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Topology/Geometry Seminar

Coisotropic submanifolds of symplectic manifolds, leafwise fixed points, and spherical nonsqueezing

Fabian Ziltener, Univ of Utrecht

Location:  ARC Rm # 203
Date & time: Tuesday, 21 January 2020 at 10:00AM - 11:00AM

My talk is partly about joint work with Dusan Joksimovic, and with Jan Swoboda.

Consider a symplectic manifold \((M,\omega)\), a closed coisotropic submanifold \(N\) of \(M\), and a Hamiltonian diffeomorphism \(\phi\) on \(M\). A leafwise fixed point for \(\phi\) is a point \(x\in N\) that under \(\phi\) is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. In classical mechanics leafwise fixed points correspond to trajectories that are changed only by a time-shift, when an autonomous mechanical system is perturbed in a time-dependent way.

J. Moser posed the following problem: Find conditions under which leafwise fixed points exist. A special case of this problem is V.I. Arnold's conjecture about fixed points of Hamiltonian diffeomorphisms.

The main result presented in this talk is that leafwise fixed points exist if the Hamiltonian diffeomorphism is the time-1-map of a Hamiltonian flow whose restriction to \(N\) stays \(C^0\)-close to the inclusion \(N\to M\). I will also mention a version of this result that is locally uniform in the symplectic form and the coisotropic submanifold.

As an application of a related result, no neighbourhood of the unit sphere symplectically embeds into the unit symplectic cylinder. This sharpens Gromov's nonsqueezing result.