On the multiplicity of eigenvalues of conformally covariant operators

Canzani, Yaiza

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 947-970.

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 fonctions$f\in C^{\infty }(M,ℝ)$ pour lesquelles $P_{e^{f}g}$ n’admet que des valeurs propres non nulles est un ensemble résiduel dans $C^{\infty }(M,ℝ)$. 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 $C^{\infty }$. Nous montrons également que les valeurs propres de $P_g$ dependent continûment de $g$ dans la topologie $C^{\infty }$ si $P_g$ est fortement elliptique. Comme applications de nos résultats, nous montrons que si $P_g$ agit sur $C^{\infty }\left(M\right)$, 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 $f\in C^{\infty }(M,ℝ)$ for which $P_{e^{f}g}$ 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 $C^{\infty }$-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 $C^{\infty }\left(M\right)$ (e.g. GJMS operators), its non-zero eigenvalues are generically simple.

Classification : 53A30, 58C40
Keywords: Multiplicity, eigenvalues, conformal geometry, conformally covariant operators, GJMS operators.
Mot clés : Multiplicité, valeurs propres, géométrie conforme, opérateur covariant conforme, opérateurs GJMS.

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