Eigenvalue pinching on ${\text{spin}}^c$ manifolds

We derive various pinching results for small Dirac eigenvalues using the classification of ${\text{spin}}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for ${\text{spin}}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a principal ${\mathbb{S}}^1$-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal ${\mathbb{S}}^1$-bundle on ${\text{spin}}^c$ manifolds with Killing spinors.