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

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