Holomorphic curves into infinite dimensional almost K\"ahler manifolds and Hamiltonian dynamics

TL;DR

This paper develops a theory of pseudo-holomorphic curves in infinite-dimensional almost Kähler manifolds and explores their applications to Hamiltonian dynamics, revealing new dynamical properties in these complex geometric settings.

Contribution

It introduces a moduli theory for pseudo-holomorphic curves in infinite-dimensional almost Kähler manifolds with high symmetry, advancing the understanding of Hamiltonian dynamics in such spaces.

Findings

Established a framework for analyzing pseudo-holomorphic curves in infinite-dimensional settings.

Derived new dynamical properties of Hamiltonian diffeomorphisms on these manifolds.

Extended finite-dimensional symplectic techniques to infinite-dimensional contexts.

Abstract

We study analysis over infinite dimensional manifolds consisted by sequences of almost K\"ahler manifolds. In particular we develop moduli theory of pseudo holomorphic curves into the spaces with high symmetry. As applications, we study Hamiltonian dynamics over the infinite dimensional manifolds, and induce some dynamical properties of Hamiltonian diffeomorphisms on such spaces.