}.$$ (3)

It is possible, by further differentiating (), to obtain a formula for $\Delta^k\log\Omega$ expressed in terms of complete contractions of $\widehat{\mbox{\sf P}}_{ab}$, its hatted derivatives, and $\Upsilon_a$. 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 $\int_MQ$ 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

$$\hbox{constant}_L \hbox{Pff}+ \hbox{divergence}_L + (\hbox{local~conformal~invariant})_L$$

where $L$ signals the dependence on $L$ then, in this decomposition for $Q,$ we have $\hbox{constant}_Q \ne 0.$ In fact we know $\hbox{constant}_Q$ exactly, since we know (the constant values of) $Q$ and $\hbox{Pff}$ 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 $f\mapsto S^a\nabla_af$ for some Riemannian invariant linear differential operator from $1$-forms to $n$-forms. Now, if $[g_{ab},\alpha_a]$ is a Weyl structure, choose a representative metric $g_{ab}$ and consider the $n$-form

$$Q-S^a\alpha_a,$$

where $Q$ is the Riemannian $Q$-curvature associated to $g_{ab}$ and $\alpha_a$ is the $1$-form associated to $g_{ab}$. If $\widehat g_{ab}=\Omega^2g_{ab}$, then

$$\widehat Q-\widehat S^a\widehat\alpha_a = Q-\widehat S^a\alpha_a.$$

In dimension 4, Eastwood and Slováck have appointed that

$$Q-S^a\alpha_a+4\nabla^a(\alpha^b\nabla_{[a}\alpha_{b]})$$

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 ${\mathbf Q}$ in general even dimensions? Presumably, this would restrict the choice of Riemannian $Q$.

Though ${\mathbf Q}$ 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:

$${\mathbf Q}=2\mathbf{P}^2-2\mathbf{P}^{ab}\mathbf{P}_{ba}-D^aD_a\mathbf{P}$$

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 $S:\mbox{1-forms}\to\mbox{4-forms}$?

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 $\widehat g=\Omega^2g$ in the same conformal class on a compact manifold $M$, one should consider the quantity

$$\mathcal{H}[\widehat g,g]=\int_M(\log\Omega)(\widehat Q+Q).$$

That it is a cocycle,

$$\mathcal{H}[\widehat{\widehat g},\widehat g]+\mathcal{H}[\widehat g,g] =\mathcal{H}[\widehat{\widehat g},g],$$

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 $\mathcal{H}[\widehat g,g]$?

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 $\int_MQ$ 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:

$$\widehat Q=Q+P\log\Omega,$$

where $P$ is a linear differential operator from functions to $n$-forms whose symbol is $\Delta ^{n/2}$ this question is equivalent to

Problem 28: When does the equation $Pf=0$ 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 $Pf=0$ 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,

$$-\log\frac{\det\hat A}{\det A} = \alpha \left\{\frac12\int\o... ...\} + \int\left(\overline{F\,dv} - F\,dv \right) + \mathcal{H},$$ (4)

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

Problem 29: ³Is () true in higher even dimensions?

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

Problem 30: If $\mathbf{S}$ is a natural $n$-form and $\int\mathbf{S}$ is conformally invariant, then

$$\mathbf{S}={\rm const}\cdot\mathbf{Q}+\mathbf{L}+\mathbf{G},$$

where $\mathbf{L}$ is a local conformal invariant and $\mathbf{G}$ has a local conformal primitive. That is, there is a local invariant $\mathbf{F}$ for which the conformal variation of $\int\mathbf{F}$ is $\int\omega\mathbf{G}$.

Problem 31: Is it possible to write any $\mathbf{S},$ as in Problem 30, in the form

$${\rm const}\cdot \mathbf{Q}+\mathbf{L}+\mathbf{V},$$

where $V$ is an exact divergence?

Problem 32: Is it possible to write any $\mathbf{S},$ as in Problem 30, in the form

$${\rm const}\cdot{\mathbf{\hbox{Pff}}}+\mathbf{L}+\mathbf{V}?$$

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

$$I^g:= \left(\begin{array}{c} 2-n \\ 0 \\ J \end{array}\right)$$

of the direct sum bundle $\mathcal{E}\oplus \mathcal{E}^1 \oplus \mathcal{E}$. In dimension 4:

$$\Box I^g = \left(\begin{array}{c} 0 \\ 0 \\ Q_4 \end{array}\right) ,$$

where

$$\Box = -\nabla^a\nabla_a+(n-2)K/(4n-4),$$

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

$$\nabla_a \left(\begin{array}{c} \sigma \\ \mu \\ \tau \end{a... ...ebox[0.4em][l]{\tiny $\vert$}}{\mbox{\sf P}}\end{array}\right)$$

on the sum bundle $\mathcal{T}:=\mathcal{E}\oplus \mathcal{E}^1 \oplus \mathcal{E}$. In any even dimension $n,$ there is a conformally invariant differential operator $\Box_{n-2}$ so that for any metric $g:$

$$\Box_{n-2} I^g= \left(\begin{array}{c} 0 \\ 0 \\ Q_{n} \end{array}\right) .$$ (5)

