19. April 2022
Stephan Mescher (MLU Halle)
Spherical complexities and closed geodesics
In the 1930s, L. Lusternik and L. Schnirelmann discovered a fundamental relation between a homotopy invariant of a manifold and the minimal number of critical function on the manifold. This relation has subsequently been generalized and extended to various settings.
In my talk, I will introduce new integer-values homotopy invariants of spaces that are well-suited to study critical points of functions on loop and sphere spaces, so called spherical complexities. After defining these invariants and outlining a corresponding Lusternik-Schnirelmann-type theorem, I will apply this result to energy functionals on free loop spaces of closed Riemannian/Finsler manifolds and discuss how additional topological arguments lead to new existence results for closed geodesics on spheres and complex projective spaces of pinched Riemannian/Finsler metrics that requires a cohomology assumption, but no nondegeneracy assumption on the metric.