Journal of the Mathematical Society of Japan

On the gap between the first eigenvalues of the Laplacian on functions and $1$-forms

Junya TAKAHASHI

Abstract

We study the first positive eigenvalue $\lambda_{1}^{(p)}$ of the Laplacian on $p$-forms for oriented closed Riemannian manifolds. It is known that the inequality $\lambda_{1}^{(1)}\leq\lambda_{1}^{(0)}$ holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality $\lambda_{1}^{(1)}<\lambda_{1}^{(0)}$ holds. We show that any oriented closed manifold $M$ with the first Betti number $b_{1}(M)=0$ whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.

Article information

Source
J. Math. Soc. Japan, Volume 53, Number 2 (2001), 307-320.

Dates
First available in Project Euclid: 9 June 2008

https://projecteuclid.org/euclid.jmsj/1213023459

Digital Object Identifier
doi:10.2969/jmsj/05320307

Mathematical Reviews number (MathSciNet)
MR1815136

Zentralblatt MATH identifier
0984.58018

Citation

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