Here $I^g$ is as above, while $\Box_{n-2}$ has the form $\Delta^{n/2-1} + {\rm lot}$ (with ${\rm lot} =$ ``lower order terms''). If $\hat{g}$ is a metric related to $g$ conformally according to $\hat{g}=e^{2\omega}g$ ($\omega$ a smooth function) then

$$I^{\hat{g}} =I^g+ D \omega,$$ (6)

where $D$ is a well known second order conformally invariant linear differential operator (the tractor $D$ operator). From this and () it follows that the $Q$-curvature $\hat{Q}_n$, for $\hat{g}$, differs from $Q_n$ by a linear conformally invariant operator acting on $\omega$. In fact

$$\Box_{n-2} D \omega = \left(\begin{array}{c} 0 \\ 0 \\ P_n \omega \end{array}\right)$$

where $P_n$ is the GJMS operator of order $n,$ recovering the property $\hat{Q}_n= Q_n + P_n\omega.$

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 $\Box_{2\ell}$.

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

$$N^{\hat{g}}= N^g + L\omega ,$$ (7)

$L$ being some universal linear differential operator. (Here $\omega$ has the usual meaning; $\hat{g}=e^{2\omega}g$.)

Problem 34: Construct other natural tensor-densities which transform according to (). (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 $\mathcal{C}^0$ (i.e. the locally constant functions) into the space of $n$-forms $\mathcal{E}^n$ (identified with $\mathcal{E}[-n]$ via the conformal Hodge $\star$). This operator has the following properties:

1. $Q:\mathcal{C}^0\to \mathcal{E}^n$ is not conformally invariant but $\hat{Q}= Q + P_n\omega$, where $P_n$ is a formally self-adjoint operator from 0-forms to $n$-forms. $P_n$ has the form $d M d$ which implies the next properties.
2. $Q:\mathcal{C}^0\to H^n(M)$ is conformally invariant and non-trivial in general.
3. If $c\in \mathcal{C}^0$ and $u\in \mathcal{N}(P_n)$ then $\int u Q c$ is conformally invariant.
4. In each choice of metric $Q:\mathcal{E}^0\to \mathcal{E}[-n]$ is formally self-adjoint.
5. $Q 1$ 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 $M_k^g:\mathcal{E}^k\to \mathcal{E}^{n-k}$ ($k\leq n/2-1$), given by a uniform construction, with the following properties:

1. $M_k^g:\mathcal{C}^k\to \mathcal{E}^{n-k}$ has the conformal transformation law $M^{\hat{g}}_k= M^g_k + L_{k}\omega$, where $L_{k}$ is a formally self-adjoint operator from $k$-forms to $(n-k)$-forms, and is a constant multiple of $d M_{k+1}^g d$. Here $\mathcal{C}^k$ is the space of closed $k$-forms,
2. $\mathcal{H}^k:= \mathcal{N}(d M^g_k: \mathcal{C}^k\to \mathcal{E}^{n-k+1} )$ is a conformally invariant subspace of $\mathcal{C}^k$ and $M_k:\mathcal{H}^k\to H^{n-k}(M)$ is conformally invariant. There are conformal manifolds on which $M_k$ is non-trivial.
3. If $c\in \mathcal{C}^k$ and $u\in \mathcal{N}(L_k)$ then
   
   $$\int \langle u, M_k^g c \rangle$$
   
   
   is conformally invariant.
4. For each choice of metric $g$, $M_k^g:\mathcal{E}^k\to \mathcal{E}^{n-k}$ is formally self-adjoint.
5. $M_0^g 1$ is the $Q$-curvature.

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

$$M^g=d\delta+2J-4{\mbox{\sf P}}\sharp :\mathcal{E}^{n/2-1}\to \mathcal{E}^{n/2-1}[-2],$$ (8)

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.

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