Duke Math. J. 167 (11), 2073-2124, (15 August 2018) DOI: 10.1215/00127094-2018-0010
Paul Laurain, Tristan Rivière
KEYWORDS: Differential geometry, elliptic PDE, Willmore surfaces, energy identity, compactness by compensation, 53A30, 35J60, 49Q10, 35R01
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 possible loss of energy in collar regions. Thanks to this residue, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of ) of the space of Willmore immersions of any arbitrary closed -dimensional oriented manifold into with uniformly bounded conformal class and energy below .