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Packing Integral Tori in Del Pezzo Surfaces

Karim Boustany

Abstract

We extend a packing result of R. Hind and E. Kerman for integral Lagrangian tori in $\mathbb{S}^{2} \times \mathbb{S}^{2}$ to the Del Pezzo surfaces $(\mathbb{D}_{n}, ω_{\mathbb{D}_{n}})$ for $n = 1, \dots, 5$. An integral torus is one whose relative area homomorphism is integer-valued, and we seek a maximal integral packing. By definition, this is a disjoint collection $\{L_{i}\}$ of integral Lagrangian tori with the following property: any other integral Lagrangian torus not in this collection must intersect at least one of the $L_{i}$. We show that one can always find such a packing consisting of only the Clifford torus.

Packing Integral Tori in Del Pezzo Surfaces

Abstract

We extend a packing result of R. Hind and E. Kerman for integral Lagrangian tori in to the Del Pezzo surfaces for . An integral torus is one whose relative area homomorphism is integer-valued, and we seek a maximal integral packing. By definition, this is a disjoint collection of integral Lagrangian tori with the following property: any other integral Lagrangian torus not in this collection must intersect at least one of the . We show that one can always find such a packing consisting of only the Clifford torus.
Paper Structure (38 sections, 50 theorems, 122 equations)

This paper contains 38 sections, 50 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Commentary
  4. Organization of the Paper
  5. Acknowledgments
  6. Background from Symplectic Topology
  7. Notation and Conventions
  8. Area and Monotonicity
  9. Pseudoholomorphic Spheres
  10. The Blowup Operation
  11. Inflation
  12. Punctured Pseudoholomorphic Spheres
  13. Cylindrical Ends
  14. Punctured Pseudoholomorphic Spheres
  15. The Cotangent Bundle
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\Lambda$ be a closed manifold and $L$ be a closed and exact Lagrangian submanifold of $(T^{\ast}\Lambda, d\lambda)$, where $\lambda$ is the canonical Liouville $1$-form on $T^{\ast}\Lambda$. Then $L$ must intersect the zero section $\Lambda$.

Theorems & Definitions (86)

  • Theorem 1.1: Gromov
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Theorem 1.4: Hind, Kerman
  • Theorem 1.5
  • Corollary 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Lemma 2.1
  • ...and 76 more