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Extremal Lagrangian tori in toric domains

Shah Faisal

TL;DR

The paper proves that extremal Lagrangian tori in the standard ball \\bar{B}^{2n}(1) lie on the boundary, establishing a dimension-wide affirmative answer to a conjecture of Cieliebak–Mohnke. The authors embed the ball into a projective setting, apply neck-stretching along a contact-type boundary, and analyze the resulting holomorphic building, invoking local tangency constraints and the Borman–Sheridan class to enforce energy bounds that force boundary containment. They extend the result to a broad class of convex toric domains, including all compact strictly convex four-dimensional toric domains, by a similar neck-stretching/rounding strategy and GH-capacity considerations. They also demonstrate that the conjecture fails for certain non-convex toric domains, via explicit counterexamples built from disjoint cylinders. The work builds a cohesive SFT/GW framework to relate extremal Lagrangian geometry to holomorphic curve counts and symplectic invariants, providing a robust toolkit for probing Lagrangian capacity in toric settings and beyond.

Abstract

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,ω)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,ω)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,ω)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),ω_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, ω_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

Extremal Lagrangian tori in toric domains

TL;DR

The paper proves that extremal Lagrangian tori in the standard ball \\bar{B}^{2n}(1) lie on the boundary, establishing a dimension-wide affirmative answer to a conjecture of Cieliebak–Mohnke. The authors embed the ball into a projective setting, apply neck-stretching along a contact-type boundary, and analyze the resulting holomorphic building, invoking local tangency constraints and the Borman–Sheridan class to enforce energy bounds that force boundary containment. They extend the result to a broad class of convex toric domains, including all compact strictly convex four-dimensional toric domains, by a similar neck-stretching/rounding strategy and GH-capacity considerations. They also demonstrate that the conjecture fails for certain non-convex toric domains, via explicit counterexamples built from disjoint cylinders. The work builds a cohesive SFT/GW framework to relate extremal Lagrangian geometry to holomorphic curve counts and symplectic invariants, providing a robust toolkit for probing Lagrangian capacity in toric settings and beyond.

Abstract

Let be a closed Lagrangian submanifold of a symplectic manifold . Cieliebak and Mohnke define the symplectic area of as the minimal positive symplectic area of a smooth -disk in with boundary on . An extremal Lagrangian torus in is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in . We prove that every extremal Lagrangian torus in the symplectic unit ball is contained entirely in the boundary . This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in , which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

Paper Structure

This paper contains 36 sections, 51 theorems, 306 equations, 9 figures.

Table of Contents

  1. Introduction and statements of results
  2. Introduction
  3. Statements of results
  4. Symplectic field theory framework
  5. Symplectic cobordisms
  6. Punctured $J$-holomorphic nodal curves
  7. Holomorphic buildings
  8. Neck-stretching and Gromov--Hofer compactness
  9. Gromov--Witten invariants with local tangency constraints
  10. Local tangency constraint
  11. Moduli spaces of rational curves with local tangency constraints
  12. Gromov--Witten invariants with local tangency constraints
  13. Enumerative descendants
  14. Punctured spheres with local tangency constraints in Liouville domains
  15. The case of convex toric domains
  16. ...and 21 more sections

Key Result

Theorem 1.4

Let $\omega_{\mathrm{FS}}$ be the Fubini--Study form on $\mathbb{CP}^n$ scaled so that $\int_{\mathbb{CP}^1}\omega_{\mathrm{FS}}=1$. Then For the standard closed ball of capacity $r>0$ (and radius $\sqrt{r/\pi}$), we have

Figures (9)

  • Figure 1: The red crosses indicate situations that Theorem \ref{['extremal-lag-ball']} does not allow
  • Figure 2: The moment map images of $X^4_{\Omega}$ and $E^{4}(a,b)$.
  • Figure 3: Moment map image of $\mathbb{CP}^2$. The red line represents $L_{\mathrm{ch}}$.
  • Figure 4: A holomorphic building of height $2$ and genus $1$ with one positive end.
  • Figure 5: Geometric Setup
  • ...and 4 more figures

Theorems & Definitions (115)

  • Definition 1.1: Cieliebak--Mohnke Cieliebak2018
  • Definition 1.2: Cieliebak--Mohnke Cieliebak2018
  • Definition 1.3: Cieliebak--Mohnke Cieliebak2018
  • Theorem 1.4: Cieliebak--Mohnke Cieliebak2018
  • Conjecture 1.5: Cieliebak--Mohnke Cieliebak2018
  • Conjecture 1.6: Cieliebak--Mohnke Cieliebak2018
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • proof : Proof of Corollary \ref{['cora-extemal']}
  • ...and 105 more