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

Joé Brendel, Joontae Kim

TL;DR

This work classifies split Lagrangian tori in $X_{\alpha}=(S^2\times S^2,\omega_{\alpha})$ with a non-monotone split form by reducing the problem to good billiard trajectories in rectangles, yielding an alpha-independent classification. The central machinery uses symmetric probes and billiard unfoldings to realize fibre equivalences and to derive Chekanov-type obstructions that constrain Hamiltonian monodromy, producing a detailed monodromy theorem across regions of the Delzant square. The results have wide-ranging applications, including a complete ball-embedding dichotomy for split tori, implications for Lagrangian packing (including infinite packings in irrational cases), and a late-Poincaré recurrences framework, together with new insights into the topology of the space of Lagrangian tori. The work also confirms that certain toric-fibration fibres in non-monotone $S^2\times S^2$ are not exotic and establishes a robust connection between billiard dynamics and symplectic topology, offering a versatile toolkit for future extensions to other toric manifolds and Hirzebruch surfaces.

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 $\mathbb{R}^4$.

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

TL;DR

This work classifies split Lagrangian tori in with a non-monotone split form by reducing the problem to good billiard trajectories in rectangles, yielding an alpha-independent classification. The central machinery uses symmetric probes and billiard unfoldings to realize fibre equivalences and to derive Chekanov-type obstructions that constrain Hamiltonian monodromy, producing a detailed monodromy theorem across regions of the Delzant square. The results have wide-ranging applications, including a complete ball-embedding dichotomy for split tori, implications for Lagrangian packing (including infinite packings in irrational cases), and a late-Poincaré recurrences framework, together with new insights into the topology of the space of Lagrangian tori. The work also confirms that certain toric-fibration fibres in non-monotone are not exotic and establishes a robust connection between billiard dynamics and symplectic topology, offering a versatile toolkit for future extensions to other toric manifolds and Hirzebruch surfaces.

Abstract

In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in , 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 .

Paper Structure

This paper contains 32 sections, 30 theorems, 90 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Hamiltonian monodromy
  4. Methods of proof
  5. Application: Space of Lagrangians
  6. Application: Lagrangian packing
  7. Application: Late Poincaré recurrence
  8. Application: Lagrangian tori from symplectic ball embeddings
  9. Structure of the paper
  10. Main theorem
  11. Symmetric probes
  12. Billiards
  13. Proof of main theorem
  14. Hamiltonian monodromy
  15. Preparations
  16. ...and 17 more sections

Key Result

Theorem 1.3

Let $T(x,y),T(x',y') \subset X_{\alpha}$ be split tori. Then $T(x,y) \cong T(x',y')$ if and only if the good billiard trajectory emanating from $(x,y)$ has any of the points $(\pm x', \pm y')$ as admissible bouncing point.

Figures (7)

  • Figure 1: A split torus $T(x,y) \subset X_{\alpha}$ on the right-hand side and its base point in the rectangle $\square_{\alpha} = [-\alpha-1,\alpha+1] \times [-\alpha,\alpha]$ on the left-hand side. The parameters $x$ and $y$ correspond to the signed areas between the respective circles and equators.
  • Figure 2: A point $(x,y) \in \square_{\alpha}$, its billiard table $\square_{r(x,y)}$, part of its good billiard trajectory (dotted line), and corresponding bouncing points (black dots). The set $\Sigma$ is indicated by dashed lines and the set $Q$ as defined in \ref{['eq:Qdefset']} in grey.
  • Figure 3: Illustration of \ref{['thm:ballcases']}. The set of ball-Clifford split tori is indicated by dashed lines. The dots represent the second set in the union \ref{['eq:nonmonotonecomplement']} i.e. the tori which are ball-Chekanov but not ball-nonmonotone. The central segment drawn by a solid line consists of the points which are of neither of the three types in \ref{['def:types']}. The set of ball-nonmonotone tori is the complement of the union of dashed lines, the central segment and the dots. The set of ball-Chekanov tori is dense in $\square_{\alpha}$ and hence cannot be meaningfully represented.
  • Figure 4: Some examples and one non-example of symmetric probes in $\square_{\alpha}$. The segments $\sigma_k$ have slope $(k,-1)$ for $k \in \{-1,0,1,2,3\}$ and $\sigma$ has slope $(1,0)$. The dashed segment $\sigma_3$ is not a symmetric probe, since its intersection with the the vertical edge is not integrally transverse.
  • Figure 5: Sketch of the folding map $F_r \colon \mathbb{R}^2 \rightarrow \square_r$. It maps the line $l_{(x,y)}$ of slope $(1,1)$ on the left-hand side to the good billiard trajectory of $(x,y)$ on the right hand side and intersection points with the grid to bouncing points.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 38 more