Calabi-Yau domains in three manifolds

We prove that for every smooth compact Riemannian three-manifold $\overline{W}$ with nonempty boundary, there exists a smooth properly embedded one-manifold $\Delta \subset W={\rm Int}(\overline{W})$, each of whose components is a simple closed curve and such that the domain ${\mathcal D} = W - \Delta$ does not admit any properly immersed open surfaces with at least one annular end, bounded mean curvature, compact boundary (possibly empty) and a complete induced Riemannian metric.