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On the least positive eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space

Published online by Cambridge University Press:  22 January 2016

Hajime Urakawa*
Affiliation:
Department of MathematicsCollege of General Education Tohoku University
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Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[1] Berger, M., Gauduchon, P. and Mazet, E., Le spectre d’une variété riemannienne, Lecture note in Math., 194, Springer-Verlag, New York, 1971.Google Scholar
[2] Gangolli, R., Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math., 12 (1968), 151192.Google Scholar
[3] Harish, -Chandra, , Spherical functions on a semisimple Lie group, I, Amer. J. Math., 80 (1958), 241310.Google Scholar
[4] Helgason, S., Differential geometry and symmetric spaces, Academic Press, New York, 1962.Google Scholar
[5] Huber, H., Über den ersten Eigenwert des Laplace-Operators auf kompakten Riemannschen Flachen, Commet. Math. Helv., 49 (1974), 251259.Google Scholar
[6] Milnor, J. and Husemoller, D., Symmetric bilinear forms, Springer-Verlag, New York, 1973.Google Scholar
[7] Moriguchi, S., Udagawa, K. and Hitotsumatsu, S., Mathematical formulas, III (in Japanese), Iwanami Shoten, Tokyo, 1960.Google Scholar
[8] Sunada, T., Spectrum of a compact flat manifold, Commet. Math. Helv., 53 (1978), 613621.Google Scholar
[9] Vilenkin, N. J., Special functions and the theory of group representations, Trans. Math. Mono., Amer. Math. Soc, 22 (1968).Google Scholar
[10] Warner, G., Harmonic analysis on semi-simple Lie groups, II, Springer-Verlag, New York, 1972.Google Scholar
[11] Cheng, S. Y., Eigenvalue comparison theorems and its geometric applications, Math. Z., 143 (1975), 289297.Google Scholar
[12] Helgason, S., A duality for symmetric spaces with applications to group representations, Advance Math., 5 (1970), 1154.Google Scholar
[13] Helgason, S., The surjectivity of invariant differential operators on symmetric spaces, I, Ann. Math., 98 (1973), 451479.Google Scholar