Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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 $ {\Delta _p}$ 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, $ R$-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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58G25, 35P15, 58G30

Retrieve articles in all journals with MSC: 58G25, 35P15, 58G30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656119-2
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society