Symplectic embeddings and the lagrangian bidisk

TL;DR

This paper establishes precise conditions under which the lagrangian bidisk can be symplectically embedded into various four-dimensional shapes, using billiard-inspired methods to identify its structure as a concave toric domain.

Contribution

It introduces a novel approach connecting billiard dynamics to symplectic geometry, providing sharp obstructions and characterizations for embeddings involving the lagrangian bidisk.

Findings

Identifies the lagrangian bidisk as a concave toric domain.

Provides sharp obstructions for embedding into balls, ellipsoids, and polydisks.

Answers a question posed by Ostrover.

Abstract

In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.