The Weinstein conjecture and theorems of nearby and almost existence

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 of 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. 1 2. The nearby existence theorems Nearby existence theorems are usually proved by Floer homological methods. Below we discuss some of these methods, focusing on the key ideas and omitting many (often quite non-trivial) technical details.

Totally Non-Coisotropic Displacement and Its Applications to Hamiltonian Dynamics

“…The almost existence theorem due to Hofer and Zehnder and to Struwe,[HZ1,HZ2,St], asserts that almost all regular level sets of a proper autonomous Hamiltonian on R 2n carry periodic orbits. A similar result has also been proved for CP n , symplectic vector bundles, subcritical Stein manifolds, and certain other symplectic manifolds; see, e.g., [FS,GG,HV,Ke2,Lu2,Sc] and also the survey [Gi3] and references therein. Here, similarly to [CGK,FS,Gi1,GG,GK1,GK2,Ke1,Ke2,Lu1,Mac,Pol2,Schl], we focus on these theorems for Hamiltonians supported in a neighborhood of a closed submanifold.…”

“…The proof in [Gi3] is set theoretic in nature and works in any setting where the selector has the properties (AS0), (AS1), (AS3) and the claimed property holds for autonomous C 2 -small functions. (See the proof of (AS2) above for the fact that σ(H) = max H when H is a C 2 -small autonomous function having no non-trivial contractible fast periodic orbits.)…”

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

“…It was interpreted (for a symplectic magnetic field) as a particular case of the generalized Weinstein-Moser theorem in [Ke1]. Referring the reader to [Gi3,Gi6,Gi7] for a detailed review and further references, we mention here only some of the results most relevant to Theorems 1.1 and 1.3.…”

Symplectic capacities of classical domains

“…Using the latter he computed Gromov symplectic width and Hofer-Zehnder symplectic capacity for many symplectic manifolds, see [6,7,8]. For more detailed study history of symplectic capacities the reader may refer to [1,3,6] and references therein.…”