Spectral isolation of bi-invariant metrics on compact Lie groups

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

Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1617-1628.

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.

Classification : 53C20, 58J50
Keywords: Laplacian, eigenvalue spectrum, Lie group, left-invariant metric, bi-invariant metric
Mot clés : opérateur de Laplace, spectre des valeurs propres, groupe de Lie, métrique invariante à gauche, métrique bi-invariante

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