The geometry of symplectic energy. (English) Zbl 0829.53025

Lalonde, François; McDuff, Dusa

Ann. Math. (2) 141, No. 2, 349-371 (1995).

Two new fundamental results in symplectic geometry/topology are proven in this paper: 1. For any symplectic manifold the generalized Hofer “energy” on the group of symplectic diffeomorphisms with compact support is a nondegenerate norm. Thus also the “energy” as defined by Hofer is a nondegenerate norm. 2. The nonsqueezing theorem for arbitrary symplectic manifolds: No symplectic manifold \(M\) can be symplectically embedded into a “thin” cylinder \(M \times B^2\), where \(B^2\) is a small 2-disc.
These deep results are obtained directly by elegant geometric arguments, which allow to avoid generalization from the Euclidean case via the technique of pseudoholomorphic curves.

Reviewer: C.Günther (Libby)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R50	Differential topological aspects of diffeomorphisms

Keywords:

symplectic energy; symplectic capacities