The Nirenberg asymptotic curve conjecture asserts that a smooth embedded annulus in 
with negative Gaussian curvature in its interior
and strictly convex planar boundary curves, whose planes
are tangent to the surface along the boundary, contains
no closed asymptotic curve (Banks et al. 2026).
It is distinct from Nirenberg's conjecture
about the Gauss map of a complete
minimal surface.
Banks et al. (2026) constructed a smooth embedded annulus satisfying these hypotheses with two distinct closed asymptotic curves. Its boundary curves lie in parallel planes. The construction gives a counterexample to the conjecture. Banks et al. (2026) report that Claude and ChatGPT helped find an initial polygonal counterexample, and that the authors checked the AI-assisted content.
