Energy quantization of Willmore surfaces with bounded index

Quantization phenomena, pioneered by Sacks-Uhlenbeck in 1981, arise in the study of compactness questions of every conformally invariant functional and requires a subtle analysis. In the context of Willmore surfaces, the first study of such behaviour has been lead by Bernard and Rivière in 2014, Laurain and Rivière in 2018. In this talk, we will show that it also occurs for Willmore surfaces with bounded index. To do so, we will study the analysis of Willmore surfaces from the point of view of Bryant's duality.