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A. Rod Gover

Publications and source records attributed to A. Rod Gover.

90 records · Page 5Linked to original sources
arXiv2007

Four dimensional conformal C-spaces

We investigate the structure of conformal $C$-spaces,a class of Riemmanian manifolds which naturally arises as aconformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension 4 we obtain a full answer to this question and also investigate the incidence of the Bach condition on this class of metrics. This is related to earlier results obtained in the Einstein-Weyl context.

math.DG↗
arXiv2006

A sub-product construction of Poincare-Einstein metrics

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincare metrics we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman-Graham) obstruction tensor vanishes.

math.DG↗
arXiv2006

A holonomy characterisation of Fefferman spaces

We prove that Fefferman spaces, associated to non--degenerate CR structures of hypersurface type, are characterised, up to local conformal isometry, by the existence of a parallel orthogonal complex structure on the standard tractor bundle. This condition can be equivalently expressed in terms of conformal holonomy. Extracting from this picture the essential consequences at the level of tensor bundles yields an improved, conformally invariant analogue of Sparling's characterisation of Fefferman spaces.

math.DG↗
arXiv2006

Variational status of a class of fully nonlinear curvature prescription problems

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

math.DG↗
arXiv2006

The conformal deformation detour complex for the obstruction tensor

On pseudo-Riemannian manifolds of even dimension $n\geq 4$, with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.

math.DG↗
arXiv2006

The conformal Killing equation on forms -- prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on $k$-forms to a twisting of the conformal Killing equation on (k - l)-forms for various integers l. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.

math.DG↗
arXiv2005

Pontrjagin forms and invariant objects related to the Q-curvature

We clarify the conformal invariance of the Pontrjagin forms by giving them a manifestly conformally invariant construction; they are shown to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the $k$-form operators $Q_k$ of \cite{ddd}, which generalise the Q-curvature, retain a key aspect of the $Q$-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the $Q_k$ give rise to conformally invariant quadratic forms $Θ_k$ on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at $k=n/2$) and the Pfaffian (at $k=0$). Using a different construction, we show that the $Q_k$ operators yield a generalisation of the period map which maps conformal structures to Lagrangian subspaces of the direct sum $H^k\oplus H_k$ (where $H_k$ is the dual of the de Rham cohomology space $H^k$). We couple the $Q_k$ operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalise to $Q_k$-operators the Q-curvature prescription problem.

math.DG↗
arXiv2005

The ambient obstruction tensor and the conformal deformation complex

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence free and vanishes for a class of metrics which (strictly) includes the class of conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman-Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to the formal problem of obtaining a Ricci-flat ambient metric and that it may equivalently be viewed as the obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian type operators that generalise the conformal Laplacians of Graham-Jenne-Mason-Sparling. As background to these issues, we give an explicit construction of the deformation complex in dimensions n>3, construct two related (detour) complexes, and establish essential properties of the operators in these.

math.DG↗
arXiv2004

Obstructions to conformally Einstein metrics in $n$ dimensions

We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally related to a metric with vanishing Cotton tensor. One set of invariants we derive generalises the set of invariants in dimension four obtained by Kozameh, Newman and Tod. For the conformally Einstein problem, another set of invariants we construct gives necessary and sufficient conditions for a wider class of metrics than covered by the invariants recently presented by M. Listing. We also show that there is an alternative characterisation of conformally Einstein metrics based on the tractor connection associated with the normal conformal Cartan bundle. This plays a key role in constructing some of the invariants. Also using this we can interpret the previously known invariants geometrically in the tractor setting and relate some of them to the curvature of the Fefferman-Graham ambient metric.

math.DG↗
arXiv2004

Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge companion operators which complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator,gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. These constructions are based on a family of operators on closed forms which generalise in a natural way the Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to tractor bundles. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order $n$ in even dimension $n$. In the case of unweighted forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.

math.DG↗
arXiv2004

Conformal de Rham Hodge theory and operators generalising the Q-curvature

We look at several problems in even dimensional conformal geometry based around the de Rham complex. A leading and motivating problem is to find a conformally invariant replacement for the usual de Rham harmonics. An obviously related problem is to find, for each order of differential form bundle, a ``gauge'' operator which completes the exterior derivative to a system which is both elliptically coercive and conformally invariant. Treating these issues involves constructing a family of new operators which, on the one hand, generalise Branson's celebrated Q-curvature and, on the other hand, compose with the exterior derivative and its formal adjoint to give operators on differential forms which generalise the critical conformal power of the Laplacian of Graham-Jenne-Mason-Sparling. We prove here that, like the critical conformal Laplacians, these conformally invariant operators are not strongly invariant. The construction draws heavily on the ambient metric of Fefferman-Graham and its relationship to the conformal tractor connection and exploring this relationship will be a central theme of the lectures.

math.DG↗
arXiv2003

Conformally invariant powers of the Laplacian -- A complete non-existence theorem

We show that on conformal manifolds of even dimension $n\geq 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian $Δ^{k}$ for $k>n/2$. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of $Δ^k$ for $1\le k\le n/2$, is sharp.

math.DG↗
arXiv2003

Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus

We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6, and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on "Standard tractors and the conformal ambient metric construction" (A. Cap and A.R. Gover, math.DG/0207016), where the relationship of the normal standard tractor bundle to the ambient construction is described.

math-ph↗
arXiv2003

CR Invariant powers of the sub-Laplacian

CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Cap, Slovak and Soucek imply in three dimensions the existence of a curved analogue of each such operator in flat space.

math.DG↗
arXiv2002

Standard Tractors and the Conformal Ambient Metric Construction

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to the Cartan bundle and the Cartan connection. This result is applied to obtain a procedure to get tractor formulae for all conformal invariants that can be obtained from the ambient metric construction. We also get information on ambient metrics which are Ricci flat to higher order than guaranteed by the results of Fefferman-Graham.

math.DG↗
arXiv2001

Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds

On Riemannian signature conformal 4-manifolds we give a conformally invariant extension of the Maxwell operator on 1-forms. We show the extension is in an appropriate sense injectively elliptic, and recovers the invariant gauge operator of Eastwood and Singer. The extension has a natural compatibility with the de Rham complex and we prove that, given a certain restriction, its conformally invariant null space is isomorphic to the first de Rham cohomology. General machinery for extending this construction is developed and as a second application we describe an elliptic extension of a natural operator on perturbations of conformal structure. This operator is closely linked to a natural sequence of invariant operators that we construct explictly. In the conformally flat setting this yields a complex known as the conformal deformation complex and for this we describe a conformally invariant Hodge theory which parallels the de Rham result.

hep-th↗
arXiv2001

A conformally invariant differential operator on Weyl tensor densities

We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in algebraic Weyl tensor densities of another weight. For oriented manifolds, this operator reverses duality: For example in the Riemannian case, it takes self-dual to anti-self-dual tensors and vice versa. We also examine the place that this operator occupies in known results on the classification of conformally invariant operators, and we examine some related operators.

hep-th↗