Event

Talk
"Coisotropic Submanifolds of Symplectic Manifolds, Leafwise Fixed Points, and a Discontinuous Capacity"
Prof. Dr. Fabian Ziltener
5 Jun 2013, 17:30 – 18:30

Consider a symplectic manifold (M; !), a coisotropic submanifold N of M, and a selfmap  of M. A leafwise xed point of  is a point in N, which under  is mapped to the isotropic leaf through itself. Symplectic manifolds naturally generalize phase space of classical mechanics. In this setting coiso-tropic submanifolds arise as energy level sets. Let  be the time-one ow of a time-dependent perturbation of a given Hamiltonian function on M. Then a leafwise xed point of  is a point on a given energy level set whose trajectory is changed only by a phase shift, under the perturbation.
The main result presented in this talk is that the number of leafwise xed points is bounded below by the sum of the Betti numbers of N, provided
that  is not too far from the identity in the Hofer sense and some other conditions are satised. As an application, one obtains a symplectic capacity by considering the minimal actions of regular closed coisotropic submanifolds of a given symplectic manifold. A version of the capacity which is based on spheres, turns out to be discontinuous in dimension four. This answers a question by K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk.