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Commentarii Mathematici Helvetici

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Volume 90, Issue 1, 2015, pp. 225–254
DOI: 10.4171/CMH/352

Published online: 2015-02-23

Periodic billiard trajectories and Morse theory on loop spaces

Kei Irie[1]

(1) Kyoto University, Japan

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 proposed by 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. Further, we recover two known results on the length of a shortest periodic billiard trajectory in a convex body: Ghomi’s inequality, and Brunn–Minkowski type inequality proposed by Artstein-Avidan–Ostrover.

Keywords: Periodic billiard trajectory, loop space

Irie Kei: Periodic billiard trajectories and Morse theory on loop spaces. Comment. Math. Helv. 90 (2015), 225-254. doi: 10.4171/CMH/352