Renormalized area, Willmore energy and boundary regularity

In previous work with Spyros Alexakis, we considered the renormalized energy of complete properly embedded minimal surfaces in $\mathbb{H}^3$ and proved several structure theorems about it. I will report on that older work as well as our new results showing how control on this renormalized area yields a certain amount of regularity of the asymptotic boundary at infinity.