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One can’t hear orientability of surfaces

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Abstract

The main result of this paper is that one cannot hear orientability of a surface with boundary. More precisely, we construct two isospectral flat surfaces with boundary with the same Neumann spectrum, one orientable, the other non-orientable. For this purpose, we apply Sunada’s and Buser’s methods in the framework of orbifolds. Choosing a symmetric tile in our construction, and adapting a folklore argument of Fefferman, we also show that the surfaces have different Dirichlet spectra. These results were announced in the C. R. Acad. Sci. Paris Sér. I Math., volume 320 in 1995, but the full proofs so far have only circulated in preprint form.

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Acknowledgements

We wish to thank Carolyn Gordon for helpful discussions, Dorothee Schüth for carefully reading a preliminary version, and Peter Doyle for discussing aspects of this work. We are grateful to Bob Brooks for communicating Fefferman’s argument. The first author acknowledges the hospitality of IMPA, where some of this research was conducted. The second author is grateful to MSRI for its support and for its congenial atmosphere. Both [6, 7] acknowledged support from NSF grants DMS-9216650 and DMS 9022140, from CNRS (France), and from CNPq (Brazil). The authors wish to thank the referee for his comments.

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Authors and Affiliations

  1. Université Grenoble Alpes and CNRS, Institut Fourier, CS 40700, 38058, Grenoble Cedex 9, France

    Pierre Bérard

  2. Darmouth College, Hanover, NH, 03755, USA

    David L. Webb

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  1. Pierre Bérard
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  2. David L. Webb
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Correspondence to Pierre Bérard.

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Bérard, P., Webb, D.L. One can’t hear orientability of surfaces. Math. Z. 300, 139–160 (2022). https://doi.org/10.1007/s00209-021-02758-y

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