Morse Index Stability and the Min-Max Sphere Eversion (Geometry Seminar)

Abstract:

In 1957, Stephen Smale showed that the space of immersions of the 2-sphere into Euclidean 3-space is path-connected. In particular, there exists a path of smooth immersions connecting the standard embedding of the 2-sphere to the antipodal embedding; such a path is called a sphere eversion. In the early 1980s, drawing inspiration from work of Robert Bryant, Robert Kusner proposed to use the Willmore energy—a conformally invariant energy of immersions into Euclidean space—as a distance function to determine the “optimal” sphere eversion, or min-max sphere eversion. More generally, the Willmore energy can be considered as a quasi-Morse function on the space of immersions of any topological surface into a fixed Riemannian manifold, and the above problem can be reformulated as finding the “best” representative of a regular homotopy class.

In geometric analysis, upper Morse index bounds are known in a variety of settings (minimal surfaces, Yang-Mills connections, etc), but lower bounds are more elusive. About a year ago (arXiv:2212.03124), Francesca Da Lio, Matilde Gianocca, and Tristan 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 with values into closed manifolds). More recently, this method was also used by Mario Gauvrit (Université Paris Cité) for Yang-Mills connections. In this talk, we will explain how to apply this theory to the Willmore energy (work in collaboration with Tristan Rivière). If time allows, we will say a word about biharmonic maps in dimension 4.