Periodic billiard trajectories and Morse theory on loop spaces

TL;DR

This paper uses Morse theory on loop spaces to establish the existence of periodic billiard trajectories and geodesic loops, providing new proofs and extending known inequalities related to shortest trajectories.

Contribution

It introduces a Morse-theoretic approach to billiard trajectories on manifolds with boundary, proving existence results via homology of loop spaces and extending classical inequalities.

Findings

Existence of periodic billiard trajectories under certain homological conditions

Short billiard trajectories and geodesic loops are guaranteed by the methods

Reproof of known inequalities on shortest billiard trajectories

Abstract

We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic billiard trajectory on a convex body: Ghomi's inequality, and Brunn-Minkowski type inequality due to Artstein-Ostrover.