Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space πM of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of πM and the ‘dual’ loop cohomology product.
If X is a homology class on πM, the ‘minimax’ critical level Cr(X) is a critical value of the length function. Gromov proved that if M is simply connected, there are positive constants k and K so that for every homology class X of degree > dim(M) on πM, k < deg(X)/Cr(X) < K. When M is a sphere, we prove there are positive constants a and b so that for every homology class X on πM, aCr(X)-b < deg(X) < aCr(X)+b. There are interesting consequences for the length spectrum.
Mark Goresky and Hans-Bert Rademacher are collaborators.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
