On the multiplicity of eigenvalues of conformally covariant operators

TL;DR

This paper demonstrates that, under certain conditions, the non-zero eigenvalues of conformally covariant operators on compact manifolds are generically simple, and their dependence on the metric is continuous.

Contribution

It proves that for conformally covariant operators without rigid eigenspaces, the set of metrics yielding only simple non-zero eigenvalues is residual, and eigenvalues depend continuously on the metric.

Findings

Non-zero eigenvalues are generically simple for a residual set of metrics.

Eigenvalues depend continuously on the metric in the $C^m$-topology.

For operators like GJMS, non-zero eigenvalues are generically simple.

Abstract

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 $f \in C^\infty(M, R)$ for which $P_{e^fg}$ has only simple non-zero eigenvalues is a residual set in $C^\infty(M,R)$. 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 $C^\infty(M)$ (e.g. GJMS operators), its non-zero eigenvalues are generically simple.