On the multiplicity of eigenvalues of conformally covariant operators
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 947-970.

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 fC (M,) for which P e f g has only simple non-zero eigenvalues is a residual set in C (M,). 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 -topology. We also prove that the eigenvalues of P g depend continuously on g in the C -topology, provided P g is strongly elliptic. As an application of our work, we show that if P g acts on C (M) (e.g. GJMS operators), its non-zero eigenvalues are generically simple.

Soit (M,g) une variété riemannienne et P g un opérateur elliptique, auto-adjoint, covariant conforme d’ordre m agissant sur les sections lisses d’un fibré sur M. Nous montrons que si P g n’admet pas d’espaces propres rigides (voir Définition 2.2), l’ensemble des fonctionsfC (M,) pour lesquelles P e f g n’admet que des valeurs propres non nulles est un ensemble résiduel dans C (M,). Ce résultat a comme conséquence que si P g n’admet pas d’espaces propres rigides pour un ensemble dense de métriques, alors toutes les valeurs propres non nulles sont simples pour un ensemble résiduel de métriques dans la topologie C . Nous montrons également que les valeurs propres de P g dependent continûment de g dans la topologie C si P g est fortement elliptique. Comme applications de nos résultats, nous montrons que si P g agit sur C (M), comme dans le cas des opérateurs GJMS, alors les valeurs propres non-nulles de cet opérateur sont génériquement simples.

DOI: 10.5802/aif.2870
Classification: 53A30,  58C40
Keywords: Multiplicity, eigenvalues, conformal geometry, conformally covariant operators, GJMS operators.
Canzani, Yaiza 1

1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada.
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Canzani, Yaiza. On the multiplicity of eigenvalues of conformally covariant operators. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 947-970. doi : 10.5802/aif.2870. http://www.numdam.org/articles/10.5802/aif.2870/

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