Spectral isolation of bi-invariant metrics on compact Lie groups

Carolyn S. Gordon[1]; Dorothee Schueth[2]; Craig J. Sutton[1]

  • [1] Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)
  • [2] Humboldt-Universität Institut für Mathematik Unter den Linden 6 10099, Berlin (Germany)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 5, page 1617-1628
  • ISSN: 0373-0956

Abstract

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We show that a bi-invariant metric on a compact connected Lie group G is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric g 0 on G there is a positive integer N such that, within a neighborhood of g 0 in the class of left-invariant metrics of at most the same volume, g 0 is uniquely determined by the first N distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where G is simple, N can be chosen to be two.

How to cite

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Gordon, Carolyn S., Schueth, Dorothee, and Sutton, Craig J.. "Spectral isolation of bi-invariant metrics on compact Lie groups." Annales de l’institut Fourier 60.5 (2010): 1617-1628. <http://eudml.org/doc/116316>.

@article{Gordon2010,
abstract = {We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two.},
affiliation = {Dartmouth College Department of Mathematics Hanover, NH 03755 (USA); Humboldt-Universität Institut für Mathematik Unter den Linden 6 10099, Berlin (Germany); Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)},
author = {Gordon, Carolyn S., Schueth, Dorothee, Sutton, Craig J.},
journal = {Annales de l’institut Fourier},
keywords = {Laplacian; eigenvalue spectrum; Lie group; left-invariant metric; bi-invariant metric; bi-invariant metric.},
language = {eng},
number = {5},
pages = {1617-1628},
publisher = {Association des Annales de l’institut Fourier},
title = {Spectral isolation of bi-invariant metrics on compact Lie groups},
url = {http://eudml.org/doc/116316},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Gordon, Carolyn S.
AU - Schueth, Dorothee
AU - Sutton, Craig J.
TI - Spectral isolation of bi-invariant metrics on compact Lie groups
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1617
EP - 1628
AB - We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two.
LA - eng
KW - Laplacian; eigenvalue spectrum; Lie group; left-invariant metric; bi-invariant metric; bi-invariant metric.
UR - http://eudml.org/doc/116316
ER -

References

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  12. Z. I. Szabo, Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1999), 185-214 Zbl0964.53026MR1675894
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