Calabi-Yau Conjecture for Minimal Surfaces

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 R^3 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.