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On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere

Part of: Spectral theory and eigenvalue problems Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Pui-Fai Leung
Pui-Fai Leung
Affiliation:
National University of Singapore10 Kent Ridge CrescentSingapore0511
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Abstract

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Let 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ … denote the sequence of eigenvalues of the Laplacian of a compact minimal submanifold in a unit sphere. Yang and Yau obtained an upper bound on λn+1 in terms of λn and the sum λ1 + … + λn. In this note we shall prove an improved version of this upper bound by using the method of Hile and Protter.

MSC classification

Secondary: 53C40: Global submanifolds 35P15: Estimation of eigenvalues, upper and lower bounds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 409 - 416
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Chavel, I., Eigenvalues in Riemanian Geometry, Academic Press, 1984.Google Scholar
[2]Hile, G. N. and Protter, M. H., ‘Inequalities for eigenvalues of the Laplacian’, Indiana Univ. Math. J. 29 (1980), 523–538.CrossRefGoogle Scholar
[3]Payne, L., Polya, G. and Weinberger, H., ‘On the ratio of consecutive eigenvalues’ J. Math. Phys. 35 (1956), 289–298.CrossRefGoogle Scholar
[4]Yang, P. C. P. and Yau, S. T., ‘Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds’, Ann. Scuola Norm. Sup. Pisa 7 (1980), 55–63.Google Scholar