Spectral rigidity of Liouville tori
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917078809706496 |
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| author | Henheik, Joscha Kaloshin, Vadim Li, Yunzhe Vig, Amir |
| author_facet | Henheik, Joscha Kaloshin, Vadim Li, Yunzhe Vig, Amir |
| contents | We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof are a noncancellation result for the wave trace and an analysis of the second order variational formula for the energy functional associated to closed geodesics. Noncancellation allows us to detect parts of the length spectrum from the Laplace spectrum and conclude rational integrability for the deformed geodesic flow (Liouville metrics are folklorically conjectured to be the only Riemannian metrics with integrable geodesic flow on the torus). We then use the second variational formula to show how the preservation of a single rational torus is sufficient to conclude triviality of the deformation, assuming linearity. We also present some evidence that our hypothesis of linearity may indeed be necessary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral rigidity of Liouville tori Henheik, Joscha Kaloshin, Vadim Li, Yunzhe Vig, Amir Differential Geometry Mathematical Physics Dynamical Systems Spectral Theory 58J42, 37J35, 37J35, 35P20, 58J40, 58J50, 37D40 We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof are a noncancellation result for the wave trace and an analysis of the second order variational formula for the energy functional associated to closed geodesics. Noncancellation allows us to detect parts of the length spectrum from the Laplace spectrum and conclude rational integrability for the deformed geodesic flow (Liouville metrics are folklorically conjectured to be the only Riemannian metrics with integrable geodesic flow on the torus). We then use the second variational formula to show how the preservation of a single rational torus is sufficient to conclude triviality of the deformation, assuming linearity. We also present some evidence that our hypothesis of linearity may indeed be necessary. |
| title | Spectral rigidity of Liouville tori |
| topic | Differential Geometry Mathematical Physics Dynamical Systems Spectral Theory 58J42, 37J35, 37J35, 35P20, 58J40, 58J50, 37D40 |
| url | https://arxiv.org/abs/2511.10398 |