Length orthospectrum of convex bodies on flat tori

TL;DR

This paper introduces a new analytical framework for studying orthogeodesics on flat tori, defining a geometric Epstein function and establishing its meromorphic continuation, with applications to spectral theory and convex geometry.

Contribution

It develops anisotropic Sobolev spaces for analyzing geodesic flows on flat tori and introduces a geometric Epstein function with meromorphic continuation and residue formulas.

Findings

Defined a geometric Epstein function for orthogeodesics

Proved meromorphic continuation of the Epstein function

Established Poisson summation relating orthogeodesic lengths and magnetic Laplacian spectrum

Abstract

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.