ON THE MULTIPLICITY OF EIGENVALUES OF CONFORMALLY COVARIANT OPERATORS

Let (M, g) be a compact Riemannian manifold and P-g an elliptic, formally self-adjoint, conformally covariant operator of order m acting on smooth sections of a bundle over M. We prove that if P-g has no rigid eigenspaces (see Definition 2.2), the set of functions f is an element of C-infinity(M,R) for which P-ef (g) has only simple non-zero eigenvalues is a residual set in C-infinity(M, R). As a consequence we prove that if P-g has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the C-infinity-topology. We also prove that the eigenvalues of P-g depend continuously on g in the C-infinity-topology, provided Pg is strongly elliptic. As an application of our work, we show that if Pg acts on C-infinity(M) (e.g. GJMS operators), its non-zero eigenvalues are generically simple.