Proof of the C2-stability conjecture for geodesic flows of closed surfaces

The paper proves the C^2-stability conjecture for geodesic flows on closed surfaces via two main ingredients: (i) a Reeb-flow criterion showing that if the closure of periodic Reeb orbits is uniformly hyperbolic and stable/unstable manifolds intersect transversely, then the flow is Anosov (Theorem D), and (ii) a C^2-generic dichotomy for surface metrics (Theorem A) stating that either there is an elliptic closed geodesic or the geodesic flow is Anosov. From these, the authors derive that any C^2-structurally stable geodesic flow on a closed surface is Anosov (Theorem B). The candidate solution summarizes precisely these pillars and the conclusion. One minor imprecision is that invoking the Reeb criterion directly for a structurally stable geodesic flow requires additional hypotheses (e.g., transversality) not guaranteed by structural stability alone; the paper instead completes the argument using CP02 to get hyperbolicity of the closure of periodic orbits and a conjugacy to an Anosov flow to deduce density of periodic orbits, yielding Anosov. Overall, both the paper and the model reach the same conclusion with essentially the same structure of ideas.