Pointwise Expansion of Degenerating Immersions of Finite Total Curvature

Generalising classical result of Müller and Šverák (J. Differ. Geom. 42(2), 229-258, 1995), we obtain a pointwise estimate of the conformal factor of sequences of conformal immersions from the unit disk of the complex plane of uniformly bounded total curvature and converging strongly outside of a concentration point towards a branched immersions for which the quantization of energy holds. We show that the multiplicity associated to the conformal parameter becomes eventually constant to an integer equal to the order of the branch point of the limiting branched immersion. Furthermore, we deduce a C0 convergence of the normal unit in the neck regions. Finally, we show that these improved energy quantizations hold for Willmore surfaces of uniformly bounded energy and precompact conformal class, and for Willmore spheres arising as solutions of min-max problems in the viscosity method.