Hofer geometry of $A_3$-configurations

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 $\mathcal{L}(L_0)$ and $\mathcal{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $({\mathbb{R}}^{\infty },\Vert \cdot {\Vert }_{\infty })$, 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 $\ddot{I}„:M\to M$ along $L_1$ the boundary depth of $CF(\ddot{I}{„}^{2\ell }(L_0),L^{\prime })$ is unbounded in $L^{\prime }\in \mathcal{L}(L_2)$ for any $\ell \in {\mathbb{N}}_0$.

35 pages, 4 figures. v2: to appear in Journal of Symplectic Geometry; revised based on the referee's comments