Complete Willmore surfaces in H3 with bounded energy: boundary regularity and bubbling

We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, 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 C1 boundary regularity. We examine the possible lack of C1 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.