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Finite energy cylinders of small area. (English) Zbl 1041.57009

\(\widetilde{J}\)-holomorphic cylinders are \(\widetilde{J}\)-holomorphic maps \([ -R, R ] \times S_{1} \to {\mathbb R} \times M\), where \(M\) is a contact manifold, and \(\widetilde{J}\) is an almost complex structure in \({\mathbb R} \times M\). It is assumed that the set of periodic orbits of the Reeb vector field in \(M\) is nonempty and that all orbits with sufficiently small period are nondegenerate. Under these assumptions, the central action of any \(\widetilde{J}\)-holomorphic cylinder of sufficiently large length and small energy must be either zero or bounded away from zero. The paper shows that long cylinders of small energy whose central action is zero are close to constant maps, while long cylinders of small energy with positive central action are close to cylinders over periodic orbits of the Reeb vector field and have a well-defined shape. This result is used to prove a compactness property for sequences of \(\widetilde{J}\)-holomorphic maps of finite energy.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
37C27 Periodic orbits of vector fields and flows
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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