Lagrangian split tori in $S^2 \times S^2$ and billiards

TL;DR

This paper classifies split Lagrangian tori in $S^2 imes S^2$ with non-monotone symplectic forms, linking the problem to billiards in rectangles, and explores applications in Lagrangian packing, topology, and symplectic embeddings.

Contribution

It provides a classification of split Lagrangian tori in $S^2 imes S^2$ and connects the problem to billiard dynamics, offering new insights and applications.

Findings

Classification of split Lagrangian tori in $S^2 imes S^2$

Connection between Lagrangian classification and billiard problems

Applications to Lagrangian packing and symplectic embeddings

Abstract

In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich--Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in $\mathbb{R}^4$.