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Packing Lagrangian tori

Richard K. Hind, Ely Kerman

Abstract

In this paper we consider the problem of packing a symplectic manifold with integral Lagrangian tori, that is Lagrangian tori whose area homomorphsims take only integer values. We prove that the Clifford torus in $S^2 \times S^2$ is a maximal integral packing, in the sense that any other integral Lagranian torus must intersect it. In the other direction, we show that in any symplectic polydisk $P(a,b)$ with $a,b>2$, there is at least one integral Lagrangian torus in the complement of the collection of standard product integral Lagrangian tori.

Packing Lagrangian tori

Abstract

In this paper we consider the problem of packing a symplectic manifold with integral Lagrangian tori, that is Lagrangian tori whose area homomorphsims take only integer values. We prove that the Clifford torus in is a maximal integral packing, in the sense that any other integral Lagranian torus must intersect it. In the other direction, we show that in any symplectic polydisk with , there is at least one integral Lagrangian torus in the complement of the collection of standard product integral Lagrangian tori.

Paper Structure

This paper contains 28 sections, 46 theorems, 128 equations.

Table of Contents

  1. Introduction.
  2. Results
  3. Overview
  4. Commentary and further questions
  5. Conventions, labels and notation
  6. Proof of Theorem \ref{['one']}.
  7. Refinement 1. We may assume that $\mathbb{L}$ is monotone.
  8. Refinement 2: We may assume that $\mathbb{L}$ lies in the complement of $S_0 \cup S_{\infty} \cup T_0 \cup T_{\infty}$
  9. Foliations of $(S^2 \times S^2) \smallsetminus L$ following rgi
  10. Refinement 3: We may assume that $\mathbb{L}$ is homologically trivial in $(S^2 \times S^2) \smallsetminus(S_0 \cup S_{\infty} \cup T_0 \cup T_{\infty})$
  11. A path to the proof of Theorem \ref{['one']}
  12. Straightened holomorphic foliations of $S^2 \times S^2 \smallsetminus (\mathbb{L} \cup L_{1,1})$, under Assumption 2
  13. Stretching scenario for class $(1,d)$, under Assumption 2.
  14. The existence of special buildings, under Assumption 2.
  15. A collection of symplectic spheres and disks, under Assumption 2.
  16. ...and 13 more sections

Key Result

Theorem 1.1

The Clifford torus $L_{1,1}$ is a maximal integral packing of $(S^2 \times S^2, \pi_1^*\omega + \pi_2^*\omega).$

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 3.4
  • Remark 3.5
  • ...and 72 more