par Hofer, Helmut;Cieliebak, Kai;Latschev, Janko;Schlenk, Félix
Référence Dynamics, ergodic theory, and geometry, Cambridge University Press, Cambridge, page (1-44)
Publication Publié, 2007
Partie d'ouvrage collectif
Résumé : A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux’s Theorem such a manifold looks locally like an open set in some ℝ2n≅ ℂnwith the standard symplectic form and so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities. Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer (although the first capacity was in fact constructed by M. Gromov). Since then, lots of new capacities have been defined and they were further studied in. Surveys on symplectic capacities are. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities.