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Hofer geometry of $A_3$-configurations

Adrian Dawid

Abstract

Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $(\mathbb{R}^\infty, \|\cdot\|_\infty)$, i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that $\text{Ham}_c(M)$ itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist $τ: M \to M$ along $L_1$ the boundary depth of $CF(τ^{2\ell}(L_0), L')$ is unbounded in $L' \in \mathscr{L}(L_2)$ for any $\ell \in \mathbb{N}_0$.

Hofer geometry of $A_3$-configurations

Abstract

Let be exact Lagrangian spheres in a Liouville domain with . If form an -configuration, we show that and endowed with the Hofer metric contain quasi-isometric embeddings of , i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist along the boundary depth of is unbounded in for any .
Paper Structure (16 sections, 19 theorems, 118 equations, 4 figures)

This paper contains 16 sections, 19 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Strategy of the proof
  3. Structure of the paper
  4. Acknowledgments
  5. Preliminaries
  6. Persistent Homology
  7. Lagrangian Floer theory
  8. Action filtration
  9. Filtered Continuation and Naturality Maps
  10. Setup
  11. Action computation
  12. Index computation
  13. Spectral invariants
  14. Finite-dimensional Hofer quasi-flats
  15. Infinite-dimensional Hofer quasi-flats
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M, \omega = d\lambda)$ be a Liouville domain with $2c_1(M) = 0$ and $L_0, L_1, L_2 \subset M$ exact Lagrangian spheres in an $A_3$-configuration. Then there is a map which is a quasi-isometric embedding with quasi-isometry constant $2$, i.e. for any $v,w \in \mathbb{R}^{\infty}$

Figures (4)

  • Figure 1: Illustration of the Dehn twists used in the proof of Theorem \ref{['thm:main_1']}.
  • Figure 2: Sketch of Lagrangians obtained by applying $(\Phi^{(\infty)} \circ \Sigma)(v)$ to $L_2$ for different $v \in \mathbb{R}^2 \subset \mathbb{R}^\infty$. The support of the Hamiltonian function generating the diffeomorphism is marked in gray. Note that the different regions now have disjoint support (of the Hamiltonian function not just the diffeomorphism) which was not the case for the earlier construction.
  • Figure 3: An illustration of the ordering of the $\check{t}^\pm_{k,m}$ and $\hat{t}^\pm_{k,m}$ for $k = 3$.
  • Figure 4: Some holomorphic disks with boundary on $L_0$ and $\varphi_{(3)}(L_2)$. All other holomorphic disks are analogous.

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4: Usher 2014
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 49 more