Length orthospectrum of convex bodies on flat tori

In analogy with the study of Pollicott–Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field associated with any translation invariant Finsler metric on the torus $\mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of $\mathbb{T}^d$ (i.e. projection of the boundaries of strictly convex bodies of $\mathbb{R}^d$). Associated with the set of lengths of such orthogeodesics, we define a geometric Epstein function and prove its meromorphic continuation. We compute its residues in terms of intrinsic volumes of the convex sets. We also prove Poisson-type summation formulae relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.

Keywords

completely integrable systems, anisotropic Sobolev spaces, zeta functions, Finsler metrics, resolvent estimates

2010 Mathematics Subject Classification

Primary 52A23, 52C07. Secondary 35P99, 58J60.

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N.V.D. acknowledges the support of the Institut Universitaire de France.

M.L. is partially supported by the Agence Nationale de la Recherche under grants SALVE (ANR-19-CE40-0004) and ADYCT (ANR-20-CE40-0017).

G.R. acknowledges the support of the Institut Universitaire de France, of the Centre Henri Lebesgue (ANR-11-LABX-0020-01), and of the PRC grants ADYCT (ANR-20-CE40-0017) and ODA (ANR-18- CE40-0020) from the Agence Nationale de la Recherche.

Received 19 July 2022

Published 29 September 2023