Complex analytic realization of Reeb's foliation of S(sup 3) (Q1113491)

We investigate the question of which codimension-one foliations of three- dimensional manifolds arise as Levi-foliations of Levi-flat real hypersurfaces in two-dimensional complex manifolds. After noting that every real-analytic orientable foliation does arise in this way we proceed to investigate the important special case of Reeb's foliation of \(S^ 3\). Here we are able to show that this foliation does arise from a Levi-flat hypersurface of Lipschitz class, but cannot arise from an infinitely differentiable Levi-flat hypersurface. The case of finite differentiability is left open, but we do show that a variant of Reeb's foliation cannot be realized by a \(C^ 1\) Levi-flat hypersurface.