Periodic Hamiltonian approximation and Oka complements in \mathbb{C^2}

Abstract

We combine periodic Hamiltonian transport with attracting basins to prove that the complements in \(\C^2\) of closed tubes \(\{z:\max_j|\im z_j|\leqδ\}\), \(δ\geq0\), and products of closed annuli with positive inner radii are Oka manifolds. Scalar approximation on \((\C^*)^2\) and complete Laurent-mode flows give uniform control on periodic closed sets with compact holomorphically convex quotient. For every prescribed positive tube width, we construct a holomorphic family of Fatou--Bieberbach domains avoiding the tube. For products of annuli, we construct entire sprays on logarithmic covers and conclude by localization. In particular, the complements of the standard real plane and the standard product torus are Oka; the real-plane case settles, together with earlier work, the question of Forstneri\v c and Wold for totally real affine subspaces.