Counting arcs in negative curvature

TL;DR

This paper surveys the asymptotic counting of common perpendiculars between convex subsets in negatively curved manifolds, highlighting recent advances, tools, and applications in number theory and circle packings.

Contribution

It provides a comprehensive overview of the latest methods and results in counting problems related to negatively curved manifolds and their arithmetic applications.

Findings

Asymptotic formulas with error terms for counting common perpendiculars.

Connections established between geometric counting and circle packings.

Applications to counting integer representations by quadratic and Hermitian forms.

Abstract

Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Huber, Herrmann, Margulis and ending with the works of the authors. We describe the relationship with counting problems in circle packings of Kontorovich, Oh, Shah. We survey the tools used to obtain the precise asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe several arithmetic applications, in particular the ones by the authors on the asymptotics of the number of representations of integers by binary quadratic, Hermitian or Hamiltonian forms.