Advertisement

The Dirac operator under collapse to a smooth limit space

  • Saskia RoosEmail author
Article
  • 23 Downloads

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.

Keywords

Collapse Dirac operator Spin geometry 

Mathematics Subject Classification

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

Notes

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.

References

  1. 1.
    Baum, H.: Eichfeldtheorie. Springer Spectrum, Springer-Lehrbuch Masterclass, 2nd edn. Springer, Berlin (2014)CrossRefGoogle Scholar
  2. 2.
    Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). (reprint of the 1987 edition)zbMATHGoogle Scholar
  3. 3.
    Bourguignon, J.-P., Hijazi, O., Milhorat, J.-L., Moroianu, A., Moroianu, S.: A Spinorial Approach to Riemannian and Conformal Geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2015)CrossRefGoogle Scholar
  4. 4.
    Cheeger, J., Fukaya, K., Gromov, M.: Nilpotent structures and invariant metrics on collapsed manifolds. J. Amer. Math. Soc. 5(2), 327–372 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. I. J. Differential Geom. 23(3), 309–346 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. II. J. Differential Geom. 32(1), 269–298 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dekimpe, K.: A Users’ Guide to Infra-nilmanifolds and Almost-Bieberbach groups. ArXiv e-prints (2017). arXiv:1603.07654v2
  8. 8.
    Fukaya, K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87(3), 517–547 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fukaya, K.: Collapsing Riemannian manifolds to ones of lower dimensions. J. Differential Geom. 25(1), 139–156 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fukaya, K.: A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J. Differential Geom. 28(1), 1–21 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fukaya, K.: Collapsing Riemannian manifolds to ones with lower dimension. II. J. Math. Soc. Japan 41(2), 333–356 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilkey, P.B.: The Geometry of Spherical Space Form Groups. Series in Pure Mathematics. With an appendix by A. Bahri and M, Bendersky, vol. 7. World Scientific Publishing Co Inc, Teaneck, NJ (1989)CrossRefGoogle Scholar
  13. 13.
    Gilkey, P.B., Leahy, J.V., Park, J.: Spectral Geometry, Riemannian Submersions, and the Gromov–Lawson Conjecture. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL (1999)zbMATHGoogle Scholar
  14. 14.
    Gromov, M.: Structures métriques pour les variétés riemanniennes. In: Lafontaine, J., Pansu, P. (eds.) Textes Mathématiques [Mathematical Texts], vol. 1. CEDIC, Paris (1981)Google Scholar
  15. 15.
    Kirby, R.C., Taylor, L.R.: Geometry of low-dimensional manifolds. 2. Symplectic manifolds and Jones-Witten theory. In: Donaldson, S.K., Thomas, C.B. (eds.) Proceedings of the symposium held in Durham, July 1989. London Mathematical Society Lecture Note Series, vol. 151. Cambridge University Press, Cambridge (1990). ISBN: 0-521-40001-557-06Google Scholar
  16. 16.
    Lawson Jr., H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)zbMATHGoogle Scholar
  17. 17.
    Lott, J.: Collapsing and Dirac-type operators. In: Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), vol. 91, pp. 175–196 (2002)Google Scholar
  18. 18.
    Lott, J.: Collapsing and the differential form Laplacian: the case of a singular limit space, Feb 2002. https://math.berkeley.edu/~lott/sing.pdf
  19. 19.
    Lott, J.: Collapsing and the differential form Laplacian: the case of a smooth limit space. Duke Math. J. 114(2), 267–306 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Maier, S.: Generic metrics and connections on Spin- and Spin\(^c\)-manifolds. Comm. Math. Phys. 188(2), 407–437 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nowaczyk, N.: Continuity of Dirac spectra. Ann. Global Anal. Geom. 44(4), 541–563 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rong, X.: On the fundamental groups of manifolds of positive sectional curvature. Ann. Math. (2) 143(2), 397–411 (1996)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Roos, S.: Dirac operators with \(W^{1, \infty }\)-potential under codimension one collapse. Manuscripta Math. 157(3–4), 387–410 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Roos, S.: The Dirac operator under collapse with bounded curvature and diameter. Ph.D. thesis. Rheinische Friedrich-Wilhelms-Universität Bonn. http://hss.ulb.uni-bonn.de/2018/5196/5196.htm (2018)
  25. 25.
    Strohmaier, A.: Computation of Eigenvalues. Spectral Zeta Functions and Zeta-Determinants on Hyperbolic surfaces. ArXiv e-prints. arXiv:1604.02722v2 (2016)

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Universität Potsdam Institut für MathematikPotsdamGermany

Personalised recommendations