Continuity of Dirac Spectra

It is a well-known fact that on a bounded spectral interval the Dirac spectrum can described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that in general these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms due to spectral flow.