Maximally degenerate laplacians

Steven Zelditch

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 2, page 547-587
  • ISSN: 0373-0956

Abstract

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The Laplacian Δ g of a compact Riemannian manifold ( M , g ) is called maximally degenerate if its eigenvalue multiplicity function m g ( k ) is of maximal growth among metrics of the same dimension and volume. Canonical spheres ( S n , can ) and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic integrals of curvature and Jacobi fields, giving a local metric condition for maximal degeneracy. In special cases (surfaces of revolution, real projective spaces) the MD metrics are shown to be CROSSes.

How to cite

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Zelditch, Steven. "Maximally degenerate laplacians." Annales de l'institut Fourier 46.2 (1996): 547-587. <http://eudml.org/doc/75188>.

@article{Zelditch1996,
abstract = {The Laplacian $\Delta _g$ of a compact Riemannian manifold $(M,g)$ is called maximally degenerate if its eigenvalue multiplicity function $m_g(k)$ is of maximal growth among metrics of the same dimension and volume. Canonical spheres $(S^n,\{\rm can\})$ and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic integrals of curvature and Jacobi fields, giving a local metric condition for maximal degeneracy. In special cases (surfaces of revolution, real projective spaces) the MD metrics are shown to be CROSSes.},
author = {Zelditch, Steven},
journal = {Annales de l'institut Fourier},
keywords = {Zoll metric; eigenvalue multiplicities; band invariants; non-commutative residues; Jacobi fields; Fourier integral operators; Toeplitz operators},
language = {eng},
number = {2},
pages = {547-587},
publisher = {Association des Annales de l'Institut Fourier},
title = {Maximally degenerate laplacians},
url = {http://eudml.org/doc/75188},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Zelditch, Steven
TI - Maximally degenerate laplacians
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 2
SP - 547
EP - 587
AB - The Laplacian $\Delta _g$ of a compact Riemannian manifold $(M,g)$ is called maximally degenerate if its eigenvalue multiplicity function $m_g(k)$ is of maximal growth among metrics of the same dimension and volume. Canonical spheres $(S^n,{\rm can})$ and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic integrals of curvature and Jacobi fields, giving a local metric condition for maximal degeneracy. In special cases (surfaces of revolution, real projective spaces) the MD metrics are shown to be CROSSes.
LA - eng
KW - Zoll metric; eigenvalue multiplicities; band invariants; non-commutative residues; Jacobi fields; Fourier integral operators; Toeplitz operators
UR - http://eudml.org/doc/75188
ER -

References

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