A counterexample to Nirenberg's conjecture on closed asymptotic curves
Harrison Banks, Mohammad Ghomi, Hongda Qiu, Matteo Raffaelli
Abstract
We construct a smooth embedded annulus in R3 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.
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