## 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 $1$-forms

#### Abstract

We study the first positive eigenvalue ${\lambda}_{1}^{\left(p\right)}$ of the Laplacian on $p$-forms for oriented closed Riemannian manifolds. It is known that the inequality ${\lambda}_{1}^{\left(1\right)}\le {\lambda}_{1}^{\left(0\right)}$ 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 $M$ with the first Betti number ${b}_{1}\left(M\right)=0$ 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