Resonance for loop homology of spheres

Nancy Hingston College of New Jersey Hans-Bert Rademacher Universit¨at Leipzig

Differential Geometry mathscidoc:1609.10295

Journal of Differential Geometry, 93, (1), 133-174, 2013
A Riemannian or Finsler metric on a compact manifoldM gives rise to a length function on the free loop space M, whose critical points are the closed geodesics in the given metric. If X is a homology class on M, 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 = (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  of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to . The proof uses the ChasSullivan product and results of Goresky-Hingston [7].
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@inproceedings{nancy2013resonance,
  title={RESONANCE FOR LOOP HOMOLOGY OF SPHERES },
  author={Nancy Hingston, and Hans-Bert Rademacher},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073305859276957},
  booktitle={Journal of Differential Geometry},
  volume={93},
  number={1},
  pages={133-174},
  year={2013},
}
Nancy Hingston, and Hans-Bert Rademacher. RESONANCE FOR LOOP HOMOLOGY OF SPHERES . 2013. Vol. 93. In Journal of Differential Geometry. pp.133-174. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073305859276957.
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