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On the multiplicity of eigenvalues of conformally covariant operators

Abstract

Let (M,g)(M,g) be a compact Riemannian manifold and PgP_g an elliptic, formally self-adjoint, conformally covariant operator of order mm acting on smooth sections of a bundle over MM. We prove that if PgP_g has no rigid eigenspaces (see Definition 2.2), the set of functions fC(M,R)f \in C^\infty(M, R) for which PefgP_{e^fg} has only simple non-zero eigenvalues is a residual set in C(M,R)C^\infty(M,R). As a consequence we prove that if PgP_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 CmC^m-topology. We also prove that the eigenvalues of PgP_g depend continuously on gg in the CmC^m-topology, provided PgP_g is strongly elliptic. As an application of our work, we show that if PgP_g acts on C(M)C^\infty(M) (e.g. GJMS operators), its non-zero eigenvalues are generically simple.Comment: To appear in Annales de l'Institut Fourie

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