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 and
in the
conformally flat case.

**Problem 3:** Is there a global ambient metric construction?

**Problem 4:** Can we explicitly write in dimension 6
uniquely as constant times
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 we have
has
as a conformal primitive,
i.e.

for all smooth function on 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 , curvature is a local invariant (of density weight ) which does not have a conformal primitive. The local invariants that have conformal primitives form a vector subspace, say of the space of local invariants Thus the quotient space is the space which measures ``how many things'' do not have a conformal primitive. There are also local conformal invariants, say.

**Problem 6:** Is
one-dimensional and
generated by the class of

**Problem 7:** On , 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 and if , then the Paneitz operator
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
plus terms
involving the Weyl
curvature, and then use this to explicitly understand relationships
between and
topology.

**Problem 9:** Given a compact manifold of even dimension show that there exists a sequence of
metrics such that
.

**Robin Graham**

**Problem 10:** If is even, is there a nonzero
scalar conformal invariant of
weight which is expressible as a linear combination of complete
contractions of the tensors
, ?

If the answer to this question is no, then the -curvature defined via the ambient metric construction is uniquely determined by its transformation law in terms of the GJMS operator and the fact that it can be written just in terms of and its derivatives. The answer is no if . 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 , as is the norm squared of the ambient obstruction tensor in higher even dimensions.

**Problem 11:** If is even, is the GJMS operator
the only natural
differential operator with principal part whose
coefficients can be expressed
purely in terms of the tensors
, , and which is
conformally invariant from
to
?

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

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

**Problem 12:** What sort of information is encoded by
and/or

**Helga Baum** On a spin manifold with spin bundle
we have two conformally
covariant operators. The Dirac operator and the twistor operator
If represents the spin connection then,

and we define and with the projection on the -th factor. Let be the dimension of (harmonic spinors) and let be the dimension of (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
with
or

**Problem 14:** How and relate to other
conformal
invariants?

**Problem 15:** Relate to the holonomy of conformal
Cartan
connections.

**Problem 16:** Relate to the dynamic of null
geodesics.

**Problem 17:** Describe conformally flat Lorentzian manifolds
with
or .

**II. Problems extracted from the document ``A Primer on
-curvature'' by M. Eastwood and
J. Slovàck.**
^{1}

In the conformally flat case, locally by setting
where is flat, then

This reduces to two facts:-

**fact 1:**- is conformally invariant on flat space.
**fact 2:**- if is itself flat, then .

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

About a formula for Eastwood and Slováck have deduce:

It is possible, by further differentiating (), to obtain a formula for expressed in terms of complete contractions of , its hatted derivatives, and . With increasing , this gets rapidly out of hand. Moreover, it is only guaranteed to give in the conformally flat case. Indeed, when this naive derivation of fails for a general metric.

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

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

**Problem 20:** Find a direct link between and the
Pfaffian in the conformally flat
case. Prove directly that is a topological invariant in this
case.

**Problem 21:** Is it true that, on a general Riemannian
manifold, 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 of
density weight has the form

where signals the dependence on then, in this decomposition for we have In fact we know exactly, since we know (the constant values of) and on the sphere.

**How is -curvature related to Weyl structures?** may
be defined for a Weyl
structure as follows. Since is a Riemannian invariant, the
differential
operator is necessarily of the form
for some
Riemannian invariant linear differential operator from -forms to
-forms.
Now, if
is a Weyl structure, choose a representative
metric
and consider the -form

where is the Riemannian -curvature associated to and is the -form associated to . If , then

In dimension 4, Eastwood and Slováck have appointed that

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

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

Though 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:

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
?

**Problem 24 a:** Can we characterise the Riemannian by
sufficiently many properties?

**Problem 24 b:** Do Weyl structures help in this regard?

Tom Branson has suggested that, for two metrics and
in the same conformal class on a compact manifold , one should consider
the
quantity

That it is a cocycle,

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

**Problem 25:** Are there any deeper properties of Branson's
cocycle
?

One possible rôle for is in a curvature prescription problem:

**Problem 26:** On a given manifold , can one find a metric
with specified ?

One can also ask this question within a given conformal class or within the realm of conformally flat metrics though, of course, if is compact, then must be as specified by the conformal class and the topology of .

**Problem 27:** When does determine the metric up to
constant rescaling within a given
conformal class?

Since we know how changes under conformal rescaling:

where is a linear differential operator from functions to -forms whose symbol is this question is equivalent to

**Problem 28:** When does the equation 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 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 -curvature'' by T. Branson and R. Gover.**
^{2}

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

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

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

**Problem 30:** If is a natural -form and
is
conformally invariant, then

where is a local conformal invariant and has a local conformal primitive. That is, there is a local invariant for which the conformal variation of is .

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

where is an exact divergence?

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

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

of the direct sum bundle . In dimension 4:

where

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

on the sum bundle . In any even dimension there is a conformally invariant differential operator so that for any metric

Here is as above, while has the form (with ``lower order terms''). If is a metric related to conformally according to ( a smooth function) then

where is a well known second order conformally invariant linear differential operator (the tractor operator). From this and () it follows that the -curvature , for , differs from by a linear conformally invariant operator acting on . In fact

where is the GJMS operator of order recovering the property

While this definition avoids dimensional continuation, there is still the issue of getting a formula for . 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 .

**Problem 33:** Give general formulae or inductive formulae
for the operators .

In another direction there is
another exercise to which already are some answers. One of the
features of the -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

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

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

**Generalizations of **
In a compact, oriented, but not necessarily connected, manifold of even
dimension
can be seen as a multiplication operator from the closed 0-forms
(i.e. the locally constant functions) into the space of -forms
(identified with
via the conformal Hodge ). This
operator has the following
properties:

- is not conformally invariant but , where is a formally self-adjoint operator from 0-forms to -forms. has the form which implies the next properties.
- is conformally invariant and non-trivial in general.
- If and then is conformally invariant.
- In each choice of metric is formally self-adjoint.
- is the -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 (), given by a uniform construction, with the following properties:

- has the conformal transformation law , where is a formally self-adjoint operator from -forms to -forms, and is a constant multiple of . Here is the space of closed -forms,
- is a conformally invariant subspace of and is conformally invariant. There are conformal manifolds on which is non-trivial.
- If
and
then

is conformally invariant. - For each choice of metric , is formally self-adjoint.
- is the -curvature.

it is clear 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 generally. Thus, from the point of view that the -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 give a genuine generalization of the -curvature to an operator on closed forms.

**Problem 35:** There are analogues for the operators
of most of the Problems in Sections II. and III. for the -curvature.

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