Embedded surfaces with Anosov geodesic flows, approximating spherical billiards

Abstract

We consider a billiard in the sphere 2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space 3 . As an application, we show that every orientable surface of genus at least 11 admits an isometric embedding into 3 (equipped with the standard metric) such that its geodesic flow is Anosov. Finally, we explain why this construction cannot provide examples of isometric embeddings of surfaces in the Euclidean 3 with Anosov geodesic flows.