Journal of Symplectic Geometry

Volume 14 (2016)

Number 4

On periodic orbits in cotangent bundles of non-compact manifolds

Pages: 1145 – 1173

DOI: https://dx.doi.org/10.4310/JSG.2016.v14.n4.a6

Authors

J.B. van den Berg (Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands)

F. Pasquotto (Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands)

T. Rot (Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands)

R.C.A.M. Vandervorst (Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands)

Abstract

This paper is concerned with the existence of periodic orbits on energy hypersurfaces in cotangent bundles of Riemannian manifolds defined by mechanical Hamiltonians. In “Closed characteristics on non-compact hypersurfaces in $\mathbb{R}^{2n}$” [J. B. van den Berg, F. Pasquotto, and R. C. Vandervorst, Mathematische Annalen 343 (2009), no. 2, 247–284], it was proved that, provided certain geometric assumptions are satisfied, regular mechanical hypersurfaces in $\mathbb{R}^{2n}$, in particular non-compact ones, contain periodic orbits if one homology group among the top half does not vanish. In the present paper we extend the above mentioned existence result to a class of hypersurfaces in cotangent bundles of Riemannian manifolds with flat ends.

Keywords

periodic orbits, Weinstein conjecture, Hamiltonian dynamics, free loop space, linking sets

2010 Mathematics Subject Classification

37J05, 37J45, 70H12

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Published 10 January 2017