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.
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
as a conformal primitive,
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?
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 .
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,
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
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
How is -curvature related to Weyl structures? may
be defined for a Weyl
structure as follows. Since is a Riemannian invariant, the
operator is necessarily of the form
Riemannian invariant linear differential operator from -forms to
is a Weyl structure, choose a representative
and consider the -form
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:
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
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:
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
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: 3Is () true in higher even dimensions?
The following conjecture would be enough to answer the previous problem.
Problem 30: If is a natural -form and
conformally invariant, then
Problem 31: Is it possible to write any as in
Problem 30, in the form
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
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
within a conformal class.
More precisely, it is an example of a natural Riemannian tensor-density
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:
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:
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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