On the multiplicity of eigenvalues of conformally covariant operators

Canzani, Yaiza

doi:10.5802/aif.2870

On the multiplicity of eigenvalues of conformally covariant operators
[Sur la multiplicité des valeurs propres d’opérateurs covariants conformes]

Canzani, Yaiza ¹

¹ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada.

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

Résumé
Abstract

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 $C^{\infty} (M)$ , 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 $f \in C^{\infty} (M, ℝ)$ 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 $C^{\infty} (M)$ (e.g. GJMS operators), its non-zero eigenvalues are generically simple.

MR Zbl

DOI : 10.5802/aif.2870

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.

Affiliations des auteurs :

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, Tome 64 (2014) no. 3, pp. 947-970. doi : 10.5802/aif.2870. https://aif.centre-mersenne.org/articles/10.5802/aif.2870/

Bibliographie
Cité par

[1] Ammann, B.; Jammes, P. The supremum of conformally covariant eigenvalues in a conformal class, Variational Problems in Differential Geometry (London Mathematical Society Lecture Note Series), Volume 394, Cambridge, 2011, pp. 1-23

[2] Bando, S.; Urakawa, H. Generic properties of the eigenvalue of the laplacian for compact riemannian manifolds, Tohoku Mathematical Journal, Volume 35 (1983) no. 2, pp. 155-172 | MR

[3] Baston, R. Verma modules and differential conformal invariants, Differential Geometry, Volume 32 (1990), pp. 851-898 | MR | Zbl

[4] Bateman, H. The transformation of the electrodynamical equations, Proceedings of the London Mathematical Society, Volume 2 (1910) no. 1, pp. 223-264 | MR

[5] Bleecker, D.; Wilson, L. Splitting the spectrum of a riemannian manifold, SIAM Journal on Mathematical Analysis, Volume 11 (1980), pp. 813 | MR | Zbl

[6] Branson, T. Conformally convariant equations on differential forms, Communications in Partial Differential Equations, Volume 7 (1982) no. 4, pp. 393-431 | MR | Zbl

[7] Branson, T. Differential operators canonically associated to a conformal structure, Mathematica scandinavica, Volume 57 (1985) no. 2, pp. 293-345 | MR | Zbl

[8] Branson, T. Sharp inequalities, the functional determinant, and the complementary series, Transactions of the American Mathematical Society, Volume 347 (1995), pp. 3671-3742 | MR | Zbl

[9] Branson, T.; Chang, S.; Yang, P. Estimates and extremal problems for the log-determinant on 4-manifolds, Communications in Mathematical Physics, Volume 149 (1992), pp. 241-262 | MR | Zbl

[10] Branson, T.; Gover, A. Conformally invariant operators, differential forms, cohomology and a generalisation of q-curvature, Communications in Partial Differential Equations, Volume 30 (2005) no. 11, pp. 1611-1669 | MR | Zbl

[11] Branson, T.; Hijazi, O. Bochner-weitzenböck formulas associated with the rarita-schwinger operator, International Journal of Mathematics, Volume 13 (2002) no. 2, pp. 137-182 | MR | Zbl

[12] Branson, T.; Ørsted, B. Conformal indices of riemannian manifolds, Compositio mathematica, Volume 60 (1986) no. 3, pp. 261-293 | Numdam | MR | Zbl

[13] Branson, T.; Ørsted, B. Generalized gradients and asymptotics of the functional trace, Odense Universitet, Institut for Mathematik og Datalogi, 1988

[14] Branson, T.; Ørsted, B. Conformal geometry and global invariants, Differential Geometry and its Applications, Volume 1 (1991) no. 3, pp. 279-308 | MR | Zbl

[15] Branson, T.; Ørsted, B. Explicit functional determinants in four dimensions, Proceedings of the American Mathematical Society (1991), pp. 669-682 | MR | Zbl

[16] Dahl, M. Dirac eigenvalues for generic metrics on three-manifolds, Annals of Global Analysis and Geometry, Volume 24 (2003) no. 1, pp. 95-100 | MR | Zbl

[17] Eastwood, M. Notes on conformal differential geometry, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Volume 43 (1996), pp. 57-76 | MR | Zbl

[18] Enciso, A.; Peralta-Salas, D. Nondegeneracy of the eigenvalues of the hodge laplacian for generic metrics on 3-manifolds, Transactions of the American Mathematical Society, Volume 364 (2012), pp. 4207-4224 | MR | Zbl

[19] Ginoux, N. The dirac spectrum, 1976, Springer Verlag, 2009 | MR | Zbl

[20] Gover, A. Conformally invariant operators of standard type, The Quarterly Journal of Mathematics, Volume 40 (1989) no. 2, pp. 197 | MR | Zbl

[21] Gover, A. Conformal de rham hodge theory and operators generalising the q-curvature, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Volume 75 (2005), pp. 109-137 | MR | Zbl

[22] Graham, C.; Jenne, R.; Mason, L.; Sparling, G. Conformally invariant powers of the laplacian, i: Existence, Journal of the London Mathematical Society, Volume 2 (1992) no. 3, pp. 557 | MR | Zbl

[23] Graham, C.; Zworski, M. Scattering matrix in conformal geometry, Inventiones Mathematicae, Volume 152 (2003) no. 1, pp. 89-118 | MR | Zbl

[24] Hitchin, N. Harmonic spinors, Advances in Mathematics, Volume 14 (1974) no. 1, pp. 55 | MR | Zbl

[25] K., Uhlenbeckn Generic properties of eigenfunctions, American Journal of Mathematics, Volume 98 (1976) no. 4, pp. 1059-1078 | MR | Zbl

[26] Kodaira, K. Complex Manifolds and Deformation of Complex Structures, 283, Springer, 1986 | MR | Zbl

[27] Kodaira, K.; Spencer, D. On deformations of complex analytic structures, iii. stability theorems for complex structures, The Annals of Mathematics, Volume 71 (1960) no. 1, pp. 43-76 | MR | Zbl

[28] Lawson, H.; Michelsohn, M. Spin geometry, 38, Princeton University Press, 1989 | MR | Zbl

[29] Paneitz, S. A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds, 1983 (preprint)

[30] Rellich, F. Perturbation theory of eigenvalue problems, Routledge, 1969 | MR | Zbl

[31] Teytel, M. How rare are multiple eigenvalues?, Communications on pure and applied mathematics, Volume 52 (1999) no. 8, pp. 917-934 | MR | Zbl

[32] Wojciechowski, K.; Booss, B. Analysis, geometry and topology of elliptic operators, World Scientific Pub Co Inc, 2006 | MR

[33] Wünsch, V. On conformally invariant differential operators, Mathematische Nachrichten, Volume 129 (1986) no. 1, pp. 269-281 | MR | Zbl

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