6,382 research outputs found
Spontaneous generation of eigenvalues
We show that the action of conformal vector fields on functions on the sphere
determines the spectrum of the Laplacian (or the conformal Laplacian), without
further input of information. The spectra of intertwining operators (both
differential and non-local) with principal part a power of the Laplacian
follows as a corollary. An application of the method is the sharp form of
Gross' entropy inequality on the sphere. The same method gives the spectrum of
the Dirac operator on the sphere, as well as of a continuous family of nonlocal
intertwinors, and an infinite family of odd-order differential intertwinors.Comment: 19 pages, LaTe
Q-Curvature, Spectral Invariants, and Representation Theory
We give an introductory account of functional determinants of elliptic
operators on manifolds and Polyakov-type formulas for their infinitesimal and
finite conformal variations. We relate this to extremal problems and to the
Q-curvature on even-dimensional conformal manifolds. The exposition is
self-contained, in the sense of giving references sufficient to allow the
reader to work through all details.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
Translation to Bundle Operators
We give explicit formulas for conformally invariant operators with leading
term an -th power of Laplacian on the product of spheres with the natural
pseudo-Riemannian product metric for all .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
Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds
On Riemannian signature conformal 4-manifolds we give a conformally invariant
extension of the Maxwell operator on 1-forms. We show the extension is in an
appropriate sense injectively elliptic, and recovers the invariant gauge
operator of Eastwood and Singer. The extension has a natural compatibility with
the de Rham complex and we prove that, given a certain restriction, its
conformally invariant null space is isomorphic to the first de Rham cohomology.
General machinery for extending this construction is developed and as a second
application we describe an elliptic extension of a natural operator on
perturbations of conformal structure. This operator is closely linked to a
natural sequence of invariant operators that we construct explictly. In the
conformally flat setting this yields a complex known as the conformal
deformation complex and for this we describe a conformally invariant Hodge
theory which parallels the de Rham result.Comment: 30 pages, LaTe
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