Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds

On pseudo-Riemannian conformal 4-manifolds we give a conformally invariant extension of the Maxwell operator on I-forms. This recovers an invariant gauge operator due to Eastwood and Singer. We show that, in the case of Riemannian signature, the extension is in an appropriate sense injectively elliptic. It 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 explicitly. 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. (C) 2002 Elsevier Science B.V. All rights reserved.