Let G be the (open) set ofḢ 1 2 (R 3 ) divergence free vector fields generating global smooth solutions to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G if the variation is slow enough. This result implies that through any point in G passes an

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... le number of arbitrarily long segments included in G.