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The Weinstein conjecture and theorems of nearby and almost existence

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The Breadth of Symplectic and Poisson Geometry

Part of the book series: Progress in Mathematics ((PM,volume 232))

Abstract

The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding some fundamental properties of Hamiltonian flows.

In this paper we survey some recently developed and well-known methods of proving various particular cases of this conjecture and the closely related almost existence theorem. We also examine differentiability and continuity properties of the Hofer-Zehnder capacity function and relate these properties to the features of the underlying Hamiltonian dynamics, e.g., to the period growth.

The work is partially supported by the NSF and by the faculty research funds of the University of California at Santa Cruz.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Viktor L. Ginzburg

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  1. Viktor L. Ginzburg
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Editors and Affiliations

  1. Department of Engineering and Applied Science Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Tudor S. Ratiu

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Dedicated to Alan Weinstein on the occasion of his 60th birthday.

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Ginzburg, V.L. (2005). The Weinstein conjecture and theorems of nearby and almost existence. In: Marsden, J.E., Ratiu, T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol 232. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4419-9_6

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