Complete Willmore surfaces in $\mathbb{H}^3$ with bounded energy: Boundary regularity and bubbling

Abstract

We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in $\mathbb{H}^3$, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls $\mathcal{C}^1$ boundary regularity. We examine the possible lack of $\mathcal{C}^1$ convergence for sequences of surfaces with bounded Willmore energy and find that the mechanism responsible for this is a bubbling phenomenon, where energy escapes to infinity.