  Consider a pair of smooth, possibly noncompact, properly immersed
hypersurfaces moving by mean curvature flow, or, more generally, a pair of weak
set flows. We prove that if the ambient space is Euclidean space and if the
distance between the two surfaces is initially nonzero, then the surfaces
remain disjoint at all subsequent times. We prove the same result when the
ambient space is a complete Riemannian manifold of nonzero injectivity radius,
provided the curvature tensor (of the ambient space) and all its derivatives
are bounded.
