Symplectic embeddings and the Lagrangian bidisk

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 [12]. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.

On the rigidity of lagrangian products

“…The results of the present paper, as well as those of [30], corroborate the connection between integrable billiards and lagrangian products admitting an integrable Hamiltonian torus action. We plan on investigating this relation further in future papers.…”

“…This is the main novelty of this paper and might be of independent interest. In spirit, it is a similar result to the existence of a symplectomorphism between the lagrangian bidisk and a concave toric domain proved in [30], although the integrable system in [30] is different from the one in the current paper. Assuming Theorem 7, Theorem 4 can be restated in terms of toric domains.…”

“…[2,29]). The main result in [30] is the computation of the optimal symplectic embeddings of the lagrangian bidisk into a ball and an ellipsoid. The novelty of [30] is to identify the lagrangian bidisk with a concave toric domain using the standard billiard in the disk, thus allowing one to use the machinery of embedded contact homology (ECH) capacities to solve the problem.…”

“…Inspired by [30], this paper studies symplectic embedding problems for a large class of lagrangian products in any dimension. The main result of the paper is that, for all lagrangian products under consideration, the corresponding symplectic embedding problems are rigid, meaning that one cannot do better than inclusion (see Theorem 4 for a precise statement).…”

“…The main result of the paper is that, for all lagrangian products under consideration, the corresponding symplectic embedding problems are rigid, meaning that one cannot do better than inclusion (see Theorem 4 for a precise statement). The strategy for the proof is similar to that employed in [30]. Firstly, the relevant lagrangian products are shown to be symplectomorphic to some toric domains (see Theorem 7).…”