Morse Index Stability of Branched Willmore Immersions

Abstract

We show that the sum of the Morse index and the nullity of Willmore immersions of bounded energy is lower semi-continuous without assuming that the limiting immersion and the bubbles are free of branch points. Our proof is based on a refined analysis of the properties of two families of fourth-order differential operators with regular singularities that depend on a parameter equal to the order of the branch points. The most technical results that justify the length of the article are Gagliardo-Nirenberg-Rellich inequalities in degenerating annuli that are necessary to show that the eigenvalues of the index operator with respect to a suitable weight are bounded from below.