118 research outputs found
Poincare-einstein holography for forms via conformal geometry in the bulk
No full textWe study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of Gover and Waldron to a conformally invariant, Poincaré-Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulæ for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of Branson and Gover (2005). This complements the results of Aubry and Guillarmou where associated conformal harmonic spaces parametrise smooth solutions
Developing a tractor calculus on null-infinity of Minkowski space
No full textFull Text is available to authenticated members of The University of Auckland only.In the usual construction of conformal tractor calculus on a pseudo-Riemannian manifold, there is an assumption of nondegeneracy of the metric. So if we were to lift this assumption, we would need to redefine the tractor machinery. This is of particular interest in the study of the asymptotics of space-times, which have metrics that are not positive-definite. These space-times are often modelled as compactified manifolds, with boundary I , referred to as null-infinity, which has the structure of a conformal manifold with a degenerate direction. Then if we want to study conformally invariant objects on I , we are inclined to develop this new tractor calculus. This is done in [13]. with the motivation of modelling the geometry of gravitational waves. We aim to adapt this definition into a more familiar setting by studying the Minkowski case and defining tractor machinery in relation to the ambient tractor bundle
Distinguished curves and submanifolds in conformal geometry
No full textConformal geometry is a weakening of Riemannian geometry where one works with a
smooth manifold equipped with an equivalence class of Riemannian metrics, where two
metrics are equivalent if and only if they define the same angles between curves. Early
interest in conformal equivalence included the question of biholomorphic equivalence of
domains in the complex plane. Interest has also been driven by physics and general relativity, since light in spacetime follows null geodesics, and these only depend on the conformal structure. Recently there has been considerable progress in the study of codimension
one submanifolds in conformal manifolds. At the other extreme, there has also been an
increased understanding of the distinguished curves in conformal manifolds. In this thesis,
we develop a complete basic tractor theory of conformal submanifolds of any codimension
and use this to define a notion of distinguished conformal submanifolds. These distinguished submanifolds coincide with conformal circles and totally umbilic hypersurfaces in
the extremal cases. We emphasize three conformal tractor objects which we show encode
equivalent submanifold data. Our notion of distinguished submanifolds admits characterizations in terms of all three invariants. Our definition immediately leads to a procedure
for proliferation of conserved quantities along these submanifolds. We also obtain a theorem which characterizes our distinguished conformal submanifolds in terms of an incidence
relation and a parallel condition. We use this to show that zero loci of certain solutions
to a conformally invariant equation are, if nonempty, distinguished submanifolds. These
results extend existing results for conformal circles
Projectively Compact Pseudo-Riemannian Manifolds: Boundary Calculus and Natural Operators
No full textWe define a projectively Klein manifold to be a projective manifold with boundary where the interior is equipped with a pseudo-Riemannian metric that is projectively compact such that the boundary value of the extension of its scalar curvature is nowhere vanishing. As a consequence,
there is a conformal structure on the boundary along which the conformal and projective tractor
bundles agree.
A Klein manifold may also be interpreted to be a projective manifold with a boundary
given by the degeneracy locus of a solution of the metrisability equation, which is a first-order
Bernstein-Gelfand-Gelfand equation. From this it follows that there exists a projectively invariant
differential operator called the splitting operator and this defines a symmetric 2-cotractor
that we term the structure tractor. This gives a metric on the projective tractor bundle.
We introduce a notion of special asymptotic boundary scales which make the splittings of
the conformal and projective tractor bundles compatible along the boundary of Klein manifold.
This important observation simplifies the construction of a boundary calculus which relates
ambient projective quantities to their boundary conformal counterparts. By modifying the
Eastwood-Matveev equation, which expresses the prolongation of the metrisability equation
using the tractor calculus, the restriction of the ambient projective tractor connection along the
boundary is explicitly linked to the boundary conformal tractor connection.
An sl(2) algebra is shown to exist on a Klein manifold which is generated by a projective
tractor Laplacian-type operator, the determinant of the solution of the metrisability equation,
and a weight operator. This algebra enables the construction of tangential Laplacian-type operators
along the boundary of the Klein manifold. The generalised conformal Yamabe operator
and generalised Paneitz operator along the boundary of a Klein manifold are calculated, and we
show that generalised Graham-Jenne-Mason-Sparling (GJMS) operators can be constructed on
the boundary of Klein manifolds more generally. These are conformally invariant Laplacian
power operators that include in their coefficients extrinsic embedding data
Invariant prolongation of the Killing tensor equation
No full textThe Killing tensor equation is a first-order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolongation to a linear connection. This connection has the property that parallel sections are in 1–1 correspondence with solutions of the Killing equation. Moreover, this connection is projectively invariant and is derived entirely using the projectively invariant tractor calculus which reveals also further invariant structures linked to the prolongation.A. Rod Gover, Thomas Leistne
The Funk transform as a Penrose transform
No full textThe Funk transform is the integral transform from the space of smooth even functions on the unit sphere S²[subset or is implied by][open face R]³ to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [open face C][open face P]₂ to [open face C][open face P]*ast;₂. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.By Toby N. Bailey Michael G. Eastwood, A. Rod Gover, and Lionel J. Maso
Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds
No full textA conformal description of Poincaré-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles
Invariant differential operators in conformal geometry
No full textFull text is available to authenticated members of The University of Auckland only.We develop a universal and algorithmic construction of invariant differential operators between irreducible bundles in conformal geometry. The classification of such operators in the flat case is well-known in terms of representation theory. The main result of the thesis is a construction of curved analogues of these. We obtain curved analogues in every case save for an exception which exists in every pattern in every even dimension. The operators are described via explicit formulae in tractor calculus. These are closely related to the usual “V-formulae” for invariant operators in Riemannian geometry. The construction follows Eastwood’s curved translation principle which we implement in the conformal tractor calculus. We work in both real and complex setting and for all signatures. Further, we use the developed calculus to study one class of these operators - the conformal Killing operator on forms - in detail. We construct invariant prolongations of the corresponding systems of partial differential equations. Using these, we obtain information about the solution space. In particular, we develop a helicity raising and lowering construction in the general setting, and also on conformally Einstein manifolds
Exceptional invariants in the parabolic invariant theory of conformal geometry
No full textWe provide a construction for the exceptional invariants of certain modules for a parabolic subgroup of a pseudo-orthogonal group. The invariant theory of these modules has applications in conformal geometry.Toby N. Bailey and A. Rod Gove
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