On the existence of minimal surfaces in infinite volume hyperbolic 3-manifolds.

We will discuss two results about the existence of closed minimal surfaces in non-compact hyperbolic 3-manifolds. In the first part, we will prove the following dichotomy: either all bounded geometry surface groups admit a minimal surface, or there exists an example with a minimal fibration. In the second part we will discuss minimal surface existence for hyperbolic handlebodies. Based in joint work with Baris Coskunuzer, Zheng Huang and Ben Lowe.