Spectral isolation of bi-invariant metrics on compact Lie groups

Gordon, Carolyn S.; Schueth, Dorothee; Sutton, Craig J.

Spectral isolation of bi-invariant metrics on compact Lie groups
[Isolation spectrale des métriques bi-invariantes sur les groupes de Lie compacts]

Gordon, Carolyn S. ; Schueth, Dorothee ; Sutton, Craig J.

Annales de l'Institut Fourier, Tome 60 (2010), p. 1617-1628 / Harvested from Numdam

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Abstract

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 $N = 2$ .

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.

Publié le : 2010-01-01

MR 2766225 | Zbl 1203.53035

DOI : https://doi.org/10.5802/aif.2567
Classification: 53C20, 58J50
Mots clés: opérateur de Laplace, spectre des valeurs propres, groupe de Lie, métrique invariante à gauche, métrique bi-invariante

@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},
     volume = {60},
     year = {2010},
     pages = {1617-1628},
     doi = {10.5802/aif.2567},
     zbl = {1203.53035},
     mrnumber = {2766225},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 60 (2010) pp. 1617-1628. doi : 10.5802/aif.2567. http://gdmltest.u-ga.fr/item/AIF_2010__60_5_1617_0/

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