Poincar{é} series for analytic convex bodies
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912587463000064 |
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| author | Dang, Nguyen Viet Bonthonneau, Yannick Guedes Léautaud, Matthieu Rivière, Gabriel |
| author_facet | Dang, Nguyen Viet Bonthonneau, Yannick Guedes Léautaud, Matthieu Rivière, Gabriel |
| contents | We study Poincar{é} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{é} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{é} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11652 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincar{é} series for analytic convex bodies Dang, Nguyen Viet Bonthonneau, Yannick Guedes Léautaud, Matthieu Rivière, Gabriel Differential Geometry Mathematical Physics Spectral Theory We study Poincar{é} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{é} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{é} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest. |
| title | Poincar{é} series for analytic convex bodies |
| topic | Differential Geometry Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2509.11652 |