Morse Index Stability of Willmore Immersions I

In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\`ere developped a new method to show upper semi-continuity results in geometric analysis, which they applied to conformally invariant Lagrangians in dimension $2$ (that include harmonic maps). In this article, we apply this method to show that the sum of the Morse index and the nullity of Willmore immersions is upper semi-continuous, provided that the limiting immersions and the bubbles are free of branch points. Our result covers the case of degenerating Riemann surfaces for which the ratio of the second residue (considered by P. Laurain and T. Rivi\`ere in their work on the energy quantization of Willmore surfaces) and the length of the minimal shrinking geodesic of the underlying sequence of Riemann surfaces is smaller than a universal constant.