Canzani, Yaiza
On the multiplicity of eigenvalues of conformally covariant operators  [ Sur la multiplicité des valeurs propres d’opérateurs covariants conformes ]
Annales de l'institut Fourier, Tome 64 (2014) no. 3 , p. 947-970
MR 3330160 | Zbl 06387297
doi : 10.5802/aif.2870
URL stable : http://www.numdam.org/item?id=AIF_2014__64_3_947_0

Classification:  53A30,  58C40
Mots clés: Multiplicité, valeurs propres, géométrie conforme, opérateur covariant conforme, opérateurs GJMS.
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.
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.

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