Continuum families of non-displaceable Lagrangian tori in $(\mathbb{C}{P}^1{)}^{2m}$

Abstract

We construct a family of Lagrangian tori ${\Theta }_s^n$ $\subset$ $(\mathbb{C}{P}^1{)}^n$, $s\in (0,1)$, where ${\Theta }_{1/2}^n={\Theta }^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n=2m$ and $s\ge 1/2$ these tori are non-displaceable. Then by considering ${\Theta }_{s_1}^{k_1}$ $\times$ $\cdots$ $\times$ ${\Theta }_{s_l}^{k_l}$ $\times$ $(S_{\mathrm{eq}}^2{)}^{n-{\sum }_i{k}_i}$ $\subset$ $(\mathbb{C}{P}^1{)}^n$, with $s_i\in [1/2,1)$ and $k_i\in 2{\mathbb{Z}}_{>0}$, ${\sum }_i{k}_i\le n$ we get several $l$-dimensional families of non-displaceable Lagrangian tori. We also show that there exists partial symplectic quasi-states ${\zeta }_{{\text{e}}_s}^{{\mathfrak{b}}_s}$ and linearly independent homogeneous Calabi quasimorphims ${\mu }_{{\text{e}}_s}^{{\mathfrak{b}}_s}$ or which ${\Theta }_s^{2m}$ are ${\zeta }_{{\text{e}}_s}^{{\mathfrak{b}}_s}$-superheavy and ${\mu }_{{\text{e}}_s}^{{\mathfrak{b}}_s}$-superheavy. We also prove a similar result for $(\mathbb{C}{P}^{2}3\overline{\mathbb{C}{P}^2},{\omega }_{\epsilon })$, where $\{{\omega }_{\epsilon };0<\epsilon <1\}$ is a family of symplectic forms in $\mathbb{C}{P}^{2}3\overline{\mathbb{C}{P}^2}$, for which ${\omega }_{1/2}$ is monotone.

arXiv comment17 pages, 1 Figure