The Dirac operator under collapse to a smooth limit space

  • Saskia RoosEmail author


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.


Collapse Dirac operator Spin geometry 

Mathematics Subject Classification

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



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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Universität Potsdam Institut für MathematikPotsdamGermany

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