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On the existence of infinitely many closed geodesics on non-compact manifolds


Authors: Luca Asselle and Marco Mazzucchelli
Journal: Proc. Amer. Math. Soc. 145 (2017), 2689-2697
MSC (2010): Primary 53C22, 58E10
DOI: https://doi.org/10.1090/proc/13398
Published electronically: November 30, 2016
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Abstract: We prove that any complete (and possibly non-compact) Riemannian manifold $ M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than $ \mathrm {dim}(M)$ and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.


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Luca Asselle
Affiliation: Ruhr Universität Bochum, Fakultät für Mathematik, Gebäude NA 4/33, D-44801 Bochum, Germany
Email: luca.asselle@ruhr-uni-bochum.de

Marco Mazzucchelli
Affiliation: CNRS, École Normale Supérieure de Lyon, UMPA, 69364 Lyon Cedex 07, France
Email: marco.mazzucchelli@ens-lyon.fr

DOI: https://doi.org/10.1090/proc/13398
Keywords: Closed geodesics, Morse theory, free loop space
Received by editor(s): April 28, 2016
Received by editor(s) in revised form: July 21, 2016, and July 22, 2016
Published electronically: November 30, 2016
Additional Notes: The first author was partially supported by the DFG grant AB 360/2-1, “Periodic orbits of conservative systems below the Mañé critical energy value”
The second author was partially supported by the ANR projects WKBHJ (ANR-12-BS01-0020) and COSPIN (ANR-13-JS01-0008-01)
Communicated by: Guofang Wei
Article copyright: © Copyright 2016 American Mathematical Society