Harmonic spinors and local deformations of the metric

TL;DR

This paper demonstrates that on a compact Riemannian spin manifold, the lower bound for harmonic spinors given by the Atiyah-Singer index theorem can be achieved through local metric deformations.

Contribution

It shows that the lower bound for the Dirac operator's kernel dimension can be realized by small local changes to the metric.

Findings

The lower bound from the Atiyah-Singer index theorem can be attained locally.

Local metric deformations can increase the dimension of harmonic spinors.

The result applies to any compact Riemannian spin manifold.

Abstract

Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.