Journal of the Mathematical Society of Japan

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

Junya TAKAHASHI

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Abstract

We study the first positive eigenvalue λ1(p) of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ1(1)λ1(0) holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ1(1)<λ1(0) holds. We show that any oriented closed manifold M with the first Betti number b1(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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213023459

Digital Object Identifier
doi:10.2969/jmsj/05320307

Mathematical Reviews number (MathSciNet)
MR1815136

Zentralblatt MATH identifier
0984.58018

Subjects
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]

Keywords
Laplacian on forms eigenvalue Einstein manifold stability

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


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