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.
Categories: math.SP math.DG