Existence of Dirac Eigenvalues of higher Multiplicity
arXiv:1504.00961 · doi:10.1007/s00209-016-1657-2
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.
23 pages, slightly improved article version of arXiv:1501.04045