On the rigidity of lagrangian products

TL;DR

This paper proves the rigidity of symplectic embeddings of symmetric lagrangian products by showing they are equivalent to toric domains, utilizing Gromov width and cube capacity for the analysis.

Contribution

It establishes a broad class of lagrangian products as symplectomorphic to toric domains, advancing understanding of symplectic rigidity in high dimensions.

Findings

Lagrangian products are symplectomorphic to toric domains

Rigidity results hold for a large class of lagrangian products

Uses Gromov width and cube capacity to prove rigidity

Abstract

Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the study of billiards. The main result of the paper is the rigidity of a large class of symplectic embedding problems of lagrangian products in any dimension. This is achieved by showing that the lagrangian products under consideration are symplectomorphic to toric domains, and by using the Gromov width and the cube capacity introduced by Gutt and Hutchings to obtain rigidity.