Energy quantization for Willmore surfaces with bounded index

We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerate in the moduli space or not. To do so, we translate the question on the conformal Gauss map’s viewpoint. In particular, we prove that in a neck or a collar region, the conformal Gauss map converges to a light-like geodesic in the de Sitter space.