On the multiplicity of eigenvalues of conformally covariant operators

[Sur la multiplicité des valeurs propres d’opérateurs covariants conformes]

Canzani, Yaiza

Annales de l'Institut Fourier, Tome 64 (2014), p. 947-970 / Harvested from Numdam

Résumé

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, \mathbb{R})$ pour lesquelles $P_{e^fg}$ n’admet que des valeurs propres non nulles est un ensemble résiduel dans $C^\infty (M,\mathbb{R})$. 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.

Abstract

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 $f\in C^\infty (M, \mathbb{R})$ for which $P_{e^fg}$ has only simple non-zero eigenvalues is a residual set in $C^\infty (M,\mathbb{R})$. 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.

Détail

Publié le : 2014-01-01

MR 3330160 | Zbl 06387297

DOI : https://doi.org/10.5802/aif.2870
Classification:  53A30,  58C40
Mots clés: Multiplicité, valeurs propres, géométrie conforme, opérateur covariant conforme, opérateurs GJMS.

Bibliographie

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

[2] Bando, S.; Urakawa, H. Generic properties of the eigenvalue of the laplacian for compact riemannian manifolds, Tohoku Mathematical Journal, Tome 35 (1983) no. 2, pp. 155-172 | MR 699924

[3] Baston, R. Verma modules and differential conformal invariants, Differential Geometry, Tome 32 (1990), pp. 851-898 | MR 1078164 | Zbl 0732.53011

[4] Bateman, H. The transformation of the electrodynamical equations, Proceedings of the London Mathematical Society, Tome 2 (1910) no. 1, pp. 223-264 | MR 1577429

[5] Bleecker, D.; Wilson, L. Splitting the spectrum of a riemannian manifold, SIAM Journal on Mathematical Analysis, Tome 11 (1980), pp. 813 | MR 586909 | Zbl 0449.58021

[6] Branson, T. Conformally convariant equations on differential forms, Communications in Partial Differential Equations, Tome 7 (1982) no. 4, pp. 393-431 | MR 652815 | Zbl 0532.53021

[7] Branson, T. Differential operators canonically associated to a conformal structure, Mathematica scandinavica, Tome 57 (1985) no. 2, pp. 293-345 | MR 832360 | Zbl 0596.53009

[8] Branson, T. Sharp inequalities, the functional determinant, and the complementary series, Transactions of the American Mathematical Society, Tome 347 (1995), pp. 3671-3742 | MR 1316845 | Zbl 0848.58047

[9] Branson, T.; Chang, S.; Yang, P. Estimates and extremal problems for the log-determinant on 4-manifolds, Communications in Mathematical Physics, Tome 149 (1992), pp. 241-262 | MR 1186028 | Zbl 0761.58053

[10] Branson, T.; Gover, A. Conformally invariant operators, differential forms, cohomology and a generalisation of q-curvature, Communications in Partial Differential Equations, Tome 30 (2005) no. 11, pp. 1611-1669 | MR 2182307 | Zbl 1226.58011

[11] Branson, T.; Hijazi, O. Bochner-weitzenböck formulas associated with the rarita-schwinger operator, International Journal of Mathematics, Tome 13 (2002) no. 2, pp. 137-182 | MR 1891206 | Zbl 1109.53306

[12] Branson, T.; Ørsted, B. Conformal indices of riemannian manifolds, Compositio mathematica, Tome 60 (1986) no. 3, pp. 261-293 | Numdam | MR 869104 | Zbl 0608.58039

[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, Tome 1 (1991) no. 3, pp. 279-308 | MR 1244447 | Zbl 0785.53025

[15] Branson, T.; Ørsted, B. Explicit functional determinants in four dimensions, Proceedings of the American Mathematical Society (1991), pp. 669-682 | MR 1050018 | Zbl 0762.47019

[16] Dahl, M. Dirac eigenvalues for generic metrics on three-manifolds, Annals of Global Analysis and Geometry, Tome 24 (2003) no. 1, pp. 95-100 | MR 1990087 | Zbl 1035.53065

[17] Eastwood, M. Notes on conformal differential geometry, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Tome 43 (1996), pp. 57-76 | MR 1463509 | Zbl 0911.53020

[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, Tome 364 (2012), pp. 4207-4224 | MR 2912451 | Zbl 1286.58021

[19] Ginoux, N. The dirac spectrum, Springer Verlag Tome 1976 (2009) | MR 2509837 | Zbl 1186.58020

[20] Gover, A. Conformally invariant operators of standard type, The Quarterly Journal of Mathematics, Tome 40 (1989) no. 2, pp. 197 | MR 997647 | Zbl 0683.53063

[21] Gover, A. Conformal de rham hodge theory and operators generalising the q-curvature, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Tome 75 (2005), pp. 109-137 | MR 2152358 | Zbl 1104.53033

[22] Graham, C.; Jenne, R.; Mason, L.; Sparling, G. Conformally invariant powers of the laplacian, i: Existence, Journal of the London Mathematical Society, Tome 2 (1992) no. 3, pp. 557 | MR 1190438 | Zbl 0726.53010

[23] Graham, C.; Zworski, M. Scattering matrix in conformal geometry, Inventiones Mathematicae, Tome 152 (2003) no. 1, pp. 89-118 | MR 1965361 | Zbl 1030.58022

[24] Hitchin, N. Harmonic spinors, Advances in Mathematics, Tome 14 (1974) no. 1, pp. 55 | MR 358873 | Zbl 0284.58016

[25] K., Uhlenbeckn Generic properties of eigenfunctions, American Journal of Mathematics, Tome 98 (1976) no. 4, pp. 1059-1078 | MR 464332 | Zbl 0355.58017

[26] Kodaira, K. Complex Manifolds and Deformation of Complex Structures, Springer Tome 283 (1986) | MR 815922 | Zbl 0581.32012

[27] Kodaira, K.; Spencer, D. On deformations of complex analytic structures, iii. stability theorems for complex structures, The Annals of Mathematics, Tome 71 (1960) no. 1, pp. 43-76 | MR 115189 | Zbl 0128.16902

[28] Lawson, H.; Michelsohn, M. Spin geometry, Princeton University Press Tome 38 (1989) | MR 1031992 | Zbl 0688.57001

[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 240668 | Zbl 0181.42002

[31] Teytel, M. How rare are multiple eigenvalues?, Communications on pure and applied mathematics, Tome 52 (1999) no. 8, pp. 917-934 | MR 1686977 | Zbl 0942.47012

[32] Wojciechowski, K.; Booss, B. Analysis, geometry and topology of elliptic operators, World Scientific Pub Co Inc (2006) | MR 2254829

[33] Wünsch, V. On conformally invariant differential operators, Mathematische Nachrichten, Tome 129 (1986) no. 1, pp. 269-281 | MR 864639 | Zbl 0619.53008