The Gromov width of 4-dimensional tori

TL;DR

This paper proves that all 4-dimensional tori with linear symplectic forms can be completely filled by symplectic balls, with specific filling conditions depending on their product structure.

Contribution

It establishes the Gromov width for 4-dimensional tori and characterizes symplectic ball fillings based on the torus's product structure.

Findings

All 4D tori with linear symplectic forms can be fully filled by one symplectic ball.

Tori not symplectomorphic to a product of equal 2D tori can be filled by multiple balls with total volume less than the torus.

The filling properties depend on the torus's product structure and volume constraints.

Abstract

We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of the 4-torus with its given linear symplectic form.