Orateur
Description
The Morse index of a critical point of a Lagrangian L is the dimension of the maximal vector space on which the second derivative D2L is negative-definite. In the classical theory of Hilbert spaces, one shows that the Morse index is lower semi-continuous, while the sum of the Morse index and nullity (the dimension of the Kernel of the differential operator associated to the second derivative) is upper semi-continuous. In a recent work (arXiv:2212.03124), Francesca Da Lio, Matilde Gianocca, and Tristian Rivière (ETH Zürich) developed a new method to show upper semi-continuity results in geometric analysis—that they applied to conformally invariant Lagrangians in dimension 2 (which include harmonic maps). The proof relies on a fine analysis of the second derivative in neck regions—that link the macroscopic surface to its “bubbles”—and a pointwise estimate of the sequence of critical points in the neck regions. In this talk, we will explain how to apply this method to the Willmore energy, a conformally invariant Lagrangian associated to immersions of a surface into Euclidean spaces. Critical points of the Willmore energy—or Willmore immersions—satisfy a non-linear fourth-order elliptic differential equation, and this extension will give rise to several new technical difficulties. If time allows, we will try to show the universal character of this method, that could address (amongst others) the Morse index stability for Ginzburg-Landau energies in dimension 2, bi-harmonic maps in dimension 4, the Yang-Mills functional in dimension 4, and also apply to min-max problems.
