# Continuity of Dirac spectra

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## Abstract

It is well known that on a bounded spectral interval the Dirac spectrum can be 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 view the spectrum of a Dirac operator as a function \(\mathbb Z \,\rightarrow \mathbb R \,\) and endow the space of all spectra with an \(\mathrm{arsinh }\)-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, due to spectral flow, these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms in general.

## Keywords

Spin Geometry Dirac Operator Spectral Geometry Dirac Spectrum Spectral Flow## Mathematics Subject Classification (2010)

53C27 58J50 35Q41## Notes

### Acknowledgments

I would like to thank my PhD supervisor Bernd Ammann very much for his continuing support. I am also grateful to our various colleagues at the University of Regensburg, in particular Ulrich Bunke, Nicolas Ginoux and Andreas Hermann for fruitful discussions. Furthermore, I am indebted to Nadine Grosse for explaining [3] to me. This research was enabled by the *Studienstiftung des deutschen Volkes*.

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