Embedded minimal surfaces

In this talk I will discuss several recent results on embedded minimal surfaces in R^3.  This is joint work with T.H. Colding. 

The first is the proof of the Calabi-Yau conjectures for embedded surfaces. The Calabi-Yau conjectures date back to the 1960s and much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true. A key ingredient in the proof is our earlier compactness theorem for sequences of embedded minimal disks. I will also discuss a very recent compactness theorem for sequences of embedded minimal surfaces with unbounded topology.