FROM CURVE SHORTENING TO FLAT LINK STABILITY AND BIRKHOFF SECTIONS OF GEODESIC FLOWS

The paper proves, in full generality, that every geodesic flow on the unit tangent bundle of a closed orientable Riemannian surface admits a Birkhoff section with a uniform hitting-time bound (Theorem D), by constructing a finite “complete system” of closed geodesics with empty limit subcollection (Theorem 6.1), assembling their Birkhoff annuli, and then applying Fried’s surgery. Crucially, the paper establishes a uniform bound on first-hit times before surgery and carries this bound through the surgery in a controlled way, yielding a genuine Birkhoff section with T < ∞ (uniform for all orbits) . By contrast, the model’s Phase-3 “C^0-approximation/stability” argument to remove genericity is not justified: C^0 control on the metric alone does not ensure the necessary control of the geodesic vector fields, transversality of the interior of the limiting surface to the flow, nor the persistence of a uniform hitting-time bound. The paper’s proof avoids this pitfall by a direct construction that uses forced existence of additional closed geodesics and a complete-system argument rather than taking limits of sections .