Existence of Dirac Eigenvalues of higher Multiplicity

TL;DR

This paper proves that on certain compact spin manifolds, there exists a metric making the Dirac operator have an eigenvalue of multiplicity at least two, using topological and surgery techniques.

Contribution

It introduces a method to construct metrics with multiple Dirac eigenvalues on specific spin manifolds, extending known results via surgery and homotopy arguments.

Findings

Existence of metrics with multiple Dirac eigenvalues on certain manifolds.

Use of loops of metrics induced by rotations to achieve eigenvalue multiplicity.

Application of surgery theory to generalize results to arbitrary manifolds.

Abstract

In this article, we prove that on any compact spin manifold of dimension m congruent 0,6,7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by catching the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.