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.