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 $f\in C^{\mathrm{\infty }}(M,R)$ for which $P_{e^{f}g}$ has only simple non-zero eigenvalues is a residual set in $C^{\mathrm{\infty }}(M,R)$. 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 $C^m$-topology. We also prove that the eigenvalues of $P_g$ depend continuously on $g$ in the $C^m$-topology, provided $P_g$ is strongly elliptic. As an application of our work, we show that if $P_g$ acts on $C^{\mathrm{\infty }}(M)$ (e.g. GJMS operators), its non-zero eigenvalues are generically simple.