Tensor tomography on asymptotically hyperbolic surfaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918154665459712 |
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| author | Eptaminitakis, Nikolas Monard, François Zou, Yuzhou Joey |
| author_facet | Eptaminitakis, Nikolas Monard, François Zou, Yuzhou Joey |
| contents | We initiate a study of the inversion of the geodesic X-ray transform $I_m$ over symmetric $m$-tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever $m\ge 1$. To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for $m$-tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafutdinov, 2007; Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators $η_\pm$ (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987; Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo $\ker I_m$ for a tensor field, which is distinct from the often-used solenoidal representative.
In the case of the Poincaré disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific $L^2$ structure in data space. For even tensor fields, we provide a full picture of the data space decomposition, in particular a range characterization of $I_{2n}$ for every $n$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of even tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor tomography on asymptotically hyperbolic surfaces Eptaminitakis, Nikolas Monard, François Zou, Yuzhou Joey Differential Geometry Analysis of PDEs We initiate a study of the inversion of the geodesic X-ray transform $I_m$ over symmetric $m$-tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever $m\ge 1$. To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for $m$-tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafutdinov, 2007; Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators $η_\pm$ (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987; Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo $\ker I_m$ for a tensor field, which is distinct from the often-used solenoidal representative. In the case of the Poincaré disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific $L^2$ structure in data space. For even tensor fields, we provide a full picture of the data space decomposition, in particular a range characterization of $I_{2n}$ for every $n$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of even tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail. |
| title | Tensor tomography on asymptotically hyperbolic surfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2510.04144 |