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

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.