Caustic-Free Regions for Billiards on Surfaces of Constant Curvature

TL;DR

This paper investigates caustic-free regions in convex billiards on hyperbolic and spherical surfaces, estimating their size based on geometry and extending known boundary caustic nonexistence results.

Contribution

It extends caustic-free region estimates and boundary caustic nonexistence theorems from Euclidean to hyperbolic and spherical billiard tables.

Findings

Caustic-free regions are quantitatively estimated based on table geometry.

No caustics exist near the boundary for tables with curvature discontinuities.

Results generalize Euclidean billiard caustic properties to curved surfaces.

Abstract

In this note we study caustic-free regions for convex billiard tables in the hyperbolic plane or the hemisphere. In particular, following a result by Gutkin and Katok in the Euclidean case, we estimate the size of such regions in terms of the geometry of the billiard table. Moreover, we extend to this setting a theorem due to Hubacher which shows that no caustics exist near the boundary of a convex billiard table whose curvature is discontinuous.