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 (B, h) 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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References
Baum, H.: Eichfeldtheorie. Springer Spectrum, Springer-Lehrbuch Masterclass, 2nd edn. Springer, Berlin (2014)
Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). (reprint of the 1987 edition)
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)
Cheeger, J., Fukaya, K., Gromov, M.: Nilpotent structures and invariant metrics on collapsed manifolds. J. Amer. Math. Soc. 5(2), 327–372 (1992)
Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. I. J. Differential Geom. 23(3), 309–346 (1986)
Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. II. J. Differential Geom. 32(1), 269–298 (1990)
Dekimpe, K.: A Users’ Guide to Infra-nilmanifolds and Almost-Bieberbach groups. ArXiv e-prints (2017). arXiv:1603.07654v2
Fukaya, K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87(3), 517–547 (1987)
Fukaya, K.: Collapsing Riemannian manifolds to ones of lower dimensions. J. Differential Geom. 25(1), 139–156 (1987)
Fukaya, K.: A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J. Differential Geom. 28(1), 1–21 (1988)
Fukaya, K.: Collapsing Riemannian manifolds to ones with lower dimension. II. J. Math. Soc. Japan 41(2), 333–356 (1989)
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)
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)
Gromov, M.: Structures métriques pour les variétés riemanniennes. In: Lafontaine, J., Pansu, P. (eds.) Textes Mathématiques [Mathematical Texts], vol. 1. CEDIC, Paris (1981)
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: 0-521-40001-557-06
Lawson Jr., H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)
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)
Lott, J.: Collapsing and the differential form Laplacian: the case of a singular limit space, Feb 2002. https://math.berkeley.edu/~lott/sing.pdf
Lott, J.: Collapsing and the differential form Laplacian: the case of a smooth limit space. Duke Math. J. 114(2), 267–306 (2002)
Maier, S.: Generic metrics and connections on Spin- and Spin\(^c\)-manifolds. Comm. Math. Phys. 188(2), 407–437 (1997)
Nowaczyk, N.: Continuity of Dirac spectra. Ann. Global Anal. Geom. 44(4), 541–563 (2013)
Rong, X.: On the fundamental groups of manifolds of positive sectional curvature. Ann. Math. (2) 143(2), 397–411 (1996)
Roos, S.: Dirac operators with \(W^{1, \infty }\)-potential under codimension one collapse. Manuscripta Math. 157(3–4), 387–410 (2018)
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)
Strohmaier, A.: Computation of Eigenvalues. Spectral Zeta Functions and Zeta-Determinants on Hyperbolic surfaces. ArXiv e-prints. arXiv:1604.02722v2 (2016)
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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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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DOI: https://doi.org/10.1007/s10455-019-09691-8