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

Junya Takahashi

doi:10.2969/jmsj/05320307

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

Junya Takahashi

INF - System Information Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We study the first positive eigenvalue λ^(p)₁ of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ⁽¹⁾₁≤λ⁽⁰⁾₁ holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ⁽¹⁾₁<λ⁽⁰⁾₁ 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.

Original language	English
Pages (from-to)	307-320
Number of pages	14
Journal	Journal of the Mathematical Society of Japan
Volume	53
Issue number	2
DOIs	https://doi.org/10.2969/jmsj/05320307
Publication status	Published - 2001

Keywords

Eigenvalue
Einstein manifold
Laplacian on forms
Stability

ASJC Scopus subject areas

Mathematics(all)

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10.2969/jmsj/05320307

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AB - We study the first positive eigenvalue λ(p)1 of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ(1)1≤λ(0)1 holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ(1)1<λ(0)1 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.

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