Eigenvalues of the Laplacian on forms
Author:
Jozef Dodziuk
Journal:
Proc. Amer. Math. Soc. 85 (1982), 437-443
MSC:
Primary 58G25; Secondary 35P15, 58G30
DOI:
https://doi.org/10.1090/S0002-9939-1982-0656119-2
MathSciNet review:
656119
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Abstract: Some bounds for eigenvalues of the Laplace operator acting on forms on a compact Riemannian manifold are derived. In case of manifolds without boundary we give upper bounds in terms of the curvature, its covariant derivative and the injectivity radius. For a small geodesic ball upper and lower bounds of eigenvalues in terms of bounds of sectional curvature are given.
- [1] R. Bishop and R. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
- [2] S. Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), 289-297. MR 0378001 (51:14170)
- [3] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience, New York, 1953. MR 0065391 (16:426a)
- [4]
J. Eichhorn, Das Spektrum von
auf offenen Riemannschen Mannigfaltigkeiten mit beschränkter Schmitt-krümmung und beschranktem Kern-Tensor, preprint.
- [5] S. Hildebrandt, H. Kaul and K. O. Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), 1-16. MR 0433502 (55:6478)
- [6] M. Gromov, Paul Lévy's isoperimetric inequality, preprint.
- [7] H. Kaul, Schranken für die Christoffelsymbol, Manuscripta Math. 19 (1976), 261-273. MR 0433351 (55:6328)
- [8] J. Kern, Das Pinchingproblem in fastriemannschen Finslerschen Mannigfaltigkeiten, Mansuscripta Math. 4 (1971), 341-350. MR 0290322 (44:7506)
- [9]
D. Ray and I. M. Singer,
-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. MR 0295381 (45:4447)
- [10] G. de Rham, Variétés différentiables, Hermann, Paris, 1955.
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DOI:
https://doi.org/10.1090/S0002-9939-1982-0656119-2
Article copyright:
© Copyright 1982
American Mathematical Society