Open Access
April, 2001 On the gap between the first eigenvalues of the Laplacian on functions and 1-forms
Junya TAKAHASHI
J. Math. Soc. Japan 53(2): 307-320 (April, 2001). DOI: 10.2969/jmsj/05320307

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.

Citation

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Junya TAKAHASHI. "On the gap between the first eigenvalues of the Laplacian on functions and 1-forms." J. Math. Soc. Japan 53 (2) 307 - 320, April, 2001. https://doi.org/10.2969/jmsj/05320307

Information

Published: April, 2001
First available in Project Euclid: 9 June 2008

MathSciNet: MR1815136
zbMATH: 0984.58018
Digital Object Identifier: 10.2969/jmsj/05320307

Subjects:
Primary: 58J50
Secondary: 35P15 , 53C25 , 53C43

Keywords: eigenvalue , Einstein manifold , Laplacian on forms , stability

Rights: Copyright © 2001 Mathematical Society of Japan

Vol.53 • No. 2 • April, 2001
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