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.