1,606 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
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 , 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
. 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 -curvature prescription problems for non-critical
-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
- …