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.