A New Proof of the Existence of Embedded Surfaces with Anosov Geodesic Flow

Abstract

We give a new proof of the existence of compact surfaces embedded in ℝ³ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.