We discuss the problem of classifying the asymptotic geometry of complete, properly embedded minimal surfaces in ℝ³ with finite topology. The techniques turn out to be dramatically different depending on whether there is one end or more than one. We focus on the former, as it has proven substantially more challenging - requiring a new theory due to Colding and Minicozzi. We outline their work and describe how it is used to prove the result - due to Meeks and Rosenberg - that the plane and helicoid are the only embedded minimal disks. In the process, we indicate how to generalize the argument to positive genus. Indeed, we conclude that such surfaces are asymptotic to a helicoid; justifying the name "genus-g helicoids". This is joint work with C. Breiner.
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