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Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded.

Is it possible to remove the assumption "finite topology"? Have there been any progress in that direction?

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Properness is expected to hold for finite genus embedded minimal surfaces while it seems likely that there are infinite genus counterexamples. Both of these claims are completely open (and are extremely difficult)

Currently the best results are in a preprint of Meeks, Perez and Ros. They assume finite genus and then make some assumptions on the ends.

The basic idea is that there is a natural topology on the space of ends. One can distinguish between limit ends and isolated ends in this topology. Here it's helpful to think of the Riemann family of examples; all the planar ends are isolated ends, but the "top" and "bottom" are limit ends as they are limits of sequences of planar ends. Notice finite topology forces all ends to be isolated.

MPR further specify that a limit end is a simple limit end if it is only approximated by isolated ends. Both limit ends in Riemann example are simple limit ends. However, it is conceivable that the topology of the ends was homeomorphic to the Cantor set then every end would be a limit end and none of them would be simple.

What MPR (essentially) show is things are nicely behaved for simple limit ends. This gives them a number of consequences--for instance if there are a countably number of ends, then the surface is properly embedded.

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