Morse Index Stability of Branched Willmore Immersions

TL;DR

This paper proves the lower semi-continuity of the sum of Morse index and nullity for bounded energy Willmore immersions, using advanced analysis of differential operators with singularities.

Contribution

It establishes Morse index stability without requiring branch point-free limits, extending previous results in the analysis of Willmore immersions.

Findings

Lower semi-continuity of Morse index and nullity proven

Analysis of fourth-order differential operators with singularities

Development of inequalities in degenerating annuli

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.