Minimizing length of billiard trajectories in hyperbolic polygons

TL;DR

This paper proves that among ideal hyperbolic polygons, the average length of closed billiard trajectories is minimized in the regular case, using properties of geodesic length functions in Teichmüller space.

Contribution

It establishes a conjecture that regular hyperbolic polygons minimize the average length of billiard trajectories, employing convexity of geodesic length functions in Teichmüller space.

Findings

Average billiard trajectory length is minimized for regular polygons.

Convexity of geodesic length functions is key to the proof.

Supports the conjecture for ideal hyperbolic polygons.

Abstract

Closed billiard trajectories in a polygon in the hyperbolic plane can be coded by the order in which they hit the sides of the polygon. In this paper, we consider the average length of cyclically related closed billiard trajectories in ideal hyperbolic polygons and prove the conjecture that this average length is minimized for regular hyperbolic polygons. The proof uses a strict convexity property of the geodesic length function in Teichm\"uller space with respect to the Weil-Petersson metric, a fundamental result established by Wolpert.