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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?

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It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).

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?

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