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Pseudo-holomorphic triangles and the median quasi-state

Pazit Haim-Kislev, Michael Khanevsky, Asaf Kislev, Daniel Rosen

Abstract

We prove that the minimal area of a holomorphic triangle whose boundary lies on the union of any three Lagrangian submanifolds is bounded from above by the Lagrangian spectral norm of any pair taken out of the three. We show a relation between this result and the median quasi-state on the 2-sphere. The median quasi-state gives rise to a measure of Poisson non-commutativity of any pair of functions. We answer a question of Entov--Polterovich--Zapolsky by giving a sharp upper bound on the ratio between this measure and the uniform norm of the Poisson bracket of the pair. The sharpness of this bound also implies a lower bound on the defect of the Calabi quasi-morphism.

Pseudo-holomorphic triangles and the median quasi-state

Abstract

We prove that the minimal area of a holomorphic triangle whose boundary lies on the union of any three Lagrangian submanifolds is bounded from above by the Lagrangian spectral norm of any pair taken out of the three. We show a relation between this result and the median quasi-state on the 2-sphere. The median quasi-state gives rise to a measure of Poisson non-commutativity of any pair of functions. We answer a question of Entov--Polterovich--Zapolsky by giving a sharp upper bound on the ratio between this measure and the uniform norm of the Poisson bracket of the pair. The sharpness of this bound also implies a lower bound on the defect of the Calabi quasi-morphism.
Paper Structure (17 sections, 7 theorems, 69 equations, 19 figures)

This paper contains 17 sections, 7 theorems, 69 equations, 19 figures.

Table of Contents

  1. Introduction and main results
  2. Bounds on holomorphic triangles
  3. Connection with the Lagrangian Hofer distance
  4. The median quasi-state on $S^2$.
  5. Organization of the paper
  6. Acknowledgements
  7. Preliminaries
  8. Lagrangian Floer homology
  9. Lagrangian intersections
  10. Hamiltonian chords
  11. The equivalence of both approaches
  12. Spectral invariants
  13. Pseudo-holomorphic triangles
  14. Triangles with small area in $S^2$
  15. The median quasi-state
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $L_1,L_2 \subset M$ be closed, monotone and Hamiltonian isotopic Lagrangian submanifolds intersecting transversely. For any idempotent $a \in QH_*(L_1)$,

Figures (19)

  • Figure 1: A chord
  • Figure 2: $L_B$ inside $\Delta$
  • Figure 3: Examples for crossings
  • Figure 4: Arcs that cut $\Delta \setminus \gamma_B$ into four disjoint triangles
  • Figure 5: Example where there are no crossings
  • ...and 14 more figures

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • ...and 10 more