Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 53, Number 2 (2001), 307-320.
On the gap between the first eigenvalues of the Laplacian on functions and -forms
We study the first positive eigenvalue of the Laplacian on -forms for oriented closed Riemannian manifolds. It is known that the inequality holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality holds. We show that any oriented closed manifold with the first Betti number whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.
J. Math. Soc. Japan, Volume 53, Number 2 (2001), 307-320.
First available in Project Euclid: 9 June 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C43: Differential geometric aspects of harmonic maps [See also 58E20]
TAKAHASHI, Junya. On the gap between the first eigenvalues of the Laplacian on functions and $1$ -forms. J. Math. Soc. Japan 53 (2001), no. 2, 307--320. doi:10.2969/jmsj/05320307. https://projecteuclid.org/euclid.jmsj/1213023459