3-circle Theorem for Willmore surfaces II--degeneration of the complex structure

Abstract

We study the compactness of Willmore surfaces without assuming the convergence of the induced complex structures. In particular, we compute the energy loss in the neck in terms of the residue and we prove that the limit of the image of the Gauss map is a geodesic in the Grassmannian $G(2,n)$ whose length can also be computed in terms of the residue. Moreover, we provide a family of explicit Willmore surfaces in $\R^3$ that illustrate the denegeration phenomenon involved in the above results.