Seminar Analysis and Mathematical Physics: Fabian Ziltener (Utrecht University)

C^0-closedness of the set of symplectomorphisms, spherical symplectic nonsqueezing, and holomorphic maps

The goal of this talk is to show in an example how analysis and symplectic geometry are related in several ways.

Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation. A (smooth) diffeomorphism on a symplectic manifold is called a symplectomorphism iff it preserves the symplectic form. This happens iff the diffeomorphism solves a certain inhomogeneous quadratic first order system of PDE's. In classical mechanics symplectomorphisms play the role of canonical transformations.

A famous result by Eliashberg and Gromov states that the set of symplectomorphisms is $C^0$-closed in the set of all diffeomorphisms. This is remarkable, since in general, the $C^0$-limit of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known proof of the Eliashberg-Gromov theorem is based on Gromov's symplectic nonsqueezing theorem for balls.

In my talk I will sketch this proof. Furthermore, I will present a symplectic nonsqueezing result for spheres that sharpens Gromov's theorem. The proof of this result is based on the existence of a holomorphic map from the (real) two-dimensional unit disk to a certain symplectic manifold, satisfying some Lagrangian boundary condition. Such a map solves the Cauchy-Riemann equation for a certain almost complex structure.

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