Eigenvalue pinching on \(\text{spin}^c\) manifolds (Q501297)

The author defines almost Killing spinors sequences on the larger class of spin\(^C\) manifolds. Then she uses it to derive pinching results on spin\(^C\) and spin manifolds with small spinorial Laplace eigenvalues or Dirac eigenvalues close to the Friedrich bound. One of the main results is the following theorem: Let \((P_j,A_j)_{j\in\mathbb N}\) be a sequence of principal \(S^1\)-bundles with connection over a fixed compact Riemannian manifold \((M,g)\). For each \(j\) let \(\Omega_j\) be the 2-form representing the curvature of \(A_j\). If there is a non-negative \(K\) such that \(|| \Omega_j||_{C^{k,\alpha}}\leq K\) for all \(j\), then for any \(\beta<\alpha\) there are a principal \(S^1\)-bundle \(P\) with a \(C^{k+1,\beta}\)-connection \(A\) and a subsequence, again denoted by \((P_j,A_j)_{j\in\mathbb N}\) together with principal bundle isomorphisms \(\Phi_j:P \to P_j\) such that \(\Phi_{j}^{*}A_{j}\) converges to \(A\) in the \(C^{k+1,\beta}\)-norm. As an application the author shows that using Theorem 3.1 from an article by \textit{X. Dai} et al. [Invent. Math. 161, No. 1, 151--176 (2005; Zbl 1075.53042)], the absolute value of the Killing number of a real Killing spinor is bounded from below by a positive constant in the class of \(n\)-dimensional Riemannian manifolds with bounded Ricci-curvature and diameter, and with injectivity radius bounded from below by a positive constant.