no. 5
Spectral isolation of bi-invariant metrics on compact Lie groups
Gordon, Carolyn S.1; Schueth, Dorothee2; Sutton, Craig J.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, Volume 60 (2010) no. 5, pp. 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: 10.5802/aif.2567
Classification: 53C20, 58J50
Keywords: Laplacian, eigenvalue spectrum, Lie group, left-invariant metric, bi-invariant metric
Mots-clés : opérateur de Laplace, spectre des valeurs propres, groupe de Lie, métrique invariante à gauche, métrique bi-invariante

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

1 Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)
2 Humboldt-Universität Institut für Mathematik Unter den Linden 6 10099, Berlin (Germany) 
@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},
     pages = {1617--1628},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {5},
     year = {2010},
     doi = {10.5802/aif.2567},
     mrnumber = {2766225},
     zbl = {1203.53035},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2567/}
}
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
SP  - 1617
EP  - 1628
VL  - 60
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2567/
DO  - 10.5802/aif.2567
LA  - en
ID  - AIF_2010__60_5_1617_0
ER  - 
%0 Journal Article
%A Gordon, Carolyn S.
%A Schueth, Dorothee
%A Sutton, Craig J.
%T Spectral isolation of bi-invariant metrics on compact Lie groups
%J Annales de l'Institut Fourier
%D 2010
%P 1617-1628
%V 60
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2567/
%R 10.5802/aif.2567
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2567/

[1] Conway, J. H.; Sloane, N. J. A. Four-dimensional lattices with the same theta series, Internat. Math. Res. Notices (1992) no. 4, pp. 93-96 | DOI | MR | Zbl

[2] Gordon, Carolyn S. Isospectral deformations of metrics on spheres, Invent. Math., Volume 145 (2001) no. 2, pp. 317-331 | DOI | MR | Zbl

[3] Gordon, Carolyn S.; Sutton, Craig J. Spectral isolation of naturally reductive metrics on simple Lie groups (Math. Z., to appear)

[4] Kuwabara, Ruishi On the characterization of flat metrics by the spectrum, Comment. Math. Helv., Volume 55 (1980) no. 3, pp. 427-444 | DOI | MR | Zbl

[5] Leininger, C. J.; McReynolds, D. B.; Neumann, W. D.; Reid, A. W. Length and eigenvalue equivalence, Int. Math. Res. Not. IMRN (2007) no. 24, pp. 24 (Art. ID rnm135, 24 pp.) | MR | Zbl

[6] McKean, H. P. Selberg’s trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math., Volume 25 (1972), pp. 225-246 | DOI | MR

[7] Milnor, J. Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., Volume 51 (1964), pp. 542 | DOI | MR | Zbl

[8] Patodi, V. K. Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc., Volume 34 (1970) no. 3-4, p. 269-285 (1971) | MR | Zbl

[9] Proctor, Emily Isospectral metrics and potentials on classical compact simple Lie groups, Michigan Math. J., Volume 53 (2005) no. 2, pp. 305-318 | DOI | MR | Zbl

[10] Schueth, Dorothee Isospectral manifolds with different local geometries, J. Reine Angew. Math., Volume 534 (2001), pp. 41-94 | DOI | MR | Zbl

[11] Schueth, Dorothee Isospectral metrics on five-dimensional spheres, J. Differential Geom., Volume 58 (2001) no. 1, pp. 87-111 | MR | Zbl

[12] Szabo, Z. I. Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 185-214 | DOI | MR | Zbl

[13] Tanno, Shǔkichi Eigenvalues of the Laplacian of Riemannian manifolds, Tǒhoku Math. J. (2), Volume 25 (1973), pp. 391-403 | DOI | MR | Zbl

[14] Tanno, Shǔkichi A characterization of the canonical spheres by the spectrum, Math. Z., Volume 175 (1980) no. 3, pp. 267-274 | DOI | EuDML | MR | Zbl

[15] Urakawa, Hajime On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan, Volume 31 (1979) no. 1, pp. 209-226 | DOI | MR | Zbl

[16] Wolpert, Scott The eigenvalue spectrum as moduli for flat tori, Trans. Amer. Math. Soc., Volume 244 (1978), pp. 313-321 | DOI | MR | Zbl

Cited by Sources: