Small eigenvalues of the rough and Hodge Laplacians under fixed volume

TL;DR

This paper constructs Riemannian metrics on closed manifolds with fixed volume that cause positive eigenvalues of the rough and Hodge Laplacians on differential p-forms to approach zero, extending previous results to higher degrees.

Contribution

It generalizes prior work by Colbois and Maerten to higher degree forms, demonstrating the existence of metrics with small eigenvalues for all degrees p.

Findings

Positive eigenvalues of Laplacians can be made arbitrarily small on fixed volume manifolds.

Construction applies to any closed manifold and all degrees p.

On the sphere, metrics with non-negative sectional curvature achieve this eigenvalue behavior.

Abstract

For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on the sphere, we can choose these Riemannian metrics as those of non-negative sectional curvature. This is a generalization of the results by Colbois and Maerten in 2010 to the case of higher degree forms.