Combinig convergence of Riemannian manifolds and the classification results of spin and spin$^c$ manifolds carrying parallel or real Killing spinors, we derive various pinching results for the spinorial Laplacian and the Dirac operator.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3050