On the topology of compact smooth three-dimensional Levi-flat hypersurfaces (Q1203500)

We study topological conditions which must be satisfied by a compact \(C^ \infty\) Levi-flat hypersurface in a two-dimensional complex manifold, showing in particular that the fundamental group must be infinite. We also study unique continuation properties of the holonomy of Levi-flat hypersurfaces. As a consequence of our work we show that no two-dimensional complex manifold admits a subdomain \(\Omega\) with compact non-empty \(C^ \infty\) boundary such that \(\Omega\cong\mathbb{C}^ 2\).