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.
As a consequence of Theorem and [Theorem 4] SZSmall , the value of the capacity @math on the ball @math lies between @math and @math . In the case @math this value can be exactly calculated, if we modify the definition of @math by restricting to Lagrangian submanifolds. Namely, the so obtained capacity @math satisfies [A_ ^+(B^4)= . ] To see this, we denote by @math the standard torus in @math . For every @math the rescaled torus @math is a Lagrangian submanifold of @math , with minimal area @math . It follows that @math . To see the opposite inequality, note that every orientable closed connected Lagrangian submanifold @math is diffeomorphic to the torus @math , since its Euler characteristic vanishes. For such an @math , K. Cieliebak and K. Mohnke proved @cite_3 that @math . The statement follows.
@cite_3
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Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed.