The Dirac operator under collapse to a smooth limit space

Abstract

Let \((M_i, g_i)_{i \in \mathbb {N}}\) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (Bh) in the Gromov–Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator \(\mathcal {D}^B\) on B. We give an explicit description of \(\mathcal {D}^B\) and characterize the special case where \(\mathcal {D}^B\) equals the Dirac operator on B.

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Acknowledgements

First, I would like to thank my supervisors Werner Ballmann and Bernd Ammann for many enlightening discussions and helpful advice. I also thank Andrei Moroianu for his invitation to Orsay and for many stimulating conversations, Alexander Strohmaier deserves acknowledgment for showing me how eigenvalues can be computed numerically. I am indebted to the referee for their helpful suggestions that lead to significant improvement of this paper. I also wish to thank the Max-Planck Institute for Mathematics in Bonn for providing excellent working conditions. This research was supported by the Hausdorff Research Institute for Mathematics in Bonn.

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Correspondence to Saskia Roos.

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Roos, S. The Dirac operator under collapse to a smooth limit space. Ann Glob Anal Geom 57, 121–151 (2020). https://doi.org/10.1007/s10455-019-09691-8

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Keywords

  • Collapse
  • Dirac operator
  • Spin geometry

Mathematics Subject Classification

  • primary 53C21
  • 53C27
  • 58J50
  • secondary 22E25
  • 53B05