On the Gromov width of complements of Lagrangian tori

TL;DR

This paper calculates the Gromov width of the complement of a union of integral product Lagrangian tori within a symplectic ball, providing insights into symplectic embedding obstructions.

Contribution

It explicitly computes the Gromov width of complements of integral product Lagrangian tori in standard symplectic space, a problem not previously addressed.

Findings

Gromov width of the complement is determined for small R.

Provides explicit values or bounds for the Gromov width in this setting.

Enhances understanding of symplectic embeddings related to Lagrangian tori.

Abstract

An integral product Lagrangian torus in the standard symplectic $\mathbb{C}^2$ is defined to be a subset $\{ \pi|z_1|^2 = k, \, \pi|z_2|^2 =l \}$ with $k,l \in \mathbb{N}$. Let $\mathcal{L}$ be the union of all integral product Lagrangian tori. We compute the Gromov width of complements $B(R) \setminus \mathcal{L}$ for some small $R$, where $B(R)$ denotes the round ball of capacity $R$.