Pseudoholomorphic discs and symplectic structures in Hilbert space

TL;DR

This paper extends the theory of pseudoholomorphic discs to infinite-dimensional Hilbert spaces with almost complex structures and proves a symplectic non-squeezing theorem applicable to Hamiltonian PDEs.

Contribution

It introduces a framework for $J$-holomorphic discs in Hilbert spaces and establishes a non-squeezing theorem in this infinite-dimensional setting.

Findings

Established a version of Gromov's non-squeezing theorem in Hilbert spaces.

Applied the theorem to short-time symplectic flows of Hamiltonian PDEs.

Developed foundational theory for pseudoholomorphic discs in infinite dimensions.

Abstract

We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time symplectic flows of a wide class of Hamiltonian PDEs.