Manifolds with Killing Spinors and Pinching of First Dirac Eigenvalues

Abstract

We consider compact manifolds with metrics of Hölder regularity <i>C</i>^{1, a} and employ the theory of convergence of Riemannian manifolds in this case, to study Riemannian spin manifolds carrying almost Killing spinors and converging in the <i>C</i>^{1, a}-topology. The existence of a Killing spinor in the limit and the smoothness of the limit metric are obtained. Pinching results are found for the first eigenvalues of the Dirac operator using these techniques. In particular, the <i>n</i>-dimensional sphere is characterized by proximity of enough eigenvalues of the Dirac operator to the value <i>n</i>/2.