A new proof of the existence of embedded surfaces with Anosov geodesic flow

TL;DR

This paper presents a novel proof demonstrating the existence of compact embedded surfaces in three-dimensional space with Anosov geodesic flows, utilizing a non-compact model and explicit perturbations.

Contribution

It introduces a new proof method for the existence of such surfaces, starting from a non-compact model and constructing compact examples via explicit maps.

Findings

Existence of compact embedded surfaces with Anosov geodesic flow

Construction method using explicit perturbations of a model surface

Verification of Anosov property via cone conditions

Abstract

We give a new proof of the existence of compact surfaces embedded in $R^3$ with Anosov geodesic flows. This proof starts with a non-compact 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.