Continuity of Dirac Spectra
Abstract
It is a wellknown fact that on a bounded spectral interval the Dirac spectrum can be described locally by a nondecreasing 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 from the integers to the reals and endow the space of all spectra with an arsinhuniform 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 nondecreasing 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2013
 arXiv:
 arXiv:1303.6561
 Bibcode:
 2013arXiv1303.6561N
 Keywords:

 Mathematics  Differential Geometry;
 53C27;
 58J50;
 35Q41
 EPrint:
 25 pages, 3 figures