Outer billiards in the complex hyperbolic plane

Abstract

Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space.