Title: Morse Index Stability of Willmore Immersions I
Authors: Alexis Michelat and Tristan Rivi\`ere
Categories: math.DG math.AP
Comments: 96 pages
MSC-class: 35J35, 35J48, 35R01, 49Q10, 53A05, 53A10, 53A30, 53C42
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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.
\\ ( https://arxiv.org/abs/2306.04608 , 68kb)