Abstract: We prove a bubble-neck decomposition and an energy quantisation result for sequences of Willmore surfaces immersed into $\mathbb{R}^{m\ge 3}$ with uniformly bounded energy and non-degenerating conformal structure. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold. This is joint-work with Tristan Rivière (ETH Zürich).
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