We prove that for @math there exists a compact subset @math of the closed ball in @math of radius @math , such that @math has Hausdorff dimension @math and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the @math -th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.
For @math the capacity @math is related to a definition recently introduced by H. Geiges and K. Zehmisch: In @cite_1 @cite_9 these authors defined, for any symplectic manifold @math , [c(V, ):= (M, ) ( ) , | , contact type embedding (M, ) (V, ) , ] where the supremum is taken over all closed contact manifolds @math , and @math denotes the infimum of all positive periods of closed orbits of the Reeb vector field @math . They showed that @math is a normalized symplectic capacity. (See [Theorem 4.5] GZWeinstein .)
@cite_9 @cite_1
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We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity. We apply the method of filling with holomorphic discs to a 4-dimensional sym- plectic cobordism with the standard contact 3-sphere as one convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dy- namics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic nonsqueezing, and the nonexistence of exact Lagrangian surfaces in standard symplectic 4-space.