Let be a compact Riemannian manifold and an elliptic, formally self-adjoint, conformally covariant operator of order acting on smooth sections of a bundle over . We prove that if has no rigid eigenspaces (see Definition 2.2), the set of functions for which has only simple non-zero eigenvalues is a residual set in . As a consequence we prove that if 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 -topology. We also prove that the eigenvalues of depend continuously on in the -topology, provided is strongly elliptic. As an application of our work, we show that if acts on (e.g. GJMS operators), its non-zero eigenvalues are generically simple.
Soit une variété riemannienne et un opérateur elliptique, auto-adjoint, covariant conforme d’ordre agissant sur les sections lisses d’un fibré sur . Nous montrons que si n’admet pas d’espaces propres rigides (voir Définition 2.2), l’ensemble des fonctions pour lesquelles n’admet que des valeurs propres non nulles est un ensemble résiduel dans . Ce résultat a comme conséquence que si 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 . Nous montrons également que les valeurs propres de dependent continûment de dans la topologie si est fortement elliptique. Comme applications de nos résultats, nous montrons que si agit sur , comme dans le cas des opérateurs GJMS, alors les valeurs propres non-nulles de cet opérateur sont génériquement simples.
Keywords: Multiplicity, eigenvalues, conformal geometry, conformally covariant operators, GJMS operators.
Mot clés : Multiplicité, valeurs propres, géométrie conforme, opérateur covariant conforme, opérateurs GJMS.
Canzani, Yaiza 1
@article{AIF_2014__64_3_947_0, author = {Canzani, Yaiza}, title = {On the multiplicity of eigenvalues of~conformally covariant operators}, journal = {Annales de l'Institut Fourier}, pages = {947--970}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {3}, year = {2014}, doi = {10.5802/aif.2870}, mrnumber = {3330160}, zbl = {06387297}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2870/} }
TY - JOUR AU - Canzani, Yaiza TI - On the multiplicity of eigenvalues of conformally covariant operators JO - Annales de l'Institut Fourier PY - 2014 SP - 947 EP - 970 VL - 64 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2870/ DO - 10.5802/aif.2870 LA - en ID - AIF_2014__64_3_947_0 ER -
%0 Journal Article %A Canzani, Yaiza %T On the multiplicity of eigenvalues of conformally covariant operators %J Annales de l'Institut Fourier %D 2014 %P 947-970 %V 64 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2870/ %R 10.5802/aif.2870 %G en %F AIF_2014__64_3_947_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2870/
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