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.Comment: 37 page

Congruence and Metrical Invariants of Zonotopes

Zonotopes are studied from the point of view of central symmetry and how volumes of facets and the angles between them determine a zonotope uniquely. New proofs are given for theorems of Shephard and McMullen characterizing a zonotope by the central symmetry of faces of a fixed dimension. When a zonotope is regarded as the Minkowski sum of line segments determined by the columns of a defining matrix, the product of the transpose of that matrix and the matrix acts as a shape matrix containing information about the edges of the zonotope and the angles between them. Congruence between zonotopes is determined by equality of shape matrices. This condition is used, together with volume computations for zonotopes and their facets, to obtain results about rigidity and about the uniqueness of a zonotope given arbitrary normal-vector and facet-volume data. These provide direct proofs in the case of zonotopes of more general theorems of Alexandrov on the rigidity of convex polytopes, and Minkowski on the uniqueness of convex polytopes given certain normal-vector and facet-volume data. For a zonotope, this information is encoded in the next-to-highest exterior power of the defining matrix.Comment: 23 pages (typeface increased to 12pts). Errors corrected include proofs of 1.5, 3.5, and 3.8. Comments welcom

Conformal invariants from nodal sets. I. Negative Eigenvalues and Curvature Prescription

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension $n\geq 3$. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in the appendix, the 2nd named author and Andrea Malchiodi study the $Q$-curvature prescription problems for non-critical $Q$-curvatures.Comment: v3: final version. To appear in IMRN. 31 page

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

Distributed feedback X-ray lasers in single crystals

There are two main obstacles in the way of obtaining laser action in the X-ray region. The first involves the pumping necessary to obtain the critical inversion. The second one is that of the optical feedback