Conformal de Rham Hodge theory and operators generalising the Qcurvature
Abstract
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 Qcurvature 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 GrahamJenneMasonSparling. 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 FeffermanGraham and its relationship to the conformal tractor connection and exploring this relationship will be a central theme of the lectures.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404004
 Bibcode:
 2004math......4004G
 Keywords:

 Mathematics  Differential Geometry;
 53A30 (primary);
 53A55;
 35Q99 (secondary)
 EPrint:
 30 pages. Instructional lectures