Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866910759112409088 |
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| author | Mårdby, Gustav Rowlett, Julie Rydell, Felix |
| author_facet | Mårdby, Gustav Rowlett, Julie Rydell, Felix |
| contents | In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori Mårdby, Gustav Rowlett, Julie Rydell, Felix Spectral Theory 58J53, 58C40, 94B05, 11E20 In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example. |
| title | Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori |
| topic | Spectral Theory 58J53, 58C40, 94B05, 11E20 |
| url | https://arxiv.org/abs/2412.16709 |