Abstract
In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the Twistor projection of a holomorphic curve into $${\mathbb{C}}{\mathbb{P}}^3$$ or the inversion of a minimal surface with planar ends in $${\mathbb{R}}^4$$ . These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors.
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