Title: Morse Index Stability of Willmore Immersions I
\nAuthors: Alexis Michelat and Tristan Rivi\\`ere
\nCategories: math.DG math.AP
\nComments: 96 pages
\nMSC-class: 35J35, 35J48, 35R01, 49Q10, 53A05, 53A10, 53A30, 53C42
\n\\\\
\n In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\\`ere developped a new
\nmethod to show upper semi-continuity results in geometric analysis, which they
\napplied to conformally invariant Lagrangians in dimension $2$ (that include
\nharmonic maps). In this article, we apply this method to show that the sum of
\nthe Morse index and the nullity of Willmore immersions is upper
\nsemi-continuous, provided that the limiting immersions and the bubbles are free
\nof branch points. Our result covers the case of degenerating Riemann surfaces
\nfor which the ratio of the second residue (considered by P. Laurain and T.
\nRivi\\`ere in their work on the energy quantization of Willmore surfaces) and
\nthe length of the minimal shrinking geodesic of the underlying sequence of
\nRiemann surfaces is smaller than a universal constant.
\n\\\\ ( https:\/\/arxiv.org\/abs\/2306.04608 , 68kb)<\/p>\n","protected":false},"excerpt":{"rendered":"