Hartogs figure and symplectic non-squeezing

TL;DR

This paper proves a filling problem for certain real tori in symplectic 4-manifolds, providing a new proof of Gromov's non-squeezing theorem and insights into symplectic rigidity.

Contribution

It introduces a novel approach to Levi-flat hypersurface filling for totally real tori and applies it to symplectic non-squeezing and rigidity results.

Findings

Simplified proof of Gromov's non-squeezing theorem in dimension 4

New results on the rigidity of symplectic structures

Advancement in understanding Levi-flat hypersurface fillings

Abstract

We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application we obtain a simple proof of Gromov's non-squeezing theorem in dimension 4 and new results on rigidity of symplectic structures.