Dr. Nancy Hingston has had a paper published in the January 2013 issue of the Journal of Differential Geometry. The title is “Resonance for loop homology of spheres.” See below for more information.
Resonance for loop homology of spheres
Abstract
A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space ΛM , whose critical points are the closed geodesics in the given metric. If X is a homology class on ΛM , the “minimax” critical level cr(X) is a critical value. Let M be a sphere of dimension >2 , and fix a metric g and a coefficient field G . We prove that the limit as deg(X) goes to infinity of cr(X)/deg(X) exists. We call this limit α ¯ ¯ =α ¯ ¯ (M,g,G) the global mean frequency of M . As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on Λ of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) α ¯ ¯ , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to α ¯ ¯ . The proof uses the Chas-Sullivan product and results of Goresky-Hingston.–