Are complete minimal submanifolds closed?

Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?

Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $$\mathbb{R}^3$$. In particular, this immersion is not proper and so the image is not a closed subset of $$\mathbb{R}^3$$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $$\mathbb{R}^3$$ (i.e. a two dimensional submanifold of $$\mathbb{R}^3$$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?