In this talk, we will show that any open, orientable surface $S$ can be nonproperly embedded in $\mathbb{H}^3$ as a complete minimal surface. We construct these surfaces by using the bridge principle at infinity due to Martin and White.

In this talk, we will start with an overview of Calabi-Yau Conjecture (CYC), and its generalizations. After Meeks-Rosenberg’s lamination closure theorem, which is true for negatively curved spaces, one can expect a possible generalization of CYC to negatively curved spaces. In this talk, we will construct an example in $\mathbb{H}^3$, which shows that CYC does not extend to this case.

In this talk, we will start with an overview of the asymptotic Plateau problem, and then study the relation between being least area and being properly embedded in hyperbolic 3-space. In particular, we show that if P is an embedded least area plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.