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Mathematische Annalen

, 343:247 | Cite as

Closed characteristics on non-compact hypersurfaces in \({\mathbb {R}^{2n}}\)

  • Jan Bouwe van den Berg
  • Federica Pasquotto
  • Robert C. Vandervorst
Open Access
Article

Abstract

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface \({M \subset (\mathbb {R}^{2n},\omega)}\) of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (\({\mathbb {R}^{2n}}\), ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces \({\mathbb {R}^{2n}}\) coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case).

Keywords

Homology Group Contact Type Smale Sequence Deformation Retract Closed Characteristic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

We thank Sigurd Angenent and Hansjörg Geiges for helpful discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2008

Authors and Affiliations

  • Jan Bouwe van den Berg
    • 1
  • Federica Pasquotto
    • 1
  • Robert C. Vandervorst
    • 1
  1. 1.Department of MathematicsVrije Universiteit AmsterdamAmsterdamThe Netherlands

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