Spectral isolation of bi-invariant metrics on compact Lie groups
Annales de l'Institut Fourier, Volume 60 (2010) no. 5, p. 1617-1628

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.

Soit G un groupe de Lie compact et connexe, et soit g 0 une métrique bi-invariante sur G. On démontre que g 0 est isolée spectralement dans la classe des métriques invariantes à gauche  : plus précisément, il existe un entier positif N tel que, dans un voisinage de g 0 dans la classe des métriques invariantes à gauche et de volume inférieur ou égal à celui de g 0 , la métrique g 0 est determinée de manière unique par les N premières valeurs propres strictement positives de son Laplacien (sans multiplicités). Si G est simple, on peut choisir N=2.

DOI : https://doi.org/10.5802/aif.2567
Classification:  53C20,  58J50
Keywords: Laplacian, eigenvalue spectrum, Lie group, left-invariant metric, bi-invariant metric
@article{AIF_2010__60_5_1617_0,
     author = {Gordon, Carolyn S. and Schueth, Dorothee and Sutton, Craig J.},
     title = {Spectral isolation of bi-invariant metrics on compact Lie groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {5},
     year = {2010},
     pages = {1617-1628},
     doi = {10.5802/aif.2567},
     mrnumber = {2766225},
     zbl = {1203.53035},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_5_1617_0}
}
Gordon, Carolyn S.; Schueth, Dorothee; Sutton, Craig J. Spectral isolation of bi-invariant metrics on compact Lie groups. Annales de l'Institut Fourier, Volume 60 (2010) no. 5, pp. 1617-1628. doi : 10.5802/aif.2567. http://www.numdam.org/item/AIF_2010__60_5_1617_0/

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