Energy Quantization of Willmore surfaces at the boundary of the Moduli Space

Abstract

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. we notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also prove compactness of Willmore immersion with bounded conformal class and energy below $12\pi$.