One can’t hear orientability of surfaces

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 i...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 300; no. 1; pp. 139 - 160
Main Authors Bérard, Pierre, Webb, David L.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2022
Springer Nature B.V
Springer
Subjects
Analysis of PDEs
Differential Geometry
Dirichlet problem
Flat surfaces
Mathematics
Mathematics and Statistics
Spectral Theory
58J32
Isospectral surfaces
Orientability
Laplacian
58J50
Spectrum
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02758-y