On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani[1]

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

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 947-970
  • ISSN: 0373-0956

Abstract

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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 ∈ C ∞ ( M , ℝ ) for which P e f g has only simple non-zero eigenvalues is a residual set in C ∞ ( 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 ∞ -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.

How to cite

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Canzani, Yaiza. "On the multiplicity of eigenvalues of conformally covariant operators." Annales de l’institut Fourier 64.3 (2014): 947-970. <http://eudml.org/doc/275432>.

@article{Canzani2014,
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.},
affiliation = {Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada.},
author = {Canzani, Yaiza},
journal = {Annales de l’institut Fourier},
keywords = {Multiplicity; eigenvalues; conformal geometry; conformally covariant operators; GJMS operators; rigid eigenspace},
language = {eng},
number = {3},
pages = {947-970},
publisher = {Association des Annales de l’institut Fourier},
title = {On the multiplicity of eigenvalues of conformally covariant operators},
url = {http://eudml.org/doc/275432},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Canzani, Yaiza
TI - On the multiplicity of eigenvalues of conformally covariant operators
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 947
EP - 970
AB - 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.
LA - eng
KW - Multiplicity; eigenvalues; conformal geometry; conformally covariant operators; GJMS operators; rigid eigenspace
UR - http://eudml.org/doc/275432
ER -

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