A counterexample to Nirenberg's conjecture on closed asymptotic curves

We construct a smooth embedded annulus in $R^3$ with negative Gauss curvature in its interior whose boundary consists of two strictly convex curves in planes tangent to the annulus, and which contains two closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre’s formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from the periodic solution of a Riccati equation.