The spectrum of the Laplacian on forms over flat manifolds

Abstract

In this article we prove that the spectrum of the Laplacian on k-forms over a non compact flat manifold is always a connected closed interval of the non negative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.

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Notes

  1. 1.

    Note for example that \(\lambda _Z(0)=0\), not the first nonzero eigenvalue of the compact manifold Z.

  2. 2.

    A special choice of f(x) is given in the proof of Proposition 4.1.

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Acknowledgements

The authors would like to thank V. Kapovich and R. Mazzeo for their feedback and useful discussions regarding the structure of flat manifolds. They are also grateful to J. Lott for helping them work out Example 2.4.

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Correspondence to Nelia Charalambous.

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The first author was partially supported by a University of Cyprus Start-Up Grant. The second author is partially supported by the DMS-1510232.

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Charalambous, N., Lu, Z. The spectrum of the Laplacian on forms over flat manifolds. Math. Z. 296, 1–12 (2020). https://doi.org/10.1007/s00209-019-02407-5

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Keywords

  • Essential spectrum
  • Hodge Laplacian
  • Flat manifolds

Mathematics Subject Classification

  • Primary 58J50
  • Secondary 53C35