In this talk, I will first give an outline of the current state of the Calabi-Yau Conjecture, which asserts that any complete, embedded minimal surface in R3 must be properly embedded. Then, I will give a construction of a counterexample in H 3. Indeed, the construction is very general and it produces nonproperly embedded minimal surfaces in H 3 with arbitrary topology.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3050