Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

TL;DR

This paper constructs Riemannian metrics on closed manifolds where specific eigenvalues of the Hodge-Laplacian tend to zero, revealing curvature constraints affecting spectral properties of differential forms.

Contribution

It introduces a method to produce metrics with small eigenvalues of the Hodge-Laplacian, demonstrating how curvature bounds influence spectral behavior.

Findings

Eigenvalues of the Hodge-Laplacian can be made arbitrarily small under certain metrics.

The construction applies to any degree p and eigenvalue index k on closed manifolds.

Results impose curvature constraints relevant to previous spectral geometry theorems.

Abstract

For each degree p and each natural number k $\ge$ 1, we construct a oneparameter family of Riemannian metrics on any oriented closed manifold with volume one and the sectional curvature bounded below such that the k-th positive eigenvalue of the Hodge-Laplacian acting on differential p-forms converges to zero. This result imposes a constraint on the sectional curvature for our previous result in [AT24].