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Yaiza Canzani
On the multiplicity of eigenvalues of conformally covariant operators
(Sur la multiplicité des valeurs propres d’opérateurs covariants conformes)
Annales de l'institut Fourier, 64 no. 3 (2014), p. 947-970, doi: 10.5802/aif.2870
Article PDF | Reviews MR 3330160 | Zbl 06387297
Class. Math.: 53A30, 58C40
Keywords: Multiplicity, eigenvalues, conformal geometry, conformally covariant operators, GJMS operators.

Résumé - 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 $f\in C^\infty (M, \mathbb{R})$ for which $P_{e^fg}$ has only simple non-zero eigenvalues is a residual set in $C^\infty (M,\mathbb{R})$. 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^\infty $-topology. We also prove that the eigenvalues of $P_g$ depend continuously on $g$ in the $C^\infty $-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.

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