58,584 research outputs found
Conformal Dirichlet-Neumann Maps and Poincar\'e-Einstein Manifolds
A conformal description of Poincare-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.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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.Comment: 30 pages. Instructional lecture
Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds
There is a class of Laplacian like conformally invariant differential
operators on differential forms which may be considered the
generalisation to differential forms of the conformally invariant powers of the
Laplacian known as the Paneitz and GJMS operators. On conformally Einstein
manifolds we give explicit formulae for these as explicit factored polynomials
in second order differential operators. In the case the manifold is not Ricci
flat we use this to provide a direct sum decomposition of the null space of the
in terms of the null spaces of mutually commuting second order
factors.Comment: minor changes; we correct typos, add further explanation and clarify
the treatment of the higher order operators in the case of even dimensions;
results unchange
Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds
On locally conformally flat manifolds, we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular, this yields all symmetries of the powers of the Laplacian Δ k , k∈Z>0, on Euclidean space En. The algebra formed by the symmetry operators is described explicitly.A.R.G. gratefully acknowledges support from the Royal Society of New Zealand via Marsden
Grant Nos. 06-UOA-029 and 10-UOA-113. J.S. was supported by the Max-Planck-Institute fur Math- ¨
ematik in Bonn and by the Grant agency of the Czech republic under the Grant No. P201/12/G028
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