Morse Index Stability of Willmore Immersions I

TL;DR

This paper proves the upper semi-continuity of the sum of Morse index and nullity for Willmore immersions under certain degenerations, extending understanding of stability in geometric analysis.

Contribution

It introduces a new application of a recent method to establish Morse index and nullity upper semi-continuity for Willmore immersions with degenerating Riemann surfaces.

Findings

Upper semi-continuity of Morse index plus nullity for Willmore immersions.

Applicable to degenerating Riemann surfaces with controlled residue-to-geodesic length ratio.

Extends stability analysis in geometric analysis of conformally invariant surfaces.

Abstract

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.