574 research outputs found
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 -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.Comment: 37 page
Equivalence of the coupled-mode and Floquet-Bloch formalisms in periodic optical waveguides
A comparison of two theories used to analyze distributed feedback lasers and periodic optical devices finds them, contrary to some claims, to be formally equivalent
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
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 , , on Euclidean space . The algebra formed by the
symmetry operators is described explicitly.Comment: 33 pages, minor revisions. To appear in J. Math. Phy
- …