Almost harmonic spinors

TL;DR

This paper demonstrates that on most closed spin manifolds, one can find metrics making the smallest non-zero Dirac eigenvalue arbitrarily close to zero, and compares the Dirac spectrum with conformal volume.

Contribution

It introduces a method to construct metrics on spin manifolds that cause the Dirac eigenvalues to tend to zero, linking spectral properties with geometric invariants.

Findings

Existence of metrics with eigenvalues approaching zero on most spin manifolds

Comparison between Dirac spectrum and conformal volume

Special case of the two-sphere not admitting such metrics

Abstract

We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.