CALABI-YAU DOMAINS IN THREE MANIFOLDS

We prove that for every smooth compact Riemannian three-manifold (W) over bar with nonempty boundary, there exists a smooth properly embedded one-manifold Delta subset of W = Int((W) over bar), each of whose components is a simple closed curve and such that the domain 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.