Generalizing symplectic topology from 1 to 2 dimensions

TL;DR

This paper extends symplectic topology concepts from 1D to 2D using harmonic maps, introducing new PDEs and proving rigidity results like non-squeezing and cuplength theorems.

Contribution

It generalizes symplectic structures and Hamiltonian equations to 2D, developing new analytical tools such as generalized Floer curves.

Findings

Established a non-squeezing theorem in the 2D symplectic setting.

Proved a cuplength inequality for quadratic Hamiltonians on cotangent bundles.

Developed foundational Fredholm and compactness results for the generalized Floer curves.

Abstract

In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps from surfaces are the natural 2-dimensional generalizations of geodesics. In this paper, we give the corresponding generalization of symplectic manifolds and Hamiltonian equations, leading to a class of partial differential equations that share properties similar to Hamiltonian (ordinary) differential equations. Two rigidity results are discussed: a non-squeezing theorem and a version of the cuplength result for quadratic Hamiltonians on cotangent bundles. The proof of the latter uses a generalization of Floer curves, for which the necessary Fredholm and compactness results will be proven.