Energy Quantization for Willmore Surfaces and Applications
  We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into an arbitrary euclidian space with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the M\"obius group action, below some energy threshold 
Yann Bernard and Tristan Rivi\`ere
math.AP math.DG