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Mathematische Annalen

, Volume 370, Issue 1–2, pp 863–915 | Cite as

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

  • Lashi BandaraEmail author
  • Alan McIntosh
  • Andreas Rosén
Article

Abstract

We prove that the Atiyah–Singer Dirac operator Open image in new window in \(\mathrm{L}^{2}_\mathrm{}\) depends Riesz continuously on \(\mathrm{L}^{\infty }_\mathrm{}\) perturbations of complete metrics \(\text {g}\) on a smooth manifold. The Lipschitz bound for the map Open image in new window depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Mathematics Subject Classification

58J05 58J37 58J30 35J46 42B37 

Notes

Acknowledgements

The first author was supported by the Knut and Alice Wallenberg foundation, KAW 2013.0322 postdoctoral program in Mathematics for researchers from outside Sweden. The second author appreciates the support of the Mathematical Sciences Institute at The Australian National University, and also the support of Chalmers University of Technology and University of Gothenburg during his visits to Gothenburg. Further he acknowledges support from the Australian Research Council. The third author was supported by Grant 621-2011-3744 from the Swedish Research Council, VR. We would like to thank Alan Carey and Krzysztof P. Wojciechowski for discussions about the relevance of our approach to open questions involving spectral flow for paths of unbounded self-adjoint operators. Unfortunately Wojciechowski became ill and died before these promising early discussions could be developed. Also, the authors would like to thank the referees for their valuable feedback that helped refine the paper to its current form.

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Copyright information

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Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdam OT GolmGermany
  2. 2.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  3. 3.Mathematical SciencesChalmers University of Technology and University of GothenburgGöteborgSweden

